Journal of Lie Theory Volume 12 (2002) 397–407 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Resolutions of Singularities of Varieties of Lie Algebras of Dimensions 3 and 4

Roberta Basili

Communicated by Karl H. Hofmann

Abstract. We will determine the singular points and a resolution of singularities of each irreducible component of the varieties of the Lie algebras of dimension 3 and 4 over ℂ.

1. Introduction

Let L_{n} be the projective variety of the Lie algebras of dimension n over ℂ. In some recent papers many results on the irreducible components of L_{n} were found for small values of n. In [2] Carles and Diakité determined the open orbits and described the irreducible components of L_{n} as orbit closures for n ≤ 7. In [6] Kirillov and Neretin determined the number of irreducible components of L_{n} and their dimension for n ≤ 6; they also determined representatives of the generic orbits of any component of L_{4}. In [1] Burde and Steinhoff gave a classification of any orbit closure of L_{4}. The variety L_{3} has two irreducible components and one of them is a linear variety; the variety L_{4} has four irreducible components. In this paper we will determine the singular points and find a resolution of singularities of each irreducible component of L_{3} and L_{4}. By using the classification of the Lie algebras of dimension 3 and 4 over ℂ, we will describe each irreducible component by giving algebraic equations of it. The first classification is well known (see [3]); the second one may be deduced from [8] and from [9] (see [1]); nevertheless we will give a short proof of it. Each resolution of singularities is a subvariety of the product of the irreducible component with a suitable grassmannian or is a resolution of singularities of a variety of this type. We observe that the results of this paper are also true over any algebraically closed field K such that char K ≠ 2.

2. Preliminaries

For any n ∈ ℕ let L_{n} be the subvariety of the projective space ℙ(Hom(ℂ^{n} ∧ ℂ^{n}, ℂ^{n}))

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag of all [α] such that α(x ∧ α(y ∧ z)) + α(y ∧ α(z ∧ x)) + α(z ∧ α(x ∧ y)) = 0 for any x, y, z ∈ ℂ^{n}, which we regard as the variety of all the Lie algebras over ℂ of dimension n. For any [α] ∈ L_{n} let L_{α} be the Lie algebra defined by α. The group GL(n, ℂ) acts on Hom(ℂ^{n}∧ ℂ^{n}, ℂ^{n}) by the relation α·G(Gx∧Gy) = G(α(x∧y)), for any G ∈ GL(n, ℂ), α ∈ Hom(ℂ^{n} ∧ ℂ^{n}, ℂ^{n}), x, y ∈ ℂ^{n} and this induces an action of GL(n, ℂ) on L_{n}; the orbits of this action are the classes of isomorphic Lie algebras. For any n, n′ ∈ ℕ let M_{n×n′}, M_{n} and S_{n} be the vector spaces of all n × n′ matrices, of all n × n matrices and of all n × n symmetric matrices respectively over ℂ. Let {e_{1}, …, e_{n}} be the canonical basis of ℂ^{n} and let us order the set {e_{i} ∧ e_{j} : i, j = 1, …, n, i < j}, writing it as {E_{1}, …, E_{m}}. For any α ∈ Hom(ℂ^{n} ∧ ℂ^{n}, ℂ^{n}) let A_{α} ∈ M_{n×m} be the matrix of α with respect to the previous bases; then A_{α·G} = GA_{α}Ĝ where Ĝ ∈ GL(m, ℂ) is the matrix whose (h, k) entry is the determinant of the 2×2 submatrix of G^{−1} obtained by choosing the rows i, j with E_{h} = e_{i} ∧ e_{j} and the columns i′, j′ with E_{k} = e_{i′} ∧ e_{j′}. If n = 3 we set E_{1} = e_{2}∧e_{3}, E_{2} = e_{3}∧e_{1}, E_{3} = e_{1}∧e_{2} and we get A_{α·G} = (det G)^{−1}GA_{α}G^{t}. Then we have

L_{3} = {[α] ∈ ℙ(Hom(ℂ^{3} ∧ ℂ^{3}, ℂ^{3})) : cof A_{α} ∈ S_{3}}

where for any A = (a_{ij}) ∈ M_{n} cof A is the matrix whose (i, j) entry is the algebraic complement of a_{ji}. We recall that, up to isomorphisms, we have the following non-abelian Lie algebras of dimension 3 over ℂ ([3]), which may also be obtained as in the proof of theorem 4.1:

l_{a} : [e_{1}, e_{2}] = e_{2}, [e_{1}, e_{3}] = ae_{3}, [e_{2}, e_{3}] = 0, a ∈ ℂ,
n_{3} : [e_{1}, e_{2}] = [e_{1}, e_{3}] = 0, [e_{2}, e_{3}] = e_{1},
r_{3} : [e_{1}, e_{2}] = e_{2}, [e_{1}, e_{3}] = e_{2} + e_{3}, [e_{2}, e_{3}] = 0,
sl(2, ℂ) : [e_{1}, e_{2}] = e_{3}, [e_{1}, e_{3}] = −2e_{1}, [e_{2}, e_{3}] = 2e_{2},

where the only pairs of isomorphic Lie algