Journal of Lie Theory Volume 9 (1999) 125–156 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

On classification of metabelian Lie algebras

L. Yu. Galitski, D. A. Timashev

Abstract. We classify metabelian Lie algebras with successive dimensions of quotients of the lower central series (m,n)=(5,5) and (6,3). The problem is reduced to describing orbits of the linear group ⋀^{2}SL_{m}⊗SL_{n}, the latter being a θ-group in both cases. The results obtained in the paper allow to complete the classification of metabelian Lie algebras of dimension up to 9.

Introduction

The classification of finite-dimensional Lie algebras divides in three parts: (1) classification of nilpotent Lie algebras; (2) description of solvable Lie algebras with given nilradical; (3) description of Lie algebras with given radical. The third problem reduces to the description of semisimple subalgebras in the algebra of derivations of a given solvable algebra [8]. The second problem reduces to the description of orbits of certain unipotent linear groups [9]. The first problem is most complicated. Just recall that the classification of all Lie algebras over ℂ is obtained in dimension up to 6, and nilpotent complex Lie algebras are classified only in dimension up to 7.

Two-step nilpotent, or metabelian, Lie algebras form the first non-trivial subclass of nilpotent algebras. However even the classification of metabelian algebras is a rather complicated problem. The solution in dimension up to 7 is given in [4]. In greater dimensions, only partial results are obtained. In this paper, we introduce an invariant-theoretic approach to this problem, which allows to complete the classification of complex metabelian Lie algebras in dimension up to 9.

Let m, n be the dimensions of successive quotients of the lower central series for a metabelian Lie algebra. We call the pair (m, n) the signature of the metabelian Lie algebra. The classification of metabelian Lie algebras with given signature reduces to the following problem of linear algebra: classify the orbits of GL_{m}(ℂ) × GL_{n}(ℂ) acting naturally on ⋀^{2}ℂ^{m} ⊗ ℂ^{n}. Moreover, the solution of the latter problem yields the classification of all metabelian Lie algebras with signature (p, q), p ≤ m, q ≤ n. The problem of classifying the orbits of the linear group ⋀^{2}GL_{m}(ℂ) ⊗ GL_{n}(ℂ) belongs to Invariant Theory and may be attacked by invariant-theoretic methods. It is more convenient to consider the respective unimodular group ⋀^{2}SL_{m}(ℂ) ⊗ SL_{n}(ℂ), since ⋀^{2}GL_{m}(ℂ) ⊗ GL_{n}(ℂ) has no polynomial invariants, and homotheties are easily controlled.

An inspection of tables in [5] yields the list of pairs (m, n) such that the group ⋀^{2}SL_{m}(ℂ) ⊗ SL_{n}(ℂ) is visible: (5, 2), (6, 2), (7, 2), (8, 2), (5, 3), (6, 3), (5, 4), (5, 5), (4, n). (A linear algebraic group is called visible if each level variety for the polynomial invariants consists of finitely many orbits.) The description of orbits for the group ⋀^{2}SL_{m}(ℂ)⊗SL_{2}(ℂ), or classes of two-dimensional subspaces of skew-symmetric matrices, can be obtained in terms of the Weierstraß–Kronecker theory of pencils, see sect. 1 and references therein. Here we classify the orbits of ⋀^{2}SL_{m}(ℂ) ⊗ SL_{n}(ℂ) for (m, n) = (5, 5) and (6, 3). This yields the solution in all remaining visible cases. If a group is not visible, then there arise “conditional” continuous invariants on some “bad” level varieties of invariant polynomials, and the classification of orbits is much more complicated if at all possible.

Our linear groups in the cases (m, n) = (5, 5) and (6, 3) are not only visible, but they are θ-groups, i. e., they integrate the adjoint action of the degreezero component of a periodically graded semisimple Lie algebra on its degree-one component. The invariant theory of θ-groups is very nice. It resembles the theory of the adjoint representation in many features. We use the theory of θ-groups developed in [13] and [14] (see sect. 2) in order to describe the invariants and classify the orbits of our linear groups.

We obtain the following results. There are three families of metabelian complex Lie algebras of signature (5, 5). The first one depends on one continuous parameter. The second family contains 7 algebras. The third family consists of those algebras, whose structure tensor can be contracted to zero by unimodular change of coordinates. It contains 43 algebras. Moreover, there are 38 algebras of signature (5, 4), 17 of (5, 3), 5 of (5, 2), 2 of (5, 1), 2 of (4, 5), 3 of (4, 4), 5 of (4, 3), 3 of (4, 2), 2 of (4, 1), 1 of (3, 3), 1 of (3, 2), 1 of (3, 1), and 1 of (2, 1). This reproduces the results of [4]. For the signature (6, 3) we have 7 families. The first one depends on 2 continuous parameters. The second and third families depend on one continuous and one arithmetic parameter. The fourth family contains 6 algebras, the fifth one contains 6 algebras, and the sixth family contains 15 algebras. The seventh family consists of those algebras, whose structure tensor can be contracted to zero by unimodular change of coordinates. It contains 61 algebras. Moreover, there are 11 algebras of signature (6, 2), and 3 of (6, 1). See sect. 3–4 and the tables therein.

The paper is organized as follows. In section 1 we recall the general theory of metabelian algebras from [4], whereas section 2 contains a brief review of θ-groups. In section 3 we study the linear θ-group ⋀^{2}SL_{5}(ℂ) ⊗ SL_{5}(ℂ), and section 4 deals with ⋀^{2}SL_{6}(ℂ) ⊗ SL_{3}(ℂ).

Acknowledgments. Both authors would like to express their gratitude to Professor E. B. Vinberg, whose encouraging help made possible to carry on the work. The second author thanks CRDF (grant RM1–206) and RFBR (grant 98–01–00598) for the support.

1. Metabelian Lie algebras

In this section we recall basic facts on metabelian Lie algebras and reduce their classification to a certain problem of linear algebra. The paper [4] will be our basic reference.

1.1. General definitions. All Lie algebras, algebraic varieties and groups are considered over ℂ. However all definitions and results remain valid over an arbitrary algebraically closed field of characteristic zero.

Definition 1.1. A finite-dimensional Lie algebra L is called metabelian if [L, [L, L]] = 0. Its signature is a pair (m, n), where m = dim L/[L, L], n = dim[L, L].

A metabelian Lie algebra structure on L is completely determined by the commutator map ⋀^{2}U → V , where V = [L, L] and U is its complement in L. Conversely, let U, V be two vector spaces of dimensions m, n. Then each skewsymmetric bilinear surjective map b : ⋀^{2}U → V determines a metabelian Lie algebra structure on L = U ⊕ V such that V = [L, L]. Two such structures are isomorphic iff the respective bilinear maps differ by a transformation from the group GL(U) × GL(V) acting naturally on the space ⋀^{2}U^{∗} ⊗ V of all such maps.

Surjective maps, or tensors of full rank, form a Zariski open GL(U) × GL(V)-stable subset in ⋀^{2}U^{∗} ⊗ V . Thus the classification of metabelian Lie algebras of signature (m, n) is equivalent to the classification of orbits in an open subset of ⋀^{2}U^{∗} ⊗ V stable under the action of GL(U) × GL(V). Moreover, if L_{0} = U_{0} ⊕ V_{0} is a metabelian algebra of signature (p, q), p ≤ m, q ≤ n, then its structure map is an element of ⋀^{2}U_{0}^{∗} ⊗ V_{0}, and embeddings U_{0}^{∗} ↪ U^{∗}, V_{0} ↪ V induce an embedding ⋀^{2}U_{0}^{∗}⊗V_{0} ↪ ⋀^{2}U^{∗}⊗V such that GL(U)×GL(V)- orbits intersect the subspace in GL(U_{0}) × GL(V_{0})-orbits. Now it is clear that the description of GL(U) × GL(V)-orbits on ⋀^{2}U^{∗} ⊗ V is the same thing as the classification of metabelian Lie algebras of signature (p, q), p ≤ m, q ≤ n. This is the problem we are interested in.

Remark 1.2. Each tensor of full rank in ⋀^{2}U_{0}^{∗}⊗V_{0}, i. e., such that its contractions with all bivectors from ⋀^{2}U_{0} generate V_{0}, represents not only a metabelian Lie algebra of signature (p, q) but also a series of metabelian algebras of signature (l, q), p < l ≤ m that are obtained by adding a (l − p)-dimensional central subalgebra.

Next we describe the automorphism group of a metabelian Lie algebra. Let L = U⊕V be a metabelian algebra defined by a structure map b : ⋀^{2}U → V . Then Aut L is a semidirect product of the subgroup of automorphisms normalizing U, which is exactly the stabilizer G(b) of b in GL(U) × GL(V), and the normal unipotent subgroup of inner automorphisms (acting trivially on V and L/V ), which is isomorphic to the vector group Hom_{ℂ}(U, V). The identity component G(b)^{0} of G(b) is a quasi-direct product of ℂ^{∗}, or (ℂ^{∗})^{2} if the SL(U) × SL(V)- orbit of b is conical, and S(b)^{0}, the identity component of the stabilizer S(b) of b in SL(U) × SL(V). The subgroups S(b)^{0} are computed (in a sense) in sect. 3–4 for all orbits in ⋀^{2}U^{∗} ⊗ V whenever (m, n) = (5, 5) or (6, 3).

Furthermore, suppose a tensor b belongs to a subspace ⋀^{2}U_{0}^{∗} ⊗ V_{0} of ⋀^{2}U^{∗} ⊗ V , and U_{0}^{∗}, V_{0} are minimal subspaces with this property. This means that U_{0}^{∗} (resp. V_{0}) is generated by all contractions of b with a vector from U and a covector from V^{∗} (resp. a bivector from ⋀^{2}U), or equivalently, that b defines a metabelian algebra structure on L_{0} = U_{0} ⊕ V_{0} such that Z(L_{0}) = [L_{0}, L_{0}] = V_{0}.

Then G(b) is a semidirect product of the stabilizer G_{0}(b) of b in GL(U_{0})×GL(V_{0}), and the subgroup of transformations from GL(U) × GL(V) acting trivially on U_{0}^{∗} and V_{0}. Thus we may compute the dimension of G_{0}(b) (or Aut L_{0}) and the type of its Levi part if we know these data for G(b), and conversely.

1.2. Duality. Another interesting thing is a natural duality on metabelian Lie algebras, cf. [4, §3]. To each surjective map b : ⋀^{2}U → V we assign the projection map b^{∨} : ⋀^{2}U^{∗} → ⋀^{2}U^{∗}/(Ker b)^{⊥}. (Here ^{⊥} denotes annihilator in the dual space.) This is a bijection between (isomorphism classes of) metabelian algebras of signature (m, n), and those of signature (m, \binom{m}{2} − n). Therefore it suffices to classify the algebras in one of the two cases, say, for 2n ≤ \binom{m}{2}.

1.3. Classification in simple cases. First we consider metabelian Lie algebras of signature (m, 1). It is clear from the above discussion that to classify them is the same thing as to classify skew-symmetric bilinear forms on U. These forms are classified by their rank 2r. It now follows that each metabelian Lie algebra of signature (m, 1) is isomorphic to one of L = H_{r}(ℂ) ⊕ ℂ^{m−2r}, where H_{r}(ℂ) is the (2r + 1)-dimensional Heisenberg algebra determined by commutator relations [e_{i}, e_{r+i}] = e_{2r+1}, 1 ≤ i ≤ r.

Now we consider the classification of metabelian algebras of signature (m, 2). This problem reduces to classifying pairs or, more precisely, two-dimen-sional subspaces of skew-symmetric bilinear forms on U. This problem can be solved in general with the aid of the Weierstraß–Kronecker theory of pencils, see [4, §6] and [3, Chap. 12].

Consider a pencil of skew-symmetric (m × m)-matrices xA + yB (x and y are indeterminates). We say that a pencil xA′ + yB′ is equivalent (congruent) to xA + yB if xA′ + yB′ = C(xA + yB)C′, where C, C′ are non-degenerate complex matrices (respectively, xA′ + yB′ = C(xA + yB)C^{T} for some non-degenerate complex matrix C).

Definition 1.3. Invariant factors of the pencil xA + yB are the homogeneous polynomials d_{i}(x, y) = \frac{Δ_{i}}{Δ_{i−1}} , i = 1, …, r, where Δ_{i}(x, y) is the g. c. d. of all (i×i)- minors of xA + yB, and r = rk(xA + yB). Each invariant factor decomposes into a product of powers of prime (linear) polynomials in x, y. These prime power factors are called elementary divisors of the pencil.

Minimal indices δ_{1}, …, δ_{m−r} of the pencil are defined by induction. Consider polynomial solutions of a linear system (xA+yB)X = 0. Each m-column X with polynomial entries satisfying the above equation is the sum of homogeneous components satisfying the equation, too. (A column X is said to be homogeneous if all its entries are homogeneous polynomials in x, y of the same degree denoted by deg X.)

Suppose we have chosen homogeneous solutions X_{i} and defined the minimal indices δ_{i} = deg X_{i} for i < k. Among all homogeneous solutions that are linearly independent of X_{i} (i < k), choose one, say X_{k}, of the lowest degree and set δ_{k} = deg X_{k}. Clearly, we have 0 ≤ δ_{1} ≤ … ≤ δ_{m−r}. It is easy to show that the sequence δ_{1}, …, δ_{m−r} does not depend on the choice of the fundamental solutions X_{i}.

Theorem 1.4. [4, §6], [3, Chap. 12] For two skew-symmetric pencils, the following assertions are equivalent:

1. The pencils are congruent.

2. The pencils are equivalent.

3. Their sets of elementary divisors and minimal indices coincide.

For a skew-symmetric pencil, elementary divisors occur with even multiplicities. One can write out a canonical form for a skew-symmetric pencil [4, §6], which generalizes both the canonical form for one skew-symmetric matrix and Jordan normal form.

If we are interested rather in classifying two-dimensional subspaces than pairs of skew-symmetric matrices, then we should allow linear substitutions of x, y. Minimal indices do not change under a linear substitution of x, y and elementary divisors transform in accordance with the change of variables. One can find a “canonical set” in each equivalence class of sets of elementary divisors by transforming three given linear forms to x, y, x+y. Thus we obtain a canonical form for a two-dimensional subspace of skew-symmetric matrices or skew bilinear forms. See [4] for precise formulations.

1.4. Invariant-theoretic approach. Now we return to our problem in its general setting: classify the orbits for the natural action GL(U)×GL(V) : ⋀^{2}U^{∗}⊗ V . It will be more convenient for us to consider the equivalent action GL(U) × GL(V) : ⋀^{2}U ⊗ V , so that each orbit determines a class of isomorphic metabelian algebra structures on L = U^{∗}⊕V . The respective linear group ⋀^{2}GL(U)⊗GL(V) contains homotheties, hence this action has no non-constant polynomial invariants. Thus it is more convenient to consider the respective unimodular linear group ⋀^{2}SL(U) ⊗ SL(V) and classify its orbits. Homotheties leave conical orbits stable and glue together one-parametric families of non-conical orbits.

The first approximation in classifying orbits of a linear algebraic group is to separate them by invariant polynomials. If each level variety of invariant polynomials contains already finitely many orbits, then a group is “good” from the invariant-theoretic point of view, and one may hope to complete the classification by introducing some arithmetic invariants on each level variety. Such algebraic linear groups are called visible. We give the precise definition.

Definition 1.5. Let H ⊆ GL(M) be a reductive algebraic linear group, π : M → M//H := Spec ℂ[M]^{H} the categorical quotient. The group H is visible if π^{−1}(z) contains finitely many H -orbits for ∀z ∈ M//H.

The visibility correlates with other good invariant-theoretic properties of a representation (equidimensionality, freeness of the algebra of invariants etc.). Visible semisimple irreducible linear groups are classified by Kac [5]. An easy inspection of tables in [5] gives the list of pairs (m, n) such that the semisimple irreducible linear group ⋀^{2}SL(U) ⊗ SL(V) is visible: (5, 2), (6, 2), (7, 2), (8, 2), (5, 3), (6, 3), (5, 4), (5, 5), (4, n). The classification of orbits in the first four cases is covered by 1.3. It is clear from the discussion in 1.1 that the five remaining cases reduce to the cases (m, n) = (5, 5) and (6, 3). (In the last case, we may assume n ≤ 3 by 1.2.)

In the latter two cases, the group ⋀^{2}SL(U)⊗SL(V) is even a θ-group, and its invariant theory is especially nice. See sect. 2 for the definition and properties of θ-groups. The case (m, n) = (5, 5) is considered in sect. 3, and the case (m, n) = (6, 3) is considered in sect. 4.

Remark 1.6. Suppose b ∈ ⋀^{2}U ⊗ V is degenerate, i. e., it lies in a subspace of the type ⋀^{2}U_{0} ⊗ V_{0}, U_{0} ⊆ U, V_{0} ⊆ V , and one of the inclusions is strict. It is easy to see that the intersection of the SL(U) × SL(V)-orbit of b with ⋀^{2}U_{0} ⊗ V_{0} is a GL(U_{0}) × GL(V_{0})-orbit. In particular, it contains λb, ∀λ ∈ ℂ^{×}. Hence (SL(U) × SL(V))b = (GL(U) × GL(V))b contains 0 in its closure, and b lies in the null-cone π^{−1}(π(0)). Elements of the null-cone are called nilpotent. Thus the classification of orbits for ⋀^{2}GL_{p}(ℂ) ⊗ GL_{q}(ℂ), where p ≤ m, q ≤ n and one of the inequalities is strict, reduces to the classification of nilpotent orbits for ⋀^{2}SL_{m}(ℂ) ⊗ SL_{n}(ℂ).

It remains to note that each metabelian algebra of dimension ≤ 9 has the signature (m, n) such that either n ≤ 2, or m, n ≤ 5, or m ≤ 6, n ≤ 3. The same holds for 10-dimensional algebras with the two exceptions: (m, n) = (7, 3) and (6, 4). Thus we obtain a complete classification of metabelian Lie algebras in dimension up to 9.

1.5. Notation and terminology. It is a good time to introduce some notation and terminology that will be used in the sequel.

We fix bases _{1}e, …,_{m}e in U and e_{1}, …, e_{n} in V , thus identifying U, V with ℂ^{m}, ℂ^{n}.

We denote dual bases of U^{∗} and V^{∗} by lifting indices: ^{i}e and e^{k}.

We denote respective tensor bases in various spaces of tensors by writing indices in a sequence, as in the “tensor notation”. For example, _{ij}e_{k} = _{i}e ∧_{j}e ⊗ e_{k}, ije^{k} = ^{i}e∧^{j}e⊗e^{k}, etc. In tables 1–8, we use the shorthand notation: ( abc ... ijk) := abe_{c} + … +_{ij}e_{k}.

Let 𝔱 ⊂ 𝔰𝔩(U) ⊕ 𝔰𝔩(V) be the standard Cartan subalgebra of operators diagonal in the chosen bases, and T ⊂ SL(U) × SL(V) be the respective Cartan subgroup.

Let _{i}ε, ε_{k} be the weights of _{i}e, e_{k} with respect to 𝔱 or T. (We will not distinguish the notation.) We have ∑_{i}ε = ∑ ε_{k} = 0. Choose _{i}ε − _{j}ε, ε_{k} − ε_{l} (i < j, k < l) as positive roots of 𝔰𝔩(U) ⊕ 𝔰𝔩(V) relative to 𝔱.

For any vector space M with a fixed basis e_{1}, …, e_{n}, we use the following notation. Let ⟨·, ·⟩ denote the natural pairing between tensor spaces M^{⊗p} and (M^{∗})^{⊗p}. Denote by C(γ) the contraction of γ ∈ M^{⊗p} ⊗ (M^{∗})^{⊗p} in the first index. We use the “big” exterior product of skew-symmetric tensors, so that ⟨v_{1}^{∗} ∧ … ∧ v_{p}^{∗}, v_{1} ∧ … ∧ v_{p}⟩ = p! det⟨v_{i}^{∗}, v_{j}⟩ (v_{i}^{∗} ∈ M^{∗}, v_{j} ∈ M). The isomorphism

ι : ⋀^{p}M ⟶^{∼} ⋀^{n−p}M^{∗} is defined by α∧β = ⟨ι(α), β⟩·e_{1} ∧…∧e_{n}, ∀α ∈ ⋀^{p}M, β ∈

⋀^{n−p}M^{∗}.

Suppose H : M is a rational representation of a reductive algebraic group, π : M → M//H is the categorical quotient. A vector v ∈ M is called nilpotent if v ∈ π(π^{−1}(0)) (or, equivalently, if all homogeneous invariants of positive degree vanish at v). A vector v ∈ M is called semisimple if the orbit Hv ⊆ M is closed. A vector that is neither semisimple nor nilpotent is called mixed.

Suppose H is an algebraic group with the Lie algebra 𝔥. Then H^{0} denotes its identity component, and H′ (𝔥′) the commutator subgroup (subalgebra) of H (resp. 𝔥). For any subset 𝔪 ⊂ 𝔥, let Z(𝔪) (𝔷(𝔪)) be the centralizer of 𝔪 in H (resp. in 𝔥).

If X is an H -variety (H -module), then H : X denotes the action of H on X, X^{H} (X^{𝔥}) is the fixed point set of H (resp. the annihilator of 𝔥) in X, and H_{x} (𝔥_{x}) denotes the stabilizer (subalgebra) of x ∈ X.

2. Generalities on θ-groups

In this section, we will give a brief review of the theory of θ-groups. For a detailed narration of this topic, see [13] and [14]. In [13], the theory of Cartan subspaces, of Weyl groups, and of invariants for a θ-group is considered. In [14], a method for classifying nilpotent elements w. r. t. the action of a θ-group is described. Papers [11] and [12] contain a shorthand exposition of main results of [13] and [14], respectively.

In [15], one of the most interesting examples of a θ-group, namely the group ⋀^{3}SL_{9}(ℂ), is considered, and the orbits and the invariants are described in that case.

2.1. Definition of a θ-group. Let G be a semisimple algebraic group, 𝔤 its Lie algebra. Assume that 𝔤 is graded modulo m:

𝔤 = ⨁_{k∈Z_{m}} 𝔤_{k}.

(Here m is any natural number or ∞; Z_{m} is the respective residue group; Z_{∞} = ℤ. We also write 𝔤_{k} = 𝔤_{k mod m} for any k ∈ ℤ.) If m < ∞, then this gradation determines an automorphism θ ∈ Aut 𝔤:

θ(x) = ω^{k}x for ∀x ∈ 𝔤_{k},

where ω = e^{\frac{2πi}{m}}. If m = ∞, then this gradation determines a one-parameter subgroup θ_{t} ∈ Aut 𝔤, t ∈ ℂ^{×}:

θ_{t}(x) = t^{k}x for ∀x ∈ 𝔤_{k}.

Conversely, each automorphism of finite order m (or a one-parameter subgroup in Aut g) determines a Z_{m}- (or ℤ-) gradation of 𝔤. We may (and will) assume that G is simply connected. Then θ (or θ_{t}) is the differential of an automorphism of G (or a one-parameter subgroup of them) denoted by the same letter.

Let G_{0} be the connected reductive subgroup of G corresponding to the subalgebra 𝔤_{0} ⊆ 𝔤. Since G is simply connected, G_{0} = G^{θ}. The restriction of the adjoint representation of G to G_{0} leaves each g_{k} stable. Let ρ : G_{0} → GL(𝔤_{1}) be the restriction of the adjoint action of G_{0} to 𝔤_{1}. The linear algebraic group ρ(G_{0}) is called a θ-group. The representation G_{0} :_{ρ} 𝔤_{1} is said to be a θ-representation.

2.2. Jordan decomposition. Each element x ∈ 𝔤 has the (unique) Jordan decomposition x = x_{s} + x_{n}, where x_{s} is a semisimple element, x_{n} is a nilpotent element, and [x_{s}, x_{n}] = 0. Recall from the theory of the adjoint representation that adjoint orbits of semisimple elements are closed, and those of nilpotent elements contain 0 in their closure. The same thing holds for any θ-group.

Theorem 2.1. [13, §§1–2] 1. Each element x ∈ 𝔤_{1} has the Jordan decomposition x = x_{s} + x_{n}, where x_{s}, x_{n} ∈ 𝔤_{1}.

2. If x ∈ 𝔤_{1} is semisimple, then the orbit G_{0}x is closed.

3. If x ∈ 𝔤_{1} is nilpotent, then \overline{G_{0}x} ∋ 0.

4. Under the assumptions of 1, G_{0}x_{s} is the unique closed G_{0}-orbit in \overline{G_{0}x}.

Remark 2.2. Many representations with nice invariant-theoretic properties are θ-representations. The theorem gives a strong motivation for the following general terminology: a vector is called semisimple w. r. t. an algebraic linear group if its orbit is closed, and is called nilpotent if its orbit contains 0 in the closure.

2.3. The Cartan subspace, the Weyl group, and invariants. Any maximal abelian subspace 𝔠 ⊆ 𝔤 that consists of commuting semisimple elements is called a Cartan subspace. This notion generalizes Cartan subalgebras. All Cartan subspaces are G_{0}-conjugated [13, §3], hence we can fix one of them, say 𝔠, for the sequel. Let Z_{0}(·) (N_{0}(·)) denote the centralizer (resp. normalizer) in G_{0}.

Theorem 2.3. [13, §§3–4,6] 1. Each semisimple element of 𝔤_{1} is G_{0}-equiva-lent to some element of 𝔠.

2. The group W = N_{0}(𝔠)/Z_{0}(𝔠) is a finite reflection group acting naturally on 𝔠. It is called the Weyl group of the θ-representation.

3. For ∀x ∈ 𝔠 : G_{0}x ∩ 𝔠 = W x.

4. The restriction of functions on 𝔤_{1} to 𝔠 induces an isomorphism

ℂ[𝔤_{1}]^{G_{0}} ⟶^{∼} ℂ[𝔠]^{W} .

Thus the classification of semisimple elements in 𝔤_{1} reduces to the description of orbits for a finite reflection group.

Furthermore, each level variety of invariants, i. e., a fiber π^{−1}(π(x)) of the quotient morphism π : 𝔤_{1} → 𝔤_{1}//G_{0}, has finitely many orbits (and exactly one closed orbit G_{0}x_{s}). In other words, each θ-group is visible.

2.4. The covering loop algebra. The description of all ℤ-gradations on a semisimple algebra 𝔤 may be obtained in the following way. Choose a maximal torus T in G containing a one-parameter subgroup θ_{t} related to the ℤ-gradation. Its Lie algebra 𝔱 is a Cartan subalgebra of 𝔤 contained in 𝔤_{0}, and each root subspace 𝔤_{α} is contained in some 𝔤_{k}. Hence each root α receives the degree deg α = k. The degree is an additive function on the root system, hence it is determined by its values on simple roots. We may choose simple roots so that their degrees are non-negative. Conversely, each non-negative integer labelling of the Dynkin diagram of 𝔤 determines a ℤ-gradation on 𝔤.

In order to describe all Z_{m}-gradations on 𝔤, it is convenient to pass to a “covering” infinite-dimensional Lie algebra. Assume 𝔤 = ⨁ 𝔤_{k} is Z_{m}-graded. Consider a twisted loop algebra

𝔊 = ⨁_{k∈ℤ}𝔊_{k} ⊆ ℂ[t, t^{−1}] ⊗ 𝔤, 𝔊_{k} = t^{k}𝔤_{k},

so that 𝔤 = 𝔊/(t^{m} − 1)𝔊. In case, where 𝔤 is simple, 𝔊 is a simple codimension 2 subquotient of an affine Kac–Moody algebra. In general, 𝔊 is a direct sum of such algebras. For the rest of this subsection, we will assume that 𝔤 is simple.

See [6, Chap. 7–8] for a detailed exposition of the theory of loop algebras. It resembles the theory of finite-dimensional (semi)simple Lie algebras. In particular, 𝔊 is graded by a free abelian finitely-generated group Q (the root lattice) so that its component 𝔱 of degree zero is a Cartan subalgebra in 𝔊_{0} ≅ 𝔤_{0} and each component 𝔊_{α} of degree α ∈ Q is contained in some 𝔊_{k}. Elements α ∈ Q such that 𝔊_{α} ≠ 0 are called roots. Thus each root α receives the degree deg α = k.

The period s of θ mod Int 𝔤 is called the index of 𝔊. There exists a unique root ν ∈ Q such that ∀α ∈ Q : t^{m}𝔊_{α} = 𝔊_{α+sν}. There exists a homomorphism Q → 𝔱^{∗}, α ↦ ᾱ, such that ∀h ∈ 𝔱, x ∈ 𝔊_{α} : [h, x] = ᾱ(h)x, and ᾱ = 0 iff α is a multiple of ν (such roots are called imaginary). For real roots (i. e., such α that ᾱ ≠ 0) we have dim 𝔊_{α} = 1.

The set Δ = { ᾱ | α is a real root } is exactly the set of nonzero weights of 𝔤 relative to 𝔱. If 𝔱 is a Cartan subalgebra of 𝔤, not only of 𝔤_{0}, or equivalently, s = 1, then Δ is just the root system of 𝔤 relative to 𝔱.

There exists a base Π of Q consisting of real roots, and we may choose it so that deg α ≥ 0 for ∀α ∈ Π. One may associate a Cartan matrix and a Dynkin diagram with Π exactly in the same way as in the theory of semisimple algebras. The diagrams thus obtained are affine Dynkin diagrams X_{l}^{(s)}, where X_{l} is the Dynkin diagram of 𝔤. We label the nodes of an affine Dynkin diagram by Kac labels a_{α}, which are the (integer mutually coprime) coefficients of the linear dependence among the columns of the respective Cartan matrix. We have

ν = ∑_{α∈Π} a_{α}α

The loop algebra is determined by its Dynkin diagram up to an isomorphism discarding the gradation. Since the degree function is additive on the roots, it is determined by its values d_{α} = deg α on α ∈ Π. Clearly we have

∑_{α∈Π} a_{α}d_{α} = \frac{m}{s} (1)

Conversely, each non-negative integer labelling of the affine Dynkin diagram satisfying (1) determines a ℤ-gradation of 𝔊 and Z_{m}-gradation of 𝔤. Simple roots α with zero labels (or, more precisely, their restrictions ᾱ to 𝔱) form a base of the root system of 𝔤_{0}, whereas the functionals ᾱ such that d_{α} = 1 are exactly the lowest weights of irreducible components of the θ-representation G_{0} : 𝔤_{1}.

Remark 2.4. If 𝔤 is not simple, then the respective θ-group is a direct product of θ-groups for simple graded algebras acting each on its own space, and the θrepresentation is a direct sum of the respective θ-representations. The covering algebra 𝔊 is a direct sum of simple twisted loop algebras, and the whole theory generalizes with minor changes.

2.5. Classification of nilpotent elements. For θ-groups, we have the following generalization of the Morozov-Jacobson theorem.

Theorem 2.5. [14, §2] Suppose e ∈ 𝔤_{1} is a nonzero nilpotent element. Then there exists a nilpotent element f ∈ 𝔤_{−1} and a semisimple element h ∈ 𝔤_{0} such that {e, f, h} is an 𝔰𝔩_{2}-triple. The element h is called a characteristic of e. It is determined up to conjugation by Z_{0}(e), whereas e is determined up to conjugation by Z_{0}(h).

Replacing a nilpotent element e by its conjugate, we may (and will) assume that a maximal torus of N_{0}(e) lies in 𝔱. In particular, h ∈ 𝔱. Moreover, h lies in the rational form 𝔱(ℚ) of 𝔱 defined by the condition that differentials of all T - characters have rational values, and we may assume that h lies in the positive Weyl chamber of 𝔱(ℚ) w. r. t. a given base of the root system. Such h is determined uniquely by the orbit G_{0}e.

We will consider ℤ-graded subalgebras 𝔰 ⊆ 𝔤. (This means that 𝔰_{k} ⊆ 𝔤_{k} for ∀k ∈ ℤ.) A ℤ-graded subalgebra is said to be regular if it is normalized by a maximal torus in G_{0}, and complete if it is maximal among all semisimple regular subalgebras of the same rank.

Definition 2.6. A support of a nilpotent element e ∈ 𝔤_{1} is a minimal complete subalgebra containing e.

A support is determined up to conjugacy and may be constructed as follows. Consider a reductive ℤ-graded subalgebra

𝔤(h) = ∑_{k∈ℤ}𝔤_{k}(h), 𝔤_{k}(h) = { x ∈ 𝔤_{k} | [h, x] = 2kx }, (2)

and let 𝔱_{e} be the centralizer of e in 𝔱. Then 𝔰 = (𝔤(h)^{𝔱_{e}})′ is the (T -regular) support of e [14, §4].

Let S_{0} be the reductive subgroup of G_{0} corresponding to 𝔰_{0}. Then all elements of 𝔰_{1} are nilpotent, S_{0} acts on 𝔰_{1} with finitely many orbits, e lies in the dense orbit, and the stabilizer (S_{0})_{e} is finite. In particular, a nilpotent element is determined by its support up to conjugacy. Conversely, if 𝔰 ⊆ 𝔤 is a complete subalgebra such that the stabilizer of the dense S_{0}-orbit in 𝔰_{1} is finite, then 𝔰 is the support of any element e from the dense orbit. Semisimple ℤ-graded algebras with the above property (or, equivalently, such that dim 𝔰_{0} = dim 𝔰_{1}) are called locally flat. Simple locally flat algebras are listed in [14, §4], and semisimple ones are their direct sums.

The above discussion gives us the following method of classifying nilpotent elements in 𝔤_{1}.

1. Classify all complete T -regular locally flat subalgebras of 𝔤 up to W_{0}-conjugacy, where W_{0} = N_{0}(T)/T is the Weyl group for G_{0}. Note that each ℤ-graded subalgebra 𝔰 ⊆ 𝔤 is canonically embedded in the covering algebra 𝔊 so that 𝔰_{k} ⊆ 𝔊_{k} for ∀k ∈ ℤ. Hence the root lattice of 𝔰 may be regarded as a sublattice Q(s) ⊆ Q. It is easy to see that 𝔰 is complete iff this sublattice is saturated or, in other words, the quotient group Q/Q(𝔰) is torsion-free. Thus we have to classify all embeddings of the root systems Δ(𝔰) of locally flat algebras, or even of their systems of simple roots Π(𝔰), in Δ which preserve the degrees and the invariant inner product, up to W_{0}-conjugacy.

2. For each complete locally flat subalgebra 𝔰 ⊆ 𝔤, take an element e from the dense S_{0}-orbit in 𝔰_{1}. These e compose a full coset for the set of G_{0}-orbits of nilpotent elements in 𝔤_{1}.

For the sequel, we need the third step of this procedure.

3. Compute characteristics h ∈ 𝔱(ℚ). Each h must lie in the Cartan subalgebra 𝔱(𝔰) of 𝔰, hence it must be orthogonal to the annihilator of Q(𝔰) in 𝔱 w. r. t. the invariant inner product. Moreover, h must have prescribed values on α ∈ Π(𝔰), since it defines the grading of 𝔰. These conditions determine h uniquely, and we may apply a transformation from W_{0} to take h into the positive Weyl chamber of 𝔱.

2.6. Classification of mixed elements. To classify vectors that are neither semisimple nor nilpotent, we first classify their semisimple parts, and then classify vectors x = u + e with a given semisimple part u. We may assume u ∈ 𝔠. The nilpotent part e belongs to the semisimple Z_{m}-graded subalgebra 𝔷(u)′. The problem is reduced to the classification of nilpotent elements in 𝔷(u)′_{1} up to Z_{0}(u)- conjugacy. The classification up to Z_{0}(u)^{0}-conjugacy may be performed using the method of the previous subsection applied to the graded algebra 𝔷(u)′. The finite component group may glue some of these orbits together in one Z_{0}(u)-orbit. Taking this into consideration completes the classification.

2.7. Computing stabilizers. The centralizer of a semisimple element u ∈ 𝔤_{1} is a graded reductive Levi subalgebra 𝔷(u) ⊆ 𝔤. We may assume u ∈ 𝔠, embed 𝔠 in a (graded) Cartan subalgebra 𝔥 ⊂ 𝔤, and determine the root system of 𝔷(u) relative to 𝔥.

The stabilizer of a mixed element x = u + e is the stabilizer of its nilpotent part e in the group Z_{0}(u). If we are interested only in the stabilizer subalgebra, then we may pass to the identity component Z_{0}(u)^{0}. The center of Z(u) stabilizes e, and it remains to determine the stabilizer of e ∈ 𝔷(u)′_{1} in (Z(u)′)_{0}^{0}. Thus the problem is reduced to computing the stabilizer of a nilpotent element.

The stabilizer subalgebra 𝔷_{0}(e) of a nilpotent element e ∈ 𝔤_{1} is spanned (as a linear space) by highest vectors of the 3-dimensional simple subalgebra ⟨e, f, h⟩ which lie in 𝔤_{0}. It follows from the theory of 𝔰𝔩_{2}-representations that eigenvalues of ad h on 𝔷_{0}(e) are non-negative, and ad e maps an eigenspace 𝔤_{0,k}(h) ⊂ 𝔤_{0} of eigenvalue k ≥ 0 onto an eigenspace 𝔤_{1,k+2}(h) ⊂ 𝔤_{1} of eigenvalue k + 2. Now it is easy to see [15, 4.5] that

(3) dim 𝔷_{0}(e) = ∑_{k=0}^{∞} dim 𝔤_{0,k}(h) − dim 𝔤_{1,k+2}(h).

The reductive subalgebra 𝔷_{0}(e) ∩ 𝔷_{0}(h) is a Levi part of 𝔷_{0}(e). It coincides with the stabilizer subalgebra of general position for the θ-representation 𝔤_{0}(h) : 𝔤_{1}(h) of the ℤ-graded algebra 𝔤(h) defined by (2). Indeed, the orbit of e in 𝔤_{1}(h) is Zariski open [15, 4.5]. The algebra 𝔤_{0}(h) = 𝔷_{0}(h) and its θ-representation in 𝔤_{1}(h) are easy to find. Then the stabilizer subalgebra of general position can be found using Elashvili’s tables [1], [2].

3. Metabelian algebras of signature (5, 5)

This section deals with the θ-group ⋀^{2}SL_{5}(ℂ) ⊗ SL_{5}(ℂ). We apply the general theory of θ-groups to classifying the orbits of this linear group. This is the same thing as to classify metabelian Lie algebras of signature (m, n), m, n ≤ 5.

3.1. Formulation of results. We retain the notation of 1.5 with m = n = 5. The classification of tensors in ⋀^{2}U ⊗ V under the action of SL(U) × SL(V) is made as follows. Each tensor has the Jordan decomposition x = u + e (see 2.2), where u is semisimple and e is nilpotent. Semisimple tensors fall into 3 families according to the type of stabilizer. For each family, we give a canonical form u of a semisimple tensor and classify all possible nilpotent parts e up to the action of the stabilizer of u. See tables 1–2 below. If we want to classify tensors under the action of GL(U) × GL(V), then the canonical form for u can be reduced by multiplying u by a nonzero scalar.

Thus the elements of ⋀^{2}U ⊗ V are divided into 3 families according to the type of the semisimple part. All elements in the first family are semisimple, all elements in the third family are nilpotent. Tensors of the first two families and tensors of full rank in the third family represent metabelian algebras of signature (5, 5). All degenerate tensors lie in the third family (Remark 1.6), and each of them represents a series of metabelian algebras of signature (l, q), p ≤ l ≤ 5 (Remark 1.2). We indicate the “minimal” signature (p, q) in table 2.

The last two columns in each table contain information on the dimension of the stabilizer and the type of its (connected reductive) Levi part. Here T_{l} denotes an algebraic torus of dimension l. Making use of these data, one easily computes the dimension and the type of Levi part for the automorphism group (or algebra of derivations) of the respective metabelian algebra (cf. 1.1).

We indicate characteristics of nilpotent elements by their indices, i. e., values of simple roots. All other information contained in the tables is explained in section 2 and in 3.2–3.6.

Here is the classification.

Family 1. This family contains only semisimple tensors. The canonical form is

u = λ_{1}u_{1} + λ_{2}u_{2}, λ_{1}λ_{2}(λ_{1}^{10} − 11λ_{1}^{5}λ_{2}^{5} − λ_{2}^{10}) ≠ 0,

Galitski, Timashev 137

where

u_{1} = _{12}e_{4} +_{34}e_{1} +_{51}e_{3} +_{23}e_{5} +_{45}e_{2},
u_{2} = _{13}e_{2} +_{52}e_{1} +_{41}e_{5} +_{35}e_{4} +_{24}e_{3}.

The coefficients λ_{i} are determined up to the action of a certain finite group W described explicitly in 3.4. Two canonical forms are equivalent iff they have the same values of

λ_{1}^{20} + λ_{2}^{20} + 228λ_{1}^{15}λ_{2}^{5} − 228λ_{1}^{5}λ_{2}^{15} + 494λ_{1}^{10}λ_{2}^{10}  and
λ_{1}^{30} + λ_{2}^{30} − 522λ_{1}^{25}λ_{2}^{5} + 522λ_{1}^{5}λ_{2}^{25} − 10005λ_{1}^{20}λ_{2}^{10} − 10005λ_{1}^{10}λ_{2}^{20}.

The stabilizer subgroup of u is finite.

Family 2. The canonical form of a semisimple part is

u = λu_{1}, λ ≠ 0,

where λ is determined up to multiplication by a 10-th root of unity.

Table 1: Nilpotent parts of elements of family 2

No. Canonical form Support Characteristic Stabilizer
dim Type
1 132 521 415 354  A_{4}  2 2 2 2  0 0
2 132 521 415  A_{3}  2 2 2 0  1 T_{1}
3 132 521 354  A_{2} + A_{1}  2 2 0 2  1 T_{1}
4 132 521  A_{2}  2 2 0 0  2 T_{2}
5 132 415  2A_{1}  2 0 2 0  2 T_{2}
6 132  A_{1}  2 0 0 0  3 T_{3}
7 0  4 T_{4}

Family 3. In this family, u = 0 and all elements are nilpotent.

Table 2: Elements of family 3

No.  Canonical form  Signature  Support  Characteristic  Stabilizer
dim  Type
1 125 144 153 234 243 252 342 351  (5, 5)  E_{8}  10 10 10 10 10 10 10 20  0  0
2 125 134 153 233 243 252 342 451  (5, 5)  E_{8}(a_{1})  10 10 0 10 10 10 10 10  1  0
3 125 135 144 152 234 242 251 343  (5, 5)  E_{8}(a_{2})  10 0 10 10 10 0 10 10  1  0
4 125 134 143 152 233 244 342 451  (5, 5)  E_{8}(b)  0 10 0 10 10 10 0 10  2  0
5 125 143 154 233 242 251 341 352  (5, 5)  E_{8}(d_{1})  10 0 10 0 0 10 0 10  2  0
6 125 132 144 153 234 243 252 351  (5, 5)  E_{8}(c_{2})  0 10 0 0 10 0 0 10  3  0
7 125 134 141 153 243 252 342 351  (5, 5)  E_{8}(d_{3})  0 0 10 0 0 0 10 0  3  0
8 125 134 143 233 252 342 451  (5, 5)  E_{7}  1 9 1 9 10 10 1 9  3  T_{1}
9 125 135 144 153 233 242 351  (5, 5)  E_{7}(a_{1})  10 0 3 7 7 3 7 3  3  T_{1}
10 135 144 152 234 243 251 341  (5, 5)  D_{7}  3 7 3 7 3 4 3 3  3  T_{1}
11 121 144 153 234 243 252 342 451  (5, 4)  E_{8}(d_{5})  0 0 0 0 0 0 0 10  4  0
12 125 134 144 152 232 243 351  (5, 5)  E_{7}(a_{2})  1 9 0 1 10 0 1 9  4  T_{1}
13 125 142 153 234 243 252 341  (5, 5)  E_{7}(b)  0 3 7 0 3 0 7 3  4  T_{1}
14 135 143 152 234 242 251 342  (5, 5)  D_{7}(a_{1})  2 0 8 2 2 2 6 2  4  T_{1}
15 134 141 153 243 252 342 351  (5, 4)  E_{7}(c_{2})  0 0 1 0 0 0 1 9  5  T_{1}
16 125 133 144 152 233 251 342  (5, 5)  D_{7}(a_{2})  0 5 0 5 0 5 0 5  5  T_{1}

138 Galitski, Timashev

Table 2: (Continued)

No.  Canonical form  Signature  Support  Characteristic  Stabilizer
dim  Type
17 135 143 152 234 242 251 341  (5, 5)  A_{7}  3 3 4 3 3 3 1 3  5  T_{1}
18 125 134 143 152 233 242 451  (5, 5)  E_{6} + A_{1}  1 8 1 1 10 1 1 8  5  T_{1}
19 125 134 143 153 233 251 342  (5, 5)  E_{6}(a_{1}) + A_{1}  5 0 5 0 0 5 5 0  5  T_{1}
20 125 143 152 234 242 351  (5, 5)  D_{6}  1 2 7 1 3 1 6 3  5  T_{2}
21 124 153 233 241 252 342 351  (5, 4)  E_{6}(b) + A_{1}  1 0 1 0 0 1 1 8  6  T_{1}
22 125 144 152 233 242 251 341  (5, 5)  A_{6} + A_{1}  3 3 1 3 3 1 2 4  6  T_{1}
23 125 134 143 152 233 251 342  (5, 5)  D_{5} + A_{2}  2 2 2 4 2 4 2 2  6  T_{1}
24 125 134 143 153 251 342  (5, 5)  E_{6}(a_{1})  6 0 4 0 0 6 4 0  6  T_{2}
25 125 144 151 233 244 342  (5, 5)  D_{6}(a_{1})  0 3 0 7 4 3 0 3  6  T_{2}
26 133 152 233 244 251 341  (5, 4)  D_{6}(a_{2})  0 1 0 1 0 1 0 9  6  T_{2}
27 125 134 143 152 233 242 251 341  (5, 5)  2A_{4}  2 2 2 2 2 2 2 2  7  0
28 134 143 152 233 242 251 342  (5, 4)  D_{5}(a_{1}) + A_{2}  1 0 1 1 1 1 1 7  7  T_{1}
29 125 144 153 232 251 341  (5, 5)  A_{6}  4 2 2 2 4 0 2 4  7  T_{2}
30 125 134 143 233 251 342  (5, 5)  D_{5} + A_{1}  1 2 1 6 4 3 1 2  7  T_{2}
31 144 152 233 242 251 341  (5, 4)  A_{5} + A_{1}  1 1 0 1 1 0 1 8  7  T_{2}
32 125 134 142 233 252 451  (5, 5)  E_{6}  0 8 2 0 10 2 0 8  7  A_{1} + T_{1}
33 135 141 152 234 243 251  (5, 5)  A_{6}  0 8 2 0 0 0 2 0  7  A_{1} + T_{1}
34 134 143 152 233 242 251 341  (5, 4)  A_{4} + A_{3}  1 1 1 1 1 1 1 6  8  T_{1}
35 124 132 153 233 242 451  (5, 4)  D_{4} + A_{2}  0 2 0 0 2 0 2 6  8  T_{2}
36 134 143 231 252 342 451  (5, 4)  E_{6}(b)  2 0 0 0 0 2 0 8  8  A_{1} + T_{1}
37 125 133 144 152 251 342  (5, 5)  D_{5} + A_{1}  3 2 0 5 3 5 0 2  8  A_{1} + T_{1}
38 135 144 151 232 243 341  (5, 5)  A_{5} + A_{1}  3 2 0 5 2 0 3 0  8  A_{1} + T_{1}
39 125 134 143 152 232 251 341  (5, 5)  A_{4} + A_{3}  3 1 2 1 3 1 2 1  9  T_{1}
40 125 134 143 151 233 242 341  (5, 5)  A_{4} + A_{2} + A_{1}  1 2 1 3 2 1 1 2  9  T_{1}
41 124 143 152 233 242 251 341  (5, 4)  A_{4} + A_{2} + A_{1}  1 1 1 1 1 1 2 4  9  T_{1}
42 123 135 152 234 242 251 341  (5, 5)  A_{4} + A_{3}  0 0 5 0 0 5 0 0  9  A_{1}
43 125 131 144 153 234 242 251  (5, 5)  A_{4} + A_{2} + A_{1}  0 5 0 0 0 0 0 5  9  A_{1}
44 124 143 152 232 242 351  (5, 4)  D_{5}(a_{1}) + A_{1}  1 1 0 1 2 1 1 6  9  T_{2}
45 123 142 151 232 243 344  (5, 4)  D_{5}(a_{1}) + A_{1}  0 0 0 3 3 0 0 7  9  A_{1} + T_{1}
46 125 133 144 251 342  (5, 5)  D_{5}  2 2 0 6 4 4 0 2  9  A_{1} + T_{2}
47 134 145 151 233 242  (5, 5)  A_{5}  3 3 0 4 1 0 3 0  9  A_{1} + T_{2}
48 144 153 232 241 351  (5, 4)  A_{5}  2 0 1 0 1 1 0 8  9  A_{1} + T_{2}
49 125 133 142 234 251 341  (5, 5)  A_{4} + A_{2}  0 2 2 2 2 2 0 2  10  T_{2}
50 134 143 152 232 251 341  (5, 4)  A_{4} + A_{2}  2 0 2 0 2 0 2 4  10  T_{2}
51 125 134 143 151 233 242  (5, 5)  A_{4} + 2A_{1}  1 3 1 1 0 1 1 3  10  T_{2}
52 123 152 234 242 251 341  (5, 4)  A_{4} + 2A_{1}  0 0 3 0 0 3 0 4  10  T_{2}
53 134 143 151 233 242 341  (5, 4)  A_{4} + 2A_{1}  1 1 1 2 1 1 1 4  10  T_{2}
54 131 144 153 234 242 251  (5, 4)  A_{4} + A_{2}  0 4 0 0 0 0 0 6  10  A_{1} + T_{1}
55 124 132 142 243 351  (5, 4)  D_{5}(a_{1})  0 1 0 2 3 0 1 6  10  T_{3}
56 124 133 152 242 251 341  (5, 4)  2A_{3}  1 1 1 1 1 2 1 3  11  T_{2}
57 125 133 152 234 241  (5, 5)  A_{4} + A_{1}  0 3 2 0 0 2 0 3  11  T_{3}
58 134 143 151 233 242  (5, 4)  A_{4} + A_{1}  1 2 1 1 0 1 1 4  11  T_{3}
59 133 142 234 251 341  (5, 4)  A_{4} + A_{1}  0 1 2 1 1 2 0 4  11  T_{3}
60 124 132 143 232 251 341  (5, 4)  D_{4}(a_{1}) + A_{2}  0 2 0 2 2 0 2 2  12  T_{2}
61 125 134 143 151 232 241  (5, 5)  A_{3} + A_{2} + A_{1}  2 1 1 2 1 1 1 1  12  T_{2}
62 121 153 233 252 342 451  (5, 3)  A_{3} + A_{2} + A_{1}  0 0 0 0 0 0 5 0  12  2A_{1}
63 123 134 142 232 451  (5, 4)  D_{4} + A_{1}  1 0 1 1 3 1 0 6  12  A_{1} + T_{2}
64 133 152 234 241  (5, 4)  A_{4}  0 2 2 0 0 2 0 4  12  T_{4}
65 124 133 142 232 251 341  (5, 4)  A_{3} + A_{2} + A_{1}  1 1 1 1 2 1 1 2  13  T_{2}
66 124 131 153 231 242  (5, 4)  D_{4}(a_{1}) + A_{1}  0 3 0 0 0 0 3 1  13  T_{3}
67 134 143 151 232 241  (5, 4)  A_{3} + A_{2}  2 0 1 2 1 1 1 2  13  T_{3}
68 141 152 233 242 351  (5, 3)  A_{3} + A_{2}  0 0 1 0 0 1 4 0  13  A_{1} + T_{2}
69 122 134 145 153 251 341  (5, 5)  2A_{3}  5 0 0 0 5 0 0 0  14  C_{2}
70 121 133 145 235 244 342  (4, 5)  2A_{2} + 2A_{1}  0 0 0 5 0 0 0 0  14  C_{2}
71 124 143 151 232 241  (5, 4)  A_{3} + 2A_{1}  1 2 0 1 0 1 2 1  14  T_{3}
72 132 143 232 251 341  (5, 3)  D_{4}(a_{1}) + A_{1}  0 1 0 1 1 0 4 0  14  A_{1} + T_{2}
73 124 133 142 151 232 241  (5, 4)  2A_{2} + 2A_{1}  1 1 1 1 1 1 1 1  15  T_{2}
74 124 132 143 251 341  (5, 4)  A_{3} + A_{2}  2 1 0 1 3 0 1 2  15  A_{1} + T_{2}
75 123 152 232 251 341  (5, 3)  A_{3} + 2A_{1}  1 0 1 0 1 1 3 0  15  A_{1} + T_{2}
76 124 135 142 153 231  (5, 5)  A_{3} + 2A_{1}  3 0 2 0 1 0 2 0  15  2A_{1} + T_{1}
77 125 131 143 234 242  (4, 5)  2A_{2} + A_{1}  0 1 0 4 0 0 0 1  15  2A_{1} + T_{1}
78 131 153 231 242  (5, 3)  D_{4}(a_{1})  0 2 0 0 0 0 4 0  15  A_{1} + T_{3}
79 124 142 153 231  (5, 4)  A_{3} + A_{1}  2 1 1 0 1 0 2 1  16  A_{1} + T_{3}
80 133 152 232 241  (5, 3)  A_{3} + A_{1}  1 1 0 1 0 1 3 0  16  A_{1} + T_{3}
81 121 133 244 342  (4, 4)  2A_{2}  0 0 0 4 0 0 0 2  16  2A_{1} + T_{2}
82 122 133 234 451  (5, 4)  D_{4}  0 0 2 0 4 0 0 6  16  A_{2} + A_{1} + T_{1}
83 124 132 151 233 241  (5, 4)  2A_{2} + A_{1}  0 1 2 0 2 0 1 1  17  A_{1} + T_{2}

Galitski, Timashev 139

Table 2: (Continued)

No.  Canonical form  Signature  Support  Characteristic  Stabilizer
dim  Type
84 123 142 151 232 341  (5, 3)  2A_{2} + A_{1}  1 0 1 1 1 1 2 0  17  A_{1} + T_{2}
85 124 131 143 233 242  (4, 4)  A_{2} + 3A_{1}  0 1 0 3 0 0 1 1  17  A_{1} + T_{2}
86 123 134 142 151 231  (5, 4)  A_{2} + 3A_{1}  2 0 1 1 1 1 0 1  18  A_{1} + T_{2}
87 122 153 251 341  (5, 3)  A_{3} + A_{1}  2 0 0 0 2 0 3 0  18  3A_{1} + T_{1}
88 123 131 152 242 251  (5, 3)  A_{2} + 3A_{1}  0 2 0 0 0 2 1 0  19  2A_{1} + T_{1}
89 123 134 141 232  (4, 4)  A_{2} + 2A_{1}  1 0 1 2 0 1 0 1  19  A_{1} + T_{3}
90 122 133 251 341  (5, 3)  2A_{2}  0 0 2 0 2 0 2 0  19  2A_{1} + T_{2}
91 142 153 231  (5, 3)  A_{3}  2 0 1 0 1 0 3 0  19  3A_{1} + T_{2}
92 123 132 151 241  (5, 3)  A_{2} + 2A_{1}  1 1 0 1 1 1 1 0  20  A_{1} + T_{3}
93 121 133 243 342  (4, 3)  A_{2} + 2A_{1}  0 0 0 3 0 0 2 0  20  3A_{1} + T_{1}
94 123 131 242  (4, 3)  A_{2} + A_{1}  0 1 0 2 0 1 1 0  21  A_{1} + T_{4}
95 131 152 242 251  (5, 2)  A_{2} + 2A_{1}  0 1 0 0 0 3 0 0  22  A_{2} + A_{1} + T_{1}
96 122 133 141 231  (4, 3)  4A_{1}  1 0 1 1 1 0 1 0  23  2A_{1} + T_{2}
97 121 132 143 154  (5, 4)  4A_{1}  3 0 0 0 0 0 0 1  24  A_{3} + T_{1}
98 122 151 341  (5, 2)  A_{2} + A_{1}  1 0 0 1 1 2 0 0  24  A_{2} + A_{1} + T_{2}
99 121 132 143  (4, 3)  3A_{1}  2 0 0 1 0 0 1 0  26  A_{2} + A_{1} + T_{2}
100 121 342  (4, 2)  A_{2}  0 0 0 2 0 2 0 0  26  A_{2} + 2A_{1} + T_{2}
101 122 131 241  (4, 2)  3A_{1}  0 1 0 1 1 1 0 0  27  A_{2} + A_{1} + T_{2}
102 121 132 233  (3, 3)  3A_{1}  0 0 2 0 0 0 1 0  27  A_{2} + 2A_{1} + T_{1}
103 121 132  (3, 2)  2A_{1}  1 0 1 0 0 1 0 0  30  A_{2} + 2A_{1} + T_{2}
104 121 341  (4, 1)  2A_{1}  0 0 0 1 2 0 0 0  34  C_{2} + A_{3} + T_{1}
105 121  (2, 1)  A_{1}  0 1 0 0 1 0 0 0  37 A_{3} + A_{2} + A_{1} + T_{1}
106 0  48  2A_{4}

3.2. The Z_{5}-graded algebra E_{8}. Let 𝔤 be a simple Lie algebra of the type E_{8}. We define its Z_{5}-grading by the following labelling of the affine Dynkin diagram E_{8}^{(1)} (cf. 2.4):

[figure: 1]

(We omit zero labels.) From the discussion in 2.4, it is clear that the Lie algebra 𝔤_{0} and its θ-representation in 𝔤_{1} is determined by the following Dynkin diagram with numerical labels of the (unique) highest weight:

[figure: 1; 1]

The precise description of the Z_{5}-graded algebra 𝔤 may be obtained as follows. We consider a direct sum

𝔤 = (U ⊗ ⋀^{2}V^{∗}) ⊕ (⋀^{2}U^{∗} ⊗ V^{∗}) ⊕ (𝔰𝔩(U) ⊕ 𝔰𝔩(V)) ⊕ (⋀^{2}U ⊗ V) ⊕ (U^{∗} ⊗ ⋀^{2}V)

and introduce a Lie algebra structure on it so that the above direct sum is a Z_{5}-grading

𝔤 = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2}.

The commutator map 𝔤_{0} × 𝔤_{i} → 𝔤_{i} is the natural action of 𝔰𝔩(U) ⊕ 𝔰𝔩(V) on the respective space of tensors. The commutator maps 𝔤_{i} × 𝔤_{j} → 𝔤_{i+j} (i, j ∈ Z_{5}) are determined by their 𝔤_{0}-equivariance uniquely up to proportionality. The coefficients of proportionality are determined uniquely by choosing the identification of 𝔤_{i} with the respective space of tensors and verifying the Jacobi identity.

The commutators between 𝔤_{i} and 𝔤_{j} are given by the formulas

[α^{∗} ⊗ v^{∗}, α ⊗ v] = (⟨α^{∗}, α⟩/5 · 1 − C(α^{∗} ⊗ α)) · ⟨v^{∗}, v⟩
⊕ ⟨α^{∗}, α⟩/2 · (⟨v^{∗}, v⟩/5 · 1 − v^{∗} ⊗ v)
[u ⊗ β^{∗}, u^{∗} ⊗ β] = (u^{∗} ⊗ u − ⟨u^{∗}, u⟩/5 · 1) · ⟨β^{∗}, β⟩/2
⊕ ⟨u^{∗}, u⟩ · (⟨β^{∗}, β⟩/5 · 1 − C(β^{∗} ⊗ β))
[u_{1} ∧ u_{2} ⊗ v_{1}, u_{3} ∧ u_{4} ⊗ v_{2}] = ι(u_{1} ∧ u_{2} ∧ u_{3} ∧ u_{4}) ⊗ v_{1} ∧ v_{2}
[u_{1} ∧ u_{2} ⊗ v, u^{∗} ⊗ v_{1} ∧ v_{2}] = −2C(u^{∗} ⊗ u_{1} ∧ u_{2}) ⊗ ι(v ∧ v_{1} ∧ v_{2})
[u_{1}^{∗} ⊗ v_{1} ∧ v_{2}, u_{2}^{∗} ⊗ v_{3} ∧ v_{4}] = −u_{1}^{∗} ∧ u_{2}^{∗} ⊗ ι(v_{1} ∧ v_{2} ∧ v_{3} ∧ v_{4})
[u_{1}^{∗} ∧ u_{2}^{∗} ⊗ v^{∗}, u^{∗} ⊗ v_{1} ∧ v_{2}] = 2ι(u_{1}^{∗} ∧ u_{2}^{∗} ∧ u^{∗}) ⊗ C(v^{∗} ⊗ v_{1} ∧ v_{2})

(where α ∈ ⋀^{2}U; β ∈ ⋀^{2}V ; α^{∗} ∈ ⋀^{2}U^{∗}; β^{∗} ∈ ⋀^{2}V^{∗}; u, u_{i} ∈ U; v, v_{i} ∈ V ; u^{∗}, u_{i}^{∗} ∈ U^{∗}; v^{∗}, v_{i}^{∗} ∈ V^{∗}), and the formulas dual to the last four. (That is, vectors and bivectors are replaced by those in dual spaces, and the terms in each commutator are swapped.)

Next we describe the root system Δ of 𝔤 relative to 𝔱. Denote by Δ_{k} the set of nonzero weights of 𝔱 in 𝔤_{k}. We have Δ_{0} = {_{i}ε−_{j}ε, ε_{k} −ε_{l}}, Δ_{1} = {_{i}ε+_{j}ε+ε_{k}}, Δ_{2} = {−_{i}ε + ε_{k} + ε_{l}}, Δ_{−1} = {−_{i}ε −_{j}ε − ε_{k}}, Δ_{−2} = {_{i}ε − ε_{k} − ε_{l}} (i ≠ j, k ≠ l).

The Weyl group W_{0} permutes _{i}ε and ε_{i} arbitrarily. We choose _{i}ε −_{i+1}ε, ε_{i} − ε_{i+1} (i = 1, …, 4) as simple roots of 𝔤_{0}. Together with _{4}ε + _{5}ε + ε_{5}, they compose an extended system of simple roots of 𝔤.

An invariant inner product on 𝔱^{∗} can be determined by the formulas (_{i}ε,_{j}ε) = (ε_{s}, ε_{t}) = −\frac{1}{5}, (_{i}ε,_{i}ε) = (ε_{s}, ε_{s}) = \frac{4}{5}, (_{i}ε, ε_{s}) = 0 (i ≠ j, s ≠ t). In particular, if ∑_{i}x = ∑ x_{s} = 0, then (∑_{i}x_{i}ε + ∑ x_{s}ε_{s}, ∑_{i}y_{i}ε + ∑ y_{s}ε_{s}) = ∑_{i}x_{i}y + ∑ x_{s}y_{s}, and the inner products of roots of degree 1 can be computed by the following rule:

(4) (_{i}ε +_{j}ε + ε_{s},_{k}ε +_{l}ε + ε_{t}) = \{ \begin{matrix}−1 & if δ = 0, \\ 0 & if δ = 1, \\ 1 & if δ = 2, \\ 2 & if δ = 3, \end{matrix}

where δ is δ_{st} plus the number of common elements in {i, j} and {k, l}.

3.3. A Cartan subspace. Consider regular Z_{5}-graded subalgebras of type A_{4} in 𝔤 such that their degree 0 component is a Cartan subalgebra, and simple roots and the lowest root have degree 1. They are classified by embeddings of the extended system of simple roots A_{4}^{(1)} in Δ_{1} preserving the inner product. It is easy to see that there is a unique such subalgebra up to W_{0}-conjugacy, say, the algebra 𝔄_{1} generated by root vectors _{12}e_{4},_{34}e_{1},_{51}e_{3},_{23}e_{5},_{45}e_{2}. The algebra 𝔄_{2} generated by _{13}e_{2},_{52}e_{1},_{41}e_{5},_{35}e_{4},_{24}e_{3} is the unique subalgebra of the same kind commuting with 𝔄_{1}. It follows that the elements u_{1} = _{12}e_{4} + _{34}e_{1} + _{51}e_{3} + _{23}e_{5} + _{45}e_{2} and u_{2} = _{13}e_{2} +_{52}e_{1} +_{41}e_{5} +_{35}e_{4} +_{24}e_{3} are semisimple and commute with each other. Hence 𝔠 = ⟨u_{1}, u_{2}⟩ is an abelian subspace in 𝔤_{1} consisting of semisimple elements. It can be included in a Z_{5}-graded Cartan subalgebra 𝔥 of 𝔤 constructed as follows. The subspace 𝔥_{k} is generated by u_{1}^{(k)}, u_{2}^{(k)}, where u_{i}^{(k)} is the sum of root vectors of degree k in 𝔄_{i}. An easy argument from [13, 3.1] shows that 𝔠 is a Cartan subspace. (Otherwise there exists a Cartan subspace of dimension ≥ 3, whence the minimal algebraic torus containing it has dimension ≥ 3 · 4 = 12, whereas rk 𝔤 = 8.)

3.4. The Weyl group. Now we want to determine the Weyl group W associated with the chosen Cartan subspace 𝔠. By definition, W is the group of linear transformations of 𝔠 generated by elements of G_{0} normalizing 𝔠. First note that if a scalar transformation α · 1 belongs to W, then α^{10} = 1. Indeed, we may regard u_{i} as operators ⋀^{2}U^{∗} → V and consider u_{1} ⊕ u_{2} : ⋀^{2}U^{∗} → V ⊕ V . An element g ∈ N_{0}(𝔠) acting on 𝔠 as α · 1 multiplies u_{1} ⊕ u_{2} by α, hence ⋀^{10}(u_{1} ⊕ u_{2}) ≠ 0 by α^{10}; but recall that g ∈ SL(U) × SL(V) acts on a one-dimensional space Hom(⋀^{10}(⋀^{2}U^{∗}), ⋀^{10}(V ⊕ V)) trivially. It is easy to pick elements from G_{0}, diagonal in the bases _{j}e, e_{j}, that multiply u_{k} by arbitrary and independent 5-th roots of unity. Also note that a pair of monomial transformations of U and V given by the cyclic permutation ( 1 2 4 3 ) of basic vectors _{j}e, e_{j} combined with the multiplication by −1 yields a transformation r ∈ W, whose matrix in the basis u_{1}, u_{2}

is (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). It already follows that W is irreducible and its center is isomorphic

to Z_{10}.

Consider the (unique) Z_{5}-graded Cartan subalgebra 𝔥 ⊃ 𝔠 constructed in 3.3. The automorphism θ acts naturally on 𝔥^{∗} and permutes the roots of 𝔤 relative to 𝔥. It is represented by an integral matrix in a base of simple roots and has eigenvalues ω^{±1}, ω^{±2} of multiplicity 2, where ω = e^{\frac{2πi}{5}} . The minimal polynomial μ_{θ}(t) = t^{4} + t^{3} + t^{2} + t + 1 is irreducible over ℚ. The restriction of θ to each invariant subspace of 𝔥^{∗} generated by roots has an integral matrix as well, whence its eigenvalues are ω^{±1}, ω^{±2} (of equal multiplicities). It follows that for each root α, rk{α, θα, θ^{2}α, θ^{3}α, θ^{4}α} = 4 and ∑_{k=0}^{4} θ^{k}α = 0. We conclude that {α, θα, θ^{2}α, θ^{3}α, θ^{4}α} is the extended system of simple roots of type A_{4}^{(1)} of a certain 𝔥-regular θ-invariant simple subalgebra 𝔄(α), whose 20 roots are θ^{k}α + … + θ^{k+l}α (k, l ≥ 0).

The 240 roots of 𝔤 relative to 𝔥 fall into 12 such subsystems of type A_{4}, hence we have 12 subalgebras 𝔄(α) of type A_{4}, 𝔄_{1} and 𝔄_{2} being among them. We have a decomposition 𝔥 = 𝔥(α) ⊕ 𝔥^{α}, where 𝔥(α) is a Cartan subalgebra in 𝔄(α) and 𝔥^{α} is the centralizer of 𝔄(α) in 𝔥. Since there are no outer automorphisms of 𝔄(α) of order 5, the restriction θ|_{𝔄(α)} is induced by an inner automorphism Ad g_{α}, where g_{α} is an element of the subgroup A(α) of G with Lie algebra 𝔄(α). Clearly, θ(g_{α}) = g_{α}, whence g_{α} ∈ G_{0}. We have Ad g_{α}|_{𝔥} = θ|_{𝔥(α)} ⊕ 1|_{𝔥^{α}}, and w_{α} = Ad g_{α}|_{𝔠} ∈ W has eigenspaces 𝔥(α)_{1} = 𝔠 ∩ 𝔥(α) and 𝔥_{1}^{α} = 𝔠 ∩ 𝔥^{α} and is

represented by the matrix ( \begin{matrix}ω & 0 \\ 0 & 1 \end{matrix} ) in the eigenbasis.

Thus W contains 48 (complex) reflections w_{α}^{k} (k = 1, …, 4) of order 5, and its center is isomorphic to Z_{10}. Among irreducible finite reflection groups acting on a 2-dimensional space, only one has these properties, namely the group No. 16 in the list of Shephard–Todd [10]. Its order is 600, and it can be represented as W = Z_{5} ×Ĩ, where Z_{5} is generated by z = ω ·1 and Ĩ is the binary icosahedral group. The above 12 reflections w_{α} are all conjugated, and there are no other reflections in W but w_{α}^{k}.

The binary icosahedral group is examined in detail in [7]. In the basis u_{1}, u_{2}, Ĩ is generated by the matrices (see remark 3.1 below)

c = ( \begin{matrix}ω & 0 \\ 0 & ω^{−1} \end{matrix} ) , r = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , and s = \frac{1}{\sqrt{5}} ( \begin{matrix}ω − ω^{−1} & ω^{2} − ω^{−2} \\ ω^{2} − ω^{−2} & ω^{−1} − ω \end{matrix} ) .

Each element w ∈ W can be expressed uniquely as w = ±z^{a}c^{b}r^{i}s^{j}c^{k}r^{l}, where a, b, k = 0, …, 4; i, j, l = 0, 1; and i = 1 ⟹ j = 0 ⟹ k = l = 0.

Let z_{1}, z_{2} be the coordinates on 𝔠 in the basis u_{1}, u_{2}. Then the basic invariants of Ĩ are

f = z_{1}z_{2}(−z_{1}^{10} + 11z_{1}^{5}z_{2}^{5} + z_{2}^{10})
g = \frac{H(f)}{121} = −z_{1}^{20} − z_{2}^{20} − 228z_{1}^{15}z_{2}^{5} + 228z_{1}^{5}z_{2}^{15} − 494z_{1}^{10}z_{2}^{10}
h = \frac{J(f, g)}{20} = z_{1}^{30} + z_{2}^{30} − 522z_{1}^{25}z_{2}^{5} + 522z_{1}^{5}z_{2}^{25} − 10005z_{1}^{20}z_{2}^{10} − 10005z_{1}^{10}z_{2}^{20}

where H(·) denotes the Hessian and J(·, ·) the Jacobian, and the basic syzygy is 1728f^{5} − g^{3} − h^{2} = 0. The polynomial f is an eigenfunction for z of eigenvalue ω^{−2}, and g, h are z-invariant. It follows that g, h are (algebraically independent) basic invariants for W : 𝔠, which separate W -orbits.

3.5. Mixed elements. The centralizer 𝔷(u) of u ∈ 𝔠 is an 𝔥-regular subalgebra spanned by 𝔥 and by the root subspaces 𝔤_{α} such that α(u) = 0, or equivalently, w_{α}u = u. The set of u ∈ 𝔠 such that 𝔷(u)′ ≠ 0 is a union of 12 lines in 𝔠 composing one projective W -orbit given by the equation f = 0 [7]. By 2.6, the semisimple part u of a mixed vector x = u + e is equivalent to a nonzero vector from one of these lines. Hence the canonical form for u is u = λu_{1}, where λ is determined up to multiplication by a 10-th root of unity.

For such u, 𝔷(u)′ = 𝔄_{2}, 𝔷(u)′_{0} is its Cartan subalgebra, and 𝔷(u)′_{1} is spanned by 5 root vectors corresponding to the extended system of simple roots (of degree 1) of 𝔄_{2} relative to 𝔷(u)′_{0}. These root vectors are just the summands in the expression for u_{2} (3.3). Consider the coordinates on 𝔷(u)′_{1} w. r. t. this basis of root vectors. Nilpotent orbits for the action Z_{0}(u)^{0} : 𝔷(u)′_{1} are given by the condition that coordinates from a certain subset are zero and other coordinates are nonzero.

A pair of monomial transformations of U, V given by the same permutation ( 1 2 3 4 5 ) fixes u_{i} and permutes basic vectors, their roots, and coordinates cyclically. Hence certain Z_{0}(u)^{0}-orbits are glued together under the action of Z_{0}(u). No other orbits are glued together, since the respective sets of roots cannot be transformed into each other via an automorphism of 𝔷(u)′. (Just look at their inner products!)

Remark 3.1. In fact, the discussion in 3.4 determines W only up to conjugation by transposition of u_{1}, u_{2}. To make the correct choice, one should determine the set of u = λ_{1}u_{1} + λ_{2}u_{2} ∈ 𝔠 such that 𝔷(u)_{0} ≠ 0 explicitly. Consider the metabelian Lie algebra L = U^{∗} ⊕ V with the structure tensor u. Then 𝔷(u)_{0} is identified with the subalgebra of Der L∩𝔰𝔩(L) leaving U^{∗} stable. These derivations are determined by their restriction to U^{∗}, and the defining relations of L impose 25 linear equations on 25 matrix entries of a derivation restricted to U^{∗}. This linear system splits into five 5 × 5 subsystems. Four of them have determinant −λ_{1}^{10} + 11λ_{1}^{5}λ_{2}^{5} + λ_{2}^{10}, and the 5-th one (involving diagonal entries) has corank 5 iff λ_{1}λ_{2} = 0, and has corank 1 (and no nonzero traceless solutions) otherwise. Thence 𝔷(u)_{0} ≠ 0 iff λ_{1}λ_{2}(−λ_{1}^{10} + 11λ_{1}^{5}λ_{2}^{5} + λ_{2}^{10}) = 0, and our choice of W in 3.4 is correct.

3.6. Nilpotent elements. Consider the covering loop algebra 𝔊 of 𝔤. The mapping α ↦ (ᾱ, deg α) is an embedding Q ↪ ℤΔ×ℤ. Under this identification, the extended system of simple roots of 𝔤 gives rise to a base (_{i}ε − _{i+1}ε, 0), (ε_{i} − ε_{i+1}, 0) (i = 1, …, 4), (_{4}ε + _{5}ε + ε_{5}, 1) of Q. As explained in 2.5, each semisimple ℤ-graded T -regular subalgebra 𝔰 ⊂ 𝔤 (given by its system of simple roots Π(𝔰) ⊂ Δ) is canonically embedded in 𝔊 (so that Π(𝔰)_{k} ↪ Δ_{k} × {k}), and 𝔰 is complete iff its root lattice Q(𝔰) = ℤΠ(𝔰) ⊂ Q is saturated.

There are 61 types of complete subalgebras of 𝔊 (given by types of various subsystems of Π). For some of these types, there are several (but no more than 11) locally flat algebras, and some of these algebras admit several (but no more than 4) non-conjugated embeddings in 𝔊. Summing up, we obtain 138 locally flat subalgebras of 𝔊. Some of them are not complete. Excluding them, we obtain, up to conjugacy, 105 supports of nilpotent elements, hence 106 nilpotent orbits of G_{0} in 𝔤_{1} (including 0).

Let us give an example of computation. Consider the locally flat algebra of type D_{4}(a_{1}). Its Dynkin diagram is given by the following picture, where the simple roots of degree 1 are indicated by white nodes, and the simple root of degree 0 is blackened:

[figure: β_{1}; β_{2}; β_{3}; β_{4}]

An embedding of its system of simple roots in Δ is given by 3 pairwise orthogonal roots of degree 1, and one root of degree 0, whose inner product with these three equals 1. Up to a permutation of indices, there exists a unique such subset in Δ, namely Π(𝔰) = { β_{1} = _{1}ε+_{5}ε+ε_{3}, β_{2} = _{2}ε−_{1}ε, β_{3} = _{1}ε+_{4}ε+ε_{2}, β_{4} = _{1}ε+_{3}ε+ε_{1} }. The coordinates of β_{i}, considered as elements of Q, in the base Π are given by the rows of the following matrix:

(\begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix} )

The invariant factors of this matrix are equal to 1. Hence the sublattice Q(𝔰) = ℤΠ(𝔰) ⊂ Q is saturated and the respective regular locally flat subalgebra 𝔰 ⊂ 𝔤 is complete.

The sum of root vectors corresponding to the roots β_{1}, β_{2} + β_{3}, β_{2} + β_{4}, β_{4} is a nilpotent element from the dense orbit of the θ-representation for the algebra of type D_{4}(a_{1}). Hence 𝔰 is a support of e = ( 153 242 231 131 ) (in the notation of 1.5).

The characteristic h = (diag(x_{1}, …, x_{5}), diag(y_{1}, …, y_{5})) of e is given by the solution of the linear system

\{ \begin{matrix}x_{2} − x_{1} & = & 0 \\ x_{1} + x_{5} + y_{3} & = & 2 \\ x_{1} + x_{4} + y_{2} & = & 2 \\ x_{1} + x_{3} + y_{1} & = & 2 \\ x_{1} + … + x_{5} & = & 0 \\ y_{1} + … + y_{5} & = & 0 \end{matrix}

orthogonal to all solutions of the corresponding homogeneous system. One finds x_{1} = x_{2} = \frac{6}{5}, x_{3} = x_{4} = x_{5} = −\frac{4}{5}, y_{1} = y_{2} = y_{3} = \frac{8}{5}, y_{4} = y_{5} = −\frac{12}{5} . Hence h lies in the positive Weyl chamber, and its indices are ( 0 2 0 0 0 0 4 0 ). (If x_{1}, …, x_{5} and/or y_{1}, …, y_{5} were not in decreasing order, then we would apply an appropriate pair of permutations of 1, …, 5 to Π(𝔰), e, and h, so as to put them in order.)

Finally, we obtain information on the stabilizer of e. All eigenspaces 𝔤_{i,k}(h) (i = 0, 1; k ≥ 0) are zero, except the following cases:

𝔤_{0,0}(h) = 𝔷_{0}(h) = \{ (( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) , ( \begin{matrix}C & 0 \\ 0 & D \end{matrix} )) | A, D ∈ M_{2}(ℂ);
B, C ∈ M_{3}(ℂ); tr A + tr B = tr C + tr D = 0 }
𝔤_{0,2}(h) = \{ (( \begin{matrix}0 & X \\ 0 & 0 \end{matrix} ) , 0 ) | X ∈ M_{2,3}(ℂ) \}
𝔤_{0,4}(h) = \{ ( 0 , ( \begin{matrix}0 & Y \\ 0 & 0 \end{matrix} )) | Y ∈ M_{3,2}(ℂ) \}
𝔤_{1,2}(h) = ⟨_{ij}e_{k} | i = 1, 2; j = 3, 4, 5; k = 1, 2, 3 ⟩
𝔤_{1,4}(h) = ⟨_{12}e_{k} | k = 1, 2, 3 ⟩

Hence the dimension formula (3) yields

dim 𝔷_{0}(e) = dim 𝔤_{0,0}(h) − dim 𝔤_{1,2}(h) + dim 𝔤_{0,2}(h) − dim 𝔤_{1,4}(h) + dim 𝔤_{0,4}(h)
= 24 − 18 + 6 − 3 + 6 = 15

The Levi part of 𝔷_{0}(e) is the stabilizer subalgebra of general position for the representation of 𝔤_{0}(h) = 𝔤_{0,0}(h) in 𝔤_{1}(h) = 𝔤_{1,2}(h). This is the natural representation of 𝔰𝔩_{2}(ℂ) ⊕ 𝔰𝔩_{3}(ℂ) ⊕ 𝔰𝔩_{3}(ℂ) ⊕ 𝔰𝔩_{2}(ℂ) ⊕ ℂ^{2} in ℂ^{2} ⊗ ℂ^{3} ⊗ ℂ^{3}, where the last 𝔰𝔩_{2}(ℂ) acts trivially and the torus ℂ^{2} acts by homotheties via the weight (5, 2). The stabilizer of general position of the semisimple part of 𝔤_{0}(h) has the type A_{1} + T_{2} and is smaller than the projective stabilizer. Hence the whole stabilizer has the type A_{1} + T_{2} + T_{1} = A_{1} + T_{3}.

4. Metabelian algebras of signature (6, 3)

In this section, we classify the orbits of the θ-group ⋀^{2}SL_{6}(ℂ) ⊗ SL_{3}(ℂ) or, equivalently, metabelian Lie algebras of signature (m, n), m ≤ 6, n ≤ 3.

4.1. Formulation of results. We retain the notation of 1.5 with m = 6, n = 3. The classification of tensors in ⋀^{2}U ⊗ V under the action of SL(U) × SL(V) is performed along the same lines as in 3.1. Here all tensors fall into 7 families according to the type of the stabilizer of their semisimple parts. The canonical forms for semisimple parts are given below, and the canonical forms for nilpotent parts are listed in tables 3–8. The notation in tables 3–8 is similar to that of tables 1–2, with two additional remarks. First, the canonical form for nilpotent parts in the 5-th family is represented in two ways, see 4.6.4 for details. Secondly, the characteristics of nilpotent parts in the 6-th family are indicated by 3 numerical indices (i. e., values of simple roots of 𝔷(u)′_{0}) and values of 3 linearly dependent weights generating the dual to the center of 𝔷(u)′_{0}, see 4.6.5 for details.

Here is the classification.

Family 1. This family contains only semisimple tensors. The canonical form is

u = λ_{1}u_{1} + λ_{2}u_{2} + λ_{3}u_{3},  (λ_{1}^{3} − λ_{2}^{3})(λ_{1}^{3} − λ_{3}^{3})(λ_{2}^{3} − λ_{3}^{3}) ≠ 0,
λ_{1}λ_{2}λ_{3}((λ_{1}^{3} + λ_{2}^{3} + λ_{3}^{3})^{3} − (3λ_{1}λ_{2}λ_{3})^{3}) ≠ 0,

where

u_{1} = _{12}e_{1} +_{34}e_{2} +_{56}e_{3},
u_{2} = _{54}e_{1} +_{16}e_{2} +_{32}e_{3},
u_{3} = _{36}e_{1} +_{52}e_{2} +_{14}e_{3}.

The coefficients λ_{i} are determined up to the action of a certain finite group W described in 4.4. Two canonical forms are equivalent iff they have the same values of

λ_{1}^{6} + λ_{2}^{6} + λ_{3}^{6} − 10(λ_{1}^{3}λ_{2}^{3} + λ_{1}^{3}λ_{3}^{3} + λ_{1}^{3}λ_{3}^{3}),
(λ_{1}^{3} + λ_{2}^{3} + λ_{3}^{3})((λ_{1}^{3} + λ_{2}^{3} + λ_{3}^{3})^{3} + (6λ_{1}λ_{2}λ_{3})^{3}), and
(λ_{1}^{3} − λ_{2}^{3})^{2}(λ_{1}^{3} − λ_{3}^{3})^{2}(λ_{2}^{3} − λ_{3}^{3})^{2}.

The stabilizer subalgebra of u is a one-dimensional torus.

Family 2. The canonical form of a semisimple part is

u = λu_{2} + μu_{3}, λ, μ, λ^{3} ± μ^{3} ≠ 0,

where λ, μ are determined up a permutation, simultaneous sign change and multiplication by two independent cubic roots of unity.

Family 3. The canonical form of a semisimple part is

u = λu_{1} + μ(u_{2} + u_{3}), λ, μ, λ^{3} − μ^{3}, λ^{3} + 8μ^{3} ≠ 0,

where λ, μ are determined up the action of a certain group of order 72 (group No. 5 in the list of Shephard–Todd [10]). Two canonical forms are equivalent iff they have the same values of

λ^{6} − 20λ^{3}μ^{3} − 8μ^{6}  and
(λ^{3} + 2μ^{3})((λ^{3} + 2μ^{3})^{3} + 108λ^{3}μ^{6}).

146 Galitski, Timashev

Table 3: Nilpotent parts of elements of family 2

No. Canonical form Support Characteristic Stabilizer
dim Type
1 121 342  A_{2}  2 2  1 T_{1}
2 121  A_{1}  2 0  2 T_{2}
3 0  3 T_{3}

Table 4: Nilpotent parts of elements of family 3

No. Canonical form Support Characteristic Stabilizer
dim Type
1 531 152 313 A_{1} 2 1 0
2 0  3 A_{1}

Family 4. The canonical form of a semisimple part is

u = λ(u_{2} + u_{3}), λ ≠ 0,

where λ is determined up to multiplication by a 6-th root of unity.

Table 5: Nilpotent parts of elements of family 4

No. Canonical form Support Characteristic Stabilizer
dim Type
1 121 342 531 152 313 A_{2} + A_{1}  2 2 2  1  0
2 121 531 152 313  2A_{1}  2 0 2  2  T_{1}
3 531 152 313  A_{1}  0 0 2  3  T_{2}
4 121 342  A_{2}  2 2 0  3  A_{1}
5 121  A_{1}  2 0 0  4 A_{1} + T_{1}
6 0  5 A_{1} + T_{2}

Family 5. The canonical form of a semisimple part is

u = λ(u_{3} − u_{2}), λ ≠ 0,

where λ is determined up to multiplication by a 6-th root of unity. Family 6. The canonical form of a semisimple part is

u = λu_{1}, λ ≠ 0,

where λ is determined up to multiplication by a 6-th root of unity. Family 7. This family consists of nilpotent elements.

Galitski, Timashev 147

Table 6: Nilpotent parts of elements of family 5

No.  Canonical form  Support Characteristic  Stabilizer
in S^{3}Q  in ⋀^{2}U ⊗ V  dim Type
1 \frac{1}{2}x^{2}z + \frac{1}{6}y^{3} 123 161 521 342  G_{2}  2 4  1  T_{1}
2 \frac{1}{2}x^{2}z + \frac{1}{2}xy^{2} 123 161 521 142 322 341  A_{2}  2 2  2  T_{1}
3 \frac{1}{6}x^{3} + \frac{1}{6}y^{3} 121 342  G_{2}(f)  0 2  3  T_{1}
4  \frac{1}{2}x^{2}y  122 141 321  A_{1}  1 1  4  T_{2}
5  \frac{1}{6}x^{3}  121  A_{1}  1 0  6 A_{1} + T_{1}
6  0  0  9 A_{2} + T_{1}

Table 7: Nilpotent parts of elements of family 6

No. Canonical form Support Characteristic Stabilizer
dim Type
1 143 162 233 252 351  A_{5}  6 6 6 8 − 4 − 4  1  0
2 143 152 233 361  A_{4}  6 6 0 2 2 − 4  2  T_{1}
3 143 152 361 451  A_{3} + A_{1} 6 0 0 − 4 2 2  2  T_{1}
4 133 152 361 451  2A_{2}  4 2 2 − 4 2 2  3  T_{1}
5 133 162 243 252  2A_{2}  0 0 0 8 − 4 − 4  3  A_{1}
6 143 162 351  A_{3}  6 1 1 − 2 1 1  3  T_{2}
7 133 152 361  A_{2} + A_{1} 3 3 0 − 1 − 1 2  4  T_{2}
8 133 361 451  A_{2} + A_{1} 2 1 1 − 4 5 − 1  4  T_{2}
9 152 361 451  A_{2} + A_{1} 2 1 1 − 4 − 1 5  4  T_{2}
10 133 152 351  3A_{1}  2 2 2 0 0 0  5  T_{2}
11 152 361  A_{2}  2 2 0 − 2 − 2 4  5  T_{3}
12 133 152  2A_{1}  2 1 1 2 − 1 − 1  6  T_{3}
13 133 243  2A_{1}  0 0 0 2 2 − 4  7 2A_{1} + T_{1}
14 133  A_{1}  1 1 0 1 1 − 2  8 A_{1} + T_{3}
15 0  11 3A_{1} + T_{2}

Table 8: Elements of family 7

No.  Canonical form  Signature  Support  Characteristic  Stabilizer
dim  Type
1 143 162 233 252 261 342 351  (6, 3)  E_{7}  6 6 6 6 6 6 12  1  0
2 153 162 233 242 252 261 341  (6, 3)  E_{7}(a_{1})  6 6 6 0 6 6 6  2  0
3 133 152 161 243 252 342 351  (6, 3)  E_{7}(a_{2})  0 6 0 6 6 6 6  2  0
4 143 161 233 242 251 341 352  (6, 3)  E_{7}(b)  6 0 6 0 6 0 6  3  0
5 143 152 233 261 342 351  (6, 3)  E_{6}  2 4 2 6 4 6 6  3  T_{1}
6 123 162 252 342 361 451  (6, 3)  E_{6}  0 6 0 6 0 6 12  3  A_{1}
7 133 142 153 161 243 252 341  (6, 3)  E_{7}(c_{1})  0 6 0 0 6 6 0  4  0
8 143 151 233 262 341 342  (6, 3)  E_{6}(a_{1})  0 6 0 6 0 0 6  4  T_{1}
9 133 152 242 261 342 451  (6, 3)  E_{6}(a_{1})  4 0 2 4 2 6 6  4  T_{1}
10 153 162 233 242 251 341  (6, 3)  D_{6}  6 1 5 1 5 1 5  4  T_{1}
11 123 141 152 242 261 351 362  (6, 3)  E_{7}(c_{2})  0 0 6 0 0 0 6  5  0
12 133 152 242 261 351 451  (6, 3)  D_{6}(a_{1})  1 5 0 1 5 6 1  5  T_{1}
13 143 152 161 233 242 351  (6, 3)  A_{6}  2 2 2 2 4 4 2  5  T_{1}
14 143 152 233 242 261 341  (6, 3)  D_{5} + A_{1}  1 5 1 1 4 5 1  5  T_{1}

148 Galitski, Timashev

Table 8: (Continued)

No.  Canonical form  Signature  Support  Characteristic  Stabilizer
dim  Type
15 143 152 161 233 242 251 341  (6, 3)  A_{5} + A_{2}  2 2 2 2 2 2 2  6  0
16 123 161 241 252 342 351  (6, 3)  E_{6}(b)  2 0 4 0 2 0 6  6  T_{1}
17 133 152 233 242 261 341  (6, 3)  D_{6}(a_{2})  0 1 5 0 1 1 5  6  T_{1}
18 142 153 232 261 341  (6, 3)  D_{5}  2 4 2 0 4 6 0  6  T_{2}
19 133 152 161 242 251 351  (6, 3)  D_{5}(a_{1}) + A_{1}  2 0 2 2 2 2 4  7  T_{1}
20 152 161 233 242 251 341  (6, 3)  A_{5} + A_{1}  1 1 4 1 1 1 4  7  T_{1}
21 122 133 151 243 261 342  (6, 3)  E_{6}(b)  0 0 0 6 0 6 0  7  A_{1}
22 133 162 242 251 341  (6, 3)  A_{5}  2 1 3 1 1 1 5  7  T_{2}
23 123 132 242 341 561  (6, 3)  D_{5}  0 6 0 0 0 6 6  7  2A_{1}
24 143 152 161 232 251 341  (6, 3)  A_{4} + A_{2}  2 2 0 2 2 2 2  8  T_{1}
25 143 152 161 233 241 342  (6, 3)  A_{5} + A_{1}  3 0 3 3 0 0 3  8  A_{1}
26 131 153 162 233 242 261  (6, 3)  A_{4} + A_{2}  0 6 0 0 0 0 0  8  A_{1}
27 121 153 233 252 342 451  (5, 3)  A_{3} + A_{2} + A_{1}  0 0 0 0 6 0 0  8  A_{1}
28 133 161 242 251 351  (6, 3)  D_{5}(a_{1})  1 0 2 3 1 3 3  8  T_{2}
29 123 161 162 252 341  (6, 3)  D_{5}(a_{1})  0 3 0 3 0 0 6  8  A_{1} + T_{1}
30 123 132 242 361 451  (6, 3)  A_{5}  0 1 0 5 0 6 1  8  A_{1} + T_{1}
31 123 152 232 261 341  (6, 3)  A_{4} + A_{1}  1 2 1 0 2 3 3  9  T_{2}
32 143 152 161 232 251  (6, 3)  A_{4} + A_{1}  1 4 0 1 1 1 1  9  T_{2}
33 143 151 232 261 341  (6, 3)  A_{4} + A_{1}  0 3 0 3 0 3 0  9  T_{2}
34 141 152 233 242 351  (5, 3)  A_{3} + A_{2}  0 0 1 0 5 0 1  9  T_{2}
35 152 161 233 241 342  (6, 3)  A_{5}  2 0 4 2 0 0 4  9  A_{1} + T_{1}
36 123 152 161 251 342  (6, 3)  D_{4} + A_{1}  1 2 0 3 1 1 5  9  A_{1} + T_{1}
37 123 152 161 232 251 341  (6, 3)  A_{3} + A_{2} + A_{1}  2 0 2 0 2 2 2  10  T_{1}
38 143 151 232 243 341  (5, 3)  D_{4}(a_{1}) + A_{1}  0 1 0 1 4 1 0  10  T_{2}
39 143 151 232 261  (6, 3)  A_{4}  0 4 0 2 0 2 0  10  T_{3}
40 133 152 161 232 241  (6, 3)  A_{3} + A_{2}  2 1 1 1 1 1 2  11  T_{2}
41 123 152 232 251 341  (5, 3)  A_{3} + 2A_{1}  1 0 1 0 4 1 1  11  T_{2}
42 131 153 242 253  (5, 3)  D_{4}(a_{1})  0 2 0 0 4 0 0  11  T_{3}
43 123 152 261 341  (6, 3)  A_{4}  2 2 0 0 2 4 2  11  A_{1} + T_{2}
44 123 151 261 342  (6, 3)  D_{4}  0 2 0 4 0 2 4  11  2A_{1} + T_{1}
45 121 342 561 562  (6, 2)  D_{4}(a_{1})  0 0 0 0 0 0 6  11  3A_{1}
46 123 142 151 232 261 341  (6, 3)  3A_{2}  0 2 0 2 0 2 2  12  A_{1}
47 133 152 232 241  (5, 3)  A_{3} + A_{1}  1 1 0 1 3 0 1  12  T_{3}
48 142 151 232 461  (6, 2)  A_{3} + A_{1}  0 0 0 1 0 1 5  12  2A_{1} + T_{1}
49 133 142 151 232 241  (5, 3)  2A_{2} + A_{1}  1 0 1 1 2 1 1  13  T_{2}
50 142 151 232 261 341  (6, 2)  2A_{2} + A_{1}  0 1 0 1 0 1 4  13  A_{1} + T_{1}
51 122 153 161 251 341  (6, 3)  A_{3} + 2A_{1}  3 0 0 0 3 3 0  13  2A_{1}
52 123 132 161 241 351  (6, 3)  2A_{2} + A_{1}  1 0 2 0 1 3 0  14  A_{1} + T_{1}
53 123 131 162 242 251  (6, 3)  2A_{2} + A_{1}  0 3 0 0 0 0 3  14  2A_{1}
54 142 153 161 231  (6, 3)  A_{3} + A_{1}  3 0 1 0 2 2 0  14  2A_{1} + T_{1}
55 122 153 251 341  (5, 3)  A_{3} + A_{1}  2 0 0 0 4 2 0  14  2A_{1} + T_{1}
56 123 131 152 242 251  (5, 3)  A_{2} + 3A_{1}  0 2 0 0 2 0 2  15  A_{1} + T_{1}
57 123 141 232 351  (5, 3)  2A_{2}  0 0 2 0 2 2 0  15  A_{1} + T_{2}
58 131 162 242 251  (6, 2)  2A_{2}  0 2 0 0 0 0 4  15  2A_{1} + T_{1}
59 142 153 231  (5, 3)  A_{3}  2 0 1 0 3 1 0  15  2A_{1} + T_{2}
60 123 142 151 231  (5, 3)  A_{2} + 2A_{1}  1 1 0 1 1 1 1  16  T_{3}
61 131 152 242 251  (5, 2)  A_{2} + 2A_{1}  0 1 0 0 2 0 3  16  A_{1} + T_{2}
62 122 341 561  (6, 2)  A_{3}  0 1 0 0 0 2 4  16 C_{2} + A_{1} + T_{1}
63 122 161 251 341  (6, 2)  A_{2} + 2A_{1}  0 1 0 1 0 2 2  17  2A_{1} + T_{1}
64 121 133 243 342  (4, 3)  A_{2} + 2A_{1}  0 0 0 3 0 0 0  17  3A_{1}
65 142 151 231  (5, 2)  A_{2} + A_{1}  1 0 0 1 1 1 2  18  A_{1} + T_{3}
66 123 131 242  (4, 3)  A_{2} + A_{1}  0 1 0 2 0 0 1  18  A_{1} + T_{3}
67 122 133 141 231  (4, 3)  4A_{1}  1 0 1 1 0 1 0  20  2A_{1} + T_{1}
68 121 342  (4, 2)  A_{2}  0 0 0 2 0 0 2  21  3A_{1} + T_{2}
69 122 131 241  (4, 2)  3A_{1}  0 1 0 1 0 1 1  22  2A_{1} + T_{2}
70 121 132 143  (4, 3)  3A_{1}  2 0 0 1 0 0 0  23 A_{2} + A_{1} + T_{1}
71 121 132 233  (3, 3)  3A_{1}  0 0 2 0 0 0 0  25  2A_{2}
72 121 341 561  (6, 1)  3A_{1}  0 0 0 0 0 3 0  26  C_{3} + A_{1}
73 121 132  (3, 2)  2A_{1}  1 0 1 0 0 0 1  26 A_{2} + A_{1} + T_{2}
74 121 341  (4, 1)  2A_{1}  0 0 0 1 0 2 0  27 C_{2} + 2A_{1} + T_{1}
75 121  (2, 1)  A_{1}  0 1 0 0 0 1 0  32 A_{3} + 2A_{1} + T_{1}
76 0  43  A_{5} + A_{2}

4.2. The Z_{3}-graded algebra E_{7}. Let 𝔤 be a simple Lie algebra of the type E_{7}. We define its Z_{3}-grading and the respective θ-representation by the following labelling of the affine Dynkin diagram E_{7}^{(1)} (cf. 2.4):

[figure: 1; 1; 1]

(We omit zero labels and blacken the unique simple root of positive degree. The upright digit indicates the degree of this root, and italic digits indicate numerical labels of the highest weight of the θ-representation.)

The precise description of the Z_{3}-graded algebra 𝔤 may be obtained as follows. We consider a direct sum

𝔤 = (⋀^{2}U^{∗} ⊗ V^{∗}) ⊕ (𝔰𝔩(U) ⊕ 𝔰𝔩(V)) ⊕ (⋀^{2}U ⊗ V)

and introduce a Lie algebra structure on it so that the above direct sum is a Z_{3}-grading

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}.

The commutator maps 𝔤_{i} × 𝔤_{j} → 𝔤_{i+j} are determined as in 3.2. Here are the precise formulas:

[α^{∗} ⊗ v^{∗}, α ⊗ v] = (⟨α^{∗}, α⟩/6 · 1 − C(α^{∗} ⊗ α)) · ⟨v^{∗}, v⟩
⊕ ⟨α^{∗}, α⟩/2 · (⟨v^{∗}, v⟩/3 · 1 − v^{∗} ⊗ v)
[u_{1} ∧ u_{2} ⊗ v_{1}, u_{3} ∧ u_{4} ⊗ v_{2}] = 2ι(u_{1} ∧ u_{2} ∧ u_{3} ∧ u_{4}) ⊗ ι(v_{1} ∧ v_{2})
[u_{1}^{∗} ∧ u_{2}^{∗} ⊗ v_{1}^{∗}, u_{3}^{∗} ∧ u_{4}^{∗} ⊗ v_{2}^{∗}] = −2ι(u_{1}^{∗} ∧ u_{2}^{∗} ∧ u_{3}^{∗} ∧ u_{4}^{∗}) ⊗ ι(v_{1}^{∗} ∧ v_{2}^{∗})

(where α ∈ ⋀^{2}U; α^{∗} ∈ ⋀^{2}U^{∗}; u_{i} ∈ U; v, v_{i} ∈ V ; u^{∗}, u_{i}^{∗} ∈ U^{∗}; v^{∗}, v_{i}^{∗} ∈ V^{∗}).

Let us describe the root system Δ of 𝔤 relative to 𝔱. Denote by Δ_{k} the set of nonzero weights of 𝔱 in 𝔤_{k}. We have Δ_{0} = {_{i}ε−_{j}ε, ε_{i} −ε_{j}}, Δ_{1} = {_{i}ε+_{j}ε+ε_{k}}, Δ_{−1} = {−_{i}ε −_{j}ε − ε_{k}} (i ≠ j).

The Weyl group W_{0} permutes _{i}ε and ε_{i} arbitrarily. We choose _{i}ε −_{i+1}ε (i = 1, …, 5), ε_{j} −ε_{j+1} (j = 1, 2) as simple roots of 𝔤_{0}. Together with _{5}ε+_{6}ε+ε_{3}, they compose an extended system of simple roots of 𝔤.

An invariant inner product on 𝔱^{∗} can be determined by the formulas (_{i}ε,_{j}ε) = −\frac{1}{6}, (_{i}ε,_{i}ε) = \frac{5}{6}, (ε_{s}, ε_{t}) = −\frac{1}{3}, (ε_{s}, ε_{s}) = \frac{2}{3}, (_{i}ε, ε_{s}) = 0 (i ≠ j, s ≠ t). In particular, we have (∑_{i}x_{i}ε+∑ x_{s}ε_{s}, ∑_{i}y_{i}ε+∑ y_{s}ε_{s}) = ∑_{i}x_{i}y+∑ x_{s}y_{s} whenever ∑_{i}x = ∑ x_{s} = 0. The inner products of roots of degree 1 can be computed by formula (4).

4.3. A Cartan subspace. Consider regular Z_{3}-graded subalgebras of type A_{2} in 𝔤 such that their degree 0 component is a Cartan subalgebra, and simple roots and the lowest root have degree 1. They are classified by embeddings of the extended system of simple roots A_{2}^{(1)} in Δ_{1} preserving the inner product. Up to W_{0}-conjugacy, there is a unique such subalgebra, say, the algebra 𝔄_{1} generated by root vectors _{12}e_{1},_{34}e_{2},_{56}e_{3}. Consider also an algebra 𝔄_{2} generated by 54e_{1},_{16}e_{2},_{32}e_{3}, and 𝔄_{3} generated by _{36}e_{1},_{52}e_{2},_{14}e_{3}. Making use of (4), it is easy to see that (𝔄_{1}, 𝔄_{2}, 𝔄_{3}) is the unique, up to W_{0}-conjugacy, triple of subalgebras of the above type commuting with each other. It follows that the elements u_{1} = _{12}e_{1} +_{34}e_{2} +_{56}e_{3}, u_{2} = _{54}e_{1} +_{16}e_{2} +_{32}e_{3}, u_{3} = _{36}e_{1} +_{52}e_{2} +_{14}e_{3} are semisimple and commute with each other. Hence 𝔠 = ⟨u_{1}, u_{2}, u_{3}⟩ is an abelian subspace in 𝔤_{1} consisting of semisimple elements. It can be included in a Z_{3}-graded Cartan subalgebra 𝔥 = 𝔠^{∗} ⊕ ⟨c⟩ ⊕ 𝔠 of 𝔤, where c = (diag(1, −1, 1, −1, 1, −1), 0) ∈ 𝔤_{0}, and 𝔠^{∗} = ⟨u^{1}, u^{2}, u^{3}⟩ ⊂ 𝔤_{−1} is generated by semisimple elements u^{i} ∈ 𝔄_{i} obtained from u_{i} by lifting the indices. (The u^{i} are sums of root vectors of degree −1 in 𝔄_{i}.) The same argument as in 3.3 shows that 𝔠 is a Cartan subspace.

4.4. The Weyl group. In order to find the Weyl group W associated with 𝔠, we first collect sufficiently many linear transformations of 𝔠 induced by elements of G_{0} normalizing 𝔠. It is easy to pick elements from G_{0}, diagonal in the bases je, e_{j}, that multiply u_{k} by arbitrary and independent cubic roots of unity. The permutation ( 1 2 )( 3 4 )( 5 6 ) of basic vectors _{j}e combined with the multiplication by i induces a linear transformation of 𝔠 transposing u_{2}, u_{3} and leaving u_{1} fixed. Other transpositions of u_{k} can be obtained in the same way. It follows that W contains a subgroup G(3, 1, 3) of monomial transformations in the basis u_{k}, whose nonzero matrix entries are cubic roots of unity. In particular, W is irreducible. Also note that a pair of monomial transformations of U and V given by permutations ( 1 2 )( 5 4 )( 3 6 ) and ( 2 3 ) of basic vectors combined with the multiplication by i and −1, respectively, yields −1 ∈ W.

Consider the root system of 𝔤 relative to 𝔥. The automorphism θ acts naturally on 𝔥^{∗} and permutes the roots. It is represented by an integral matrix in a base of simple roots and has eigenvalues 1, ω, ω^{−1}, where ω = e^{\frac{2πi}{3}} . Moreover, the restriction of θ to each invariant subspace of 𝔥^{∗} generated by roots has an integral matrix, too, whence eigenvalues ω, ω^{−1} occur with equal multiplicities.

The root system is divided in two subclasses. The first one consists of roots α such that α(c) = 0. Since dim 𝔷(c) = 79, there are 72 such roots. The second subclass consists of remaining 54 roots β, which satisfy β(c) ≠ 0. Each root α in the first subclass has nonzero projections on θ-eigenspaces of eigenvalues ω^{±1}, and α + θα + θ^{2}α = 0. It follows that rk{α, θα, θ^{2}α} = 2 and {α, θα, θ^{2}α} is the extended system of simple roots of type A_{2}^{(1)} of a certain 𝔥-regular θ-invariant simple subalgebra 𝔄(α), whose 6 roots are ±θ^{k}α (k = 0, 1, 2).

The 72 roots of the first subclass fall into 12 such subsystems of type A_{2}, hence we obtain 12 subalgebras 𝔄(α) of type A_{2}, including 𝔄_{1}, 𝔄_{2}, 𝔄_{3}. We have a decomposition 𝔥 = 𝔥(α) ⊕ 𝔥^{α}, where 𝔥(α) is a Cartan subalgebra in 𝔄(α) and 𝔥^{α} is the centralizer of 𝔄(α) in 𝔥. (Note that c ∈ 𝔥^{α}.) The same reasoning as in 3.4 shows that there exists an element g_{α} ∈ G_{0} such that Ad g_{α}|_{𝔥} = θ|_{𝔥(α)} ⊕ 1|_{𝔥^{α}}, and w_{α} = Ad g_{α}|_{𝔠} ∈ W has eigenspaces 𝔥(α)_{1} = 𝔠 ∩ 𝔥(α) and 𝔥_{1}^{α} = 𝔠 ∩ 𝔥^{α} = Ker α|_{𝔠} and is represented by the matrix

( \begin{matrix}ω & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

in the eigenbasis.

Thus W contains 24 reflections w_{α}^{±1} of order 3, and −1. Among irreducible finite reflection groups acting on a 3-dimensional space, only one has these properties, namely the group No. 26 in the list of Shephard–Todd. The order of W is 1296, and it is generated by −1, the subgroup G(3, 1, 3), and a 3-fold reflection

s = \frac{−i}{\sqrt{3}} ( \begin{matrix}ω & ω^{2} & ω^{2} \\ ω^{2} & ω & ω^{2} \\ ω^{2} & ω^{2} & ω \end{matrix} )

(in the basis u_{k}), see [10]. The above 12 reflections w_{α} are conjugated, and there are no 3-fold reflections in W other than w_{α}^{±1}.

However, W contains in addition 9 reflections of order 2. They can be described as follows. Let β be a root from the second subclass. Suppose that β|_{𝔠^{∗}⊕𝔠} ≠ 0; then β has nonzero projection on each eigenspace of θ. It follows that rk{β, θβ, θ^{2}β} = 3. We claim that θ^{k}β are pairwise orthogonal. Indeed, if their inner products are equal to −1, then rk{β, θβ, θ^{2}β} = 2. If their inner products are equal to 1, then { θ^{k}β −θ^{k+l}β | l = 1, 2 } is a root subsystem of type A_{2} in the first subclass. Since all such root subsystems are conjugated (by elements of G_{0} normalizing 𝔠), each one of them originates from a subsystem {±β, ±θβ, ±θ^{2}β} of the second subclass, as above. Hence 12 subsystems of type A_{2} in the first subclass give rise to at least 12 · 6 = 72 roots in the second subclass, whereas it contains only 54 roots. Our claim is proved. It follows that {±θ^{k}β} is a root system of type 3A_{1}, where θ acts by a cyclic permutation of components.

The 54 roots of the second subclass fall into 9 such subsystems. (If there is a root β such that β|_{𝔠^{∗}⊕𝔠} = 0, then there are exactly two such roots. But this cannot happen, because the number of roots in the second subclass not belonging to these subsystems is divisible by 6.) Hence we obtain 9 subalgebras 𝔅(β) of type 3A_{1}, and θ acts on each 𝔅(β) by a cyclic permutation of its components. As above, we have a decomposition 𝔥 = 𝔥(β)⊕𝔥^{β}, where 𝔥(β) is a Cartan subalgebra in 𝔅(β) and 𝔥^{β} is the centralizer of 𝔅(β) in 𝔥. (Note that c ∈ 𝔥(β).) Let B(β) be the subgroup of G corresponding to 𝔅(β). Take a θ-invariant element g_{β} from the normalizer of 𝔥(β) in B(β) such that Ad g_{β}|_{𝔥(β)} = −1, Ad g_{β}|_{𝔥^{β}} = 1. We have g_{β} ∈ G_{0}, and r_{β} = Ad g_{β}|_{𝔠} ∈ W is a reflection of order two. These 9 reflections r_{β} are conjugated and all 2-fold reflections in W are obtained in this way.

Among 𝔅(β), there is a subalgebra 𝔅_{0} constructed as follows. Let E_{ij} denote (the linear operator on U given in the basis _{i}e by) the (i, j)-th matrix unit. Put

\begin{matrix}e_{0} = & E_{12} + E_{34} + E_{56}, & f_{0} = & E_{21} + E_{43} + E_{65}, & h_{0} = & c, \\ e_{1} = & _{53}e_{1} +_{15}e_{2} +_{31}e_{3}, & f_{1} = & _{46}e_{1} +_{62}e_{2} +_{24}e_{3}, & h_{1} = & u_{3} − u_{2}, \\ e_{−1} = & ^{46}e^{1} +^{62}e^{2} +^{24}e^{3}, & f_{−1} = & ^{53}e^{1} +^{15}e^{2} +^{31}e^{3}, & h_{−1} = & u^{3} − u^{2}. \end{matrix}

Then 𝔅_{0} = ⟨ e_{k}, f_{k}, h_{k} | k = 0, ±1 ⟩ is a graded 𝔥-regular semisimple subalgebra of type 3A_{1}, deg e_{k} = deg f_{k} = deg h_{k} = k ∈ Z_{3}, and the representations (𝔅_{0})_{0} : (𝔅_{0})_{±1} are isomorphic to the adjoint representation of (𝔅_{0})_{0} ≅ 𝔰𝔩_{2}(ℂ) under the identifications e_{0} ↦ e_{±1}, f_{0} ↦ f_{±1}, h_{0} ↦ h_{±1}.

Let z_{1}, z_{2}, z_{3} be the coordinates on 𝔠 in the basis u_{1}, u_{2}, u_{3}. The basic projective invariants of W are

f = z_{1}^{6} + z_{2}^{6} + z_{3}^{6} − 10(z_{1}^{3}z_{2}^{3} + z_{1}^{3}z_{3}^{3} + z_{2}^{3}z_{3}^{3})

g = (z_{1}^{3} + z_{2}^{3} + z_{3}^{3})(z_{1}^{3} + z_{2}^{3} + z_{3}^{3} + 6z_{1}z_{2}z_{3})
× (z_{1}^{3} + z_{2}^{3} + z_{3}^{3} + 6ωz_{1}z_{2}z_{3})(z_{1}^{3} + z_{2}^{3} + z_{3}^{3} + 6ω^{2}z_{1}z_{2}z_{3})
p = (z_{1}^{3} − z_{2}^{3})(z_{1}^{3} − z_{3}^{3})(z_{2}^{3} − z_{3}^{3})
q = z_{1}z_{2}z_{3}(z_{1} + z_{2} + z_{3})(z_{1} + ωz_{2} + ω^{2}z_{3})(z_{1} + ω^{2}z_{2} + ωz_{3})
× (z_{1} + z_{2} + ωz_{3})(z_{1} + ωz_{2} + z_{3})(ωz_{1} + z_{2} + z_{3})
× (z_{1} + z_{2} + ω^{2}z_{3})(z_{1} + ω^{2}z_{2} + z_{3})(ω^{2}z_{1} + z_{2} + z_{3})

(see [10]) connected with the basic syzygy

6912q^{3} − 4g^{3} + 186624p^{4} + f^{6} − 6f^{4}g − 864f^{3}p^{2} + 9f^{2}g^{2} + 2592fp^{2}g = 0. (5)

The polynomials p, q are products of mirror functionals of all 2-fold and 3-fold reflections, respectively. It is easily shown that p is invariant under all w_{α}, and each r_{β} multiplies it by −1. Similarly, q is invariant under all r_{β}, and each w_{α} multiplies it by ω^{−1}. It follows that f, g, h = p^{2} are (algebraically independent) basic invariants for W : 𝔠, which separate W -orbits.

4.5. Nilpotent elements. Our next task is to classify nilpotent elements. We proceed as in 3.6 using the methods of 2.5. The root lattice Q of the covering loop algebra 𝔊 is embedded in ℤΔ × ℤ, so that the extended system of simple roots of 𝔤 gives rise to a base Π = { (_{i}ε −_{i+1}ε, 0), (ε_{j} − ε_{j+1}, 0), (_{5}ε +_{6}ε + ε_{3}, 1) | i = 1, …, 5; j = 1, 2 } of Q. There are 37 types of complete subalgebras of 𝔊 (given by types of various subsystems of Π). For some of these types, there are several (but no more than 6) locally flat algebras, and some of these algebras admit several (but no more than 4) non-conjugated embeddings in 𝔊. Summing up, we obtain 93 locally flat subalgebras of 𝔊. Some of them are not complete. Excluding them, we obtain, up to conjugacy, 75 supports of nilpotent elements, hence 76 nilpotent orbits of G_{0} in 𝔤_{1} (including 0).

4.6. Mixed elements. The centralizer 𝔷(u) of u ∈ 𝔠 is an 𝔥-regular subalgebra spanned by 𝔥 and the root subspaces 𝔤_{α} such that α(u) = 0. The latter equality is equivalent to w_{α}u = u or r_{α}u = u if α belongs to the first or second subclass of the root system, respectively. Therefore u is a semisimple part of a mixed element iff u lies on mirror hyperplanes of some reflections from W.

In this subsection, we use the following shorthand terminology. The mirror hyperplane of an s-fold reflection will be called an s-mirror. Linear functionals defining 2- or 3-mirrors in 𝔠 (mirror functionals) are easily obtained by decomposing p and q into linear factors.

The collection of mirrors containing u may belong to one of the following types.

4.6.1. One 3-mirror. The set of such u is determined by the conditions p(u) ≠ 0, q(u) = 0, g(u) ≠ 0. Passing to a conjugate, we may assume that the equation of the mirror is z_{1} = 0 ⟹ 𝔷(u)′ = 𝔄_{1}, 𝔷(u)′_{0} is its Cartan subalgebra, and 𝔷(u)′_{1} is spanned by three summands of u_{1}, which are the root vectors corresponding to the extended system of simple roots (of degree 1) of 𝔄_{1} relative to 𝔷(u)′_{0}. These roots and root vectors are permuted cyclically by a pair of monomial transformations of U, V given by permutations ( 1 3 5 )( 2 4 6 ), ( 1 2 3 ). The respective element of G_{0} fixes u and glues together some nilpotent orbits of the action Z_{0}(u)^{0} : 𝔷(u)′_{1}. Finally, nilpotent orbits for Z_{0}(u) : 𝔷(u)′_{1} are determined by the condition that a certain number of coordinates (in the basis of root vectors) vanish and other coordinates are nonzero.

4.6.2. One 2-mirror. The respective subset of u ∈ 𝔠 is determined by the conditions p(u) = 0, q(u) ≠ 0. We may assume that the equation of the mirror is z_{2} = z_{3} ⟹ 𝔷(u)′ = 𝔅_{0}, 𝔷(u)′_{0} = ⟨e_{0}, f_{0}, h_{0}⟩ ≅ 𝔰𝔩_{2}(ℂ), and 𝔷(u)′_{1} = ⟨e_{1}, f_{1}, h_{1}⟩ is isomorphic to the adjoint representation of 𝔰𝔩_{2}(ℂ). Hence there is a unique nilpotent orbit for the action Z_{0}(u) : 𝔷(u)′_{1}—the orbit of e_{1}.

4.6.3. One 3-mirror and one 2-mirror. The respective subset in 𝔠 is determined by the equations p(u) = 0, q(u) = 0, f^{2}(u) = 4g(u). We may assume that the equations of the mirrors are z_{1} = 0, z_{2} = z_{3} ⟹ 𝔷(u)′ = 𝔄_{1} ⊕ 𝔅_{0}, and nilpotent orbits for the action Z_{0}(u) : 𝔷(u)′_{1} are direct products (or sums, is a reader prefers) of nilpotent orbits of the above two types.

Remark 4.1. If p = q = 0, then the syzygy (5) transforms into (f^{2} − g)^{2}(f^{2} − 4g) = 0, so that either f^{2} = 4g or f^{2} = g. The first possibility has been just considered, and the second one is considered below (4.6.5).

4.6.4. Four 3-mirrors. The respective subset of 𝔠 is determined by the conditions p(u) ≠ 0, q(u) = 0, g(u) = 0. We may assume that the equations of the mirrors are z_{1} = 0, z_{1} + z_{2} + z_{3} = 0, ωz_{1} + z_{2} + z_{3} = 0, ω^{2}z_{1} + z_{2} + z_{3} = 0 ⟹ u = λ(u_{3} −u_{2}). The centralizer 𝔤_{u} = 𝔷(u) is a graded reductive subalgebra of type D_{4} +T_{3} described as follows. First, its center is 𝔷(𝔤_{u}) = ⟨h_{−1}, c, h_{1}⟩. Further, consider a 3-dimensional space Q = ⟨x, y, z⟩ and let ι_{1}, ι_{2}, ι_{3} be three embeddings of Q in U ⊕ V mapping the triple (x, y, z) to (_{1}e,_{3}e,_{5}e), (_{2}e,_{4}e,_{6}e), and (e_{1}, e_{2}, e_{3}), respectively. Then 𝔰𝔩(Q) is embedded diagonally in 𝔰𝔩(U) ⊕ 𝔰𝔩(V) via ι_{1}, ι_{2}, ι_{3}, and S^{3}Q is embedded in ⋀^{2}U ⊗ V by mapping, for q_{1}, q_{2}, q_{3} ∈ Q,

q_{1}q_{2}q_{3} ↦ ∑_{(i_{1},i_{2},i_{3})} ι_{1}(q_{i_{1}}) ∧ ι_{2}(q_{i_{2}}) ⊗ ι_{3}(q_{i_{3}}),

where the sum is taken over all permutations of (1, 2, 3). Similarly, S^{3}Q^{∗} is embedded in ⋀^{2}U^{∗} ⊗V^{∗}. One verifies that 𝔷(u)′ = S^{3}Q^{∗} ⊕𝔰𝔩(Q)⊕S^{3}Q is a graded subalgebra of type D_{4}, whose grading is determined by the following labelling of the affine Dynkin diagram D_{4}^{(3)}:

[figure: 1; <]

It is known (and easy to verify using the methods of 2.5, see [15, 5.3], [14, §5]) that there are 5 nonzero nilpotent orbits for the action (Z(u)′)_{0}^{0} : 𝔷(u)′_{1}, whose characteristics cannot be transformed into each other via an automorphism of 𝔷(u)′_{0}, see table 6. Hence these orbits are not glued together under the action of Z_{0}(u) and are exactly nilpotent orbits for Z_{0}(u) : 𝔷(u)′_{1}.

4.6.5. Two 3-mirrors and three 2-mirrors. The respective subset of 𝔠 is determined by the equations p(u) = 0, q(u) = 0, f^{2}(u) = g(u). We may assume that the equations of the mirrors are z_{2} = 0, z_{3} = 0, z_{2} − z_{3} = 0, z_{2} − ωz_{3} = 0, z_{2} − ω^{2}z_{3} = 0 ⟹ u = λu_{1}. The centralizer 𝔤_{u} = 𝔷(u) is a graded reductive subalgebra of type A_{5} + T_{2} described as follows. For its center, we have 𝔷(𝔤_{u}) = ⟨u^{1}, u_{1}⟩. Let U_{1} = ⟨_{1}e,_{2}e⟩, U_{2} = ⟨_{3}e,_{4}e⟩, U_{3} = ⟨_{5}e,_{6}e⟩. We identify V with ⋀^{2}U_{1}^{∗} ⊕ ⋀^{2}U_{2}^{∗} ⊕ ⋀^{2}U_{3}^{∗} by mapping e_{1} ↦ ^{12}e, e_{2} ↦ ^{34}e, e_{3} ↦ ^{56}e and thus embed 𝔤𝔩(U_{1}) ⊕ 𝔤𝔩(U_{2}) ⊕ 𝔤𝔩(U_{3}) in 𝔤𝔩(U) ⊕ 𝔤𝔩(V). There are also natural embeddings

⨁_{(i,j,k)} U_{i} ⊗ U_{j} ⊗ ⋀^{2}U_{k}^{∗} ↪ ⋀^{2}U ⊗ V and ⨁_{(i,j,k)} U_{i}^{∗} ⊗ U_{j}^{∗} ⊗ ⋀^{2}U_{k} ↪ ⋀^{2}U^{∗} ⊗ V^{∗}

(the direct sums are taken over all cyclic permutations of (1, 2, 3)). One verifies that

𝔷(u)′ = ⨁_{(i,j,k)} U_{i}^{∗} ⊗ U_{j}^{∗} ⊗ ⋀^{2}U_{k} ⊕ ((𝔰𝔩(U) ⊕ 𝔰𝔩(V)) ∩ ⨁ 𝔤𝔩(U_{i}))

⊕ ⨁_{(i,j,k)} U_{i} ⊗ U_{j} ⊗ ⋀^{2}U_{k}^{∗}

is a 𝔱-regular subalgebra of type A_{5}, whose grading is determined by the following labelling of the affine Dynkin diagram A_{5}^{(1)}:

[figure: 1; 1; 1]

We may assume that characteristics of nilpotent orbits for Z_{0}(u)^{0} : 𝔷(u)′_{1} lie in the positive Weyl chamber of 𝔱, and we indicate them by 6 indices, where the first triple is given by the values of the simple roots _{1}ε −_{2}ε, _{3}ε −_{4}ε, _{5}ε −_{6}ε of 𝔷(u)′_{0} and the second triple is composed of the values of the weights _{1}ε +_{2}ε − 2ε_{1}, _{3}ε + _{4}ε − 2ε_{2}, _{5}ε + _{6}ε − 2ε_{3}, which generate the dual to the 2-dimensional center of 𝔷(u)′_{0}.

A pair of monomial transformations of U, V determined by permutations ( 1 3 5 )( 2 4 6 ), ( 1 2 3 ) fixes u and permutes simple components 𝔰𝔩(U_{i}) of 𝔷(u)′_{0} and both triples in each characteristic cyclically. On the contrary, an outer automorphism transposing two simple components of 𝔷(u)′_{0} cannot be induced by an element of Z_{0}(u). (That is why the elements No. 8 and 9 in table 7 are not equivalent.) Indeed, suppose for example that a pair (a, b) ∈ SL(U) × SL(V) fixes u and permutes U_{1} and U_{2} leaving U_{3} stable. Then we must have

\begin{matrix}a(_{12}e) = & λ_{34}e, & a(_{34}e) = & μ_{12}e, & a(_{56}e) = & ν_{56}e, \\ b(e_{1}) = & λ^{−1}e_{2}, & b(e_{2}) = & μ^{−1}e_{1}, & b(e_{3}) = & ν^{−1}e_{3} \end{matrix}

⟹ det a = λμν = 1, det b = −(λμν)^{−1} = 1. A contradiction! It is also clear that an element of Z_{0}(u) normalizing each 𝔰𝔩(U_{i}) leaves each U_{i} stable, whence belongs to Z_{0}(u)^{0}. Therefore two nilpotent orbits for the action Z_{0}(u)^{0} : 𝔷(u)′_{1} are glued together under the action of Z_{0}(u) iff their characteristics are obtained from each other by a cyclic permutation of both triples. This remark allows to complete the classification of nilpotent orbits for the action Z_{0}(u) : 𝔷(u)′_{1}.

References

[1] Elashvili, A. G., Canonical forms and stationary subalgebras of points in general position for simple linear Lie groups, Funk. Anal. i Prilozhen. 6 (1972), 51–62 (Russian), English translation: Funct. Anal. Appl. 6 (1972), 44–53.

[2] —, Stationary subalgebras of points in general position for irreducible linear Lie groups, Funk. Anal. i Prilozhen. 6 (1972) 65–78 (Russian), English translation: Funct. Anal. Appl. 6 (1972), 139–148.

[3] Gantmacher, F. R., “The theory of matrices,” 3-rd edn., Nauka, Moscow, 1967 (Russian), English translation of the 1-st edn.: Chelsea, New York, 1959.

[4] Gauger, M., On the classification of metabelian Lie algebras, Trans. Amer. Math. Soc. 179 (1973), 293–329.

[5]  Kac, V., Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190–213.

[6] —, “Infinite dimensional Lie algebras,” 3-rd edn., Cambridge Univ. Press, Cambridge, 1990.

[7] Klein, F., ,,Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade“, Teubner, Leipzig, 1884.

[8] Maltsev, A. I., On semisimple subgroups of Lie groups, Izvestiya AN SSSR, ser. matem. 8 (1944), 143–174 (Russian).

[9] —, On solvable Lie algebras, Izvestiya AN SSSR, ser. matem. 9 (1945), No. 5, 329–356 (Russian), English translation: Amer. Math. Soc. Transl.

(1) 9 (1962), 228–262. [10] Shephard, G. C., and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.

[11] Vinberg, E. B., On the linear groups associated to periodic automorphisms of semisimple algebraic groups, Dokl. AN SSSR 221 (1975), 767–770 (Russian), English translation: Soviet Math. Dokl. 16 (1975), 406–409.

[12] —, On the classification of the nilpotent elements of graded Lie algebras, Dokl. AN SSSR 225 (1975), 745–748 (Russian), English translation: Soviet Math. Dokl. 16 (1975), 1517–1520.

[13] —, The Weyl group of a graded Lie algebra, Izvestiya AN SSSR, ser. matem. 40 (1976), 488–526 (Russian), English translation: Math. USSR Izvestiya 10 (1976), 463–495.

[14] —, Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Trudy Sem. Vektor. Tenzor. Anal. 19 (1979), 155–177 (Russian), English translation: Selecta Math. Sovietica 6 (1987), 15–35.

[15] Vinberg, E. B., and A. G. Elashvili, Classification of trivectors of a 9dimensional space, Trudy Sem. Vektor. Tenzor. Anal. 18 (1978), 197–233 (Russian), English translation: Selecta Math. Sovietica 7 (1988), 63–98.

Philippinenhöfer Weg, 26 Kassel, 34127, Deutschland Chair of Algebra Department of Mathematics Moscow State University Moscow, 119899, Russia

                            timashev@mech.math.msu.su

Received January 28, 1998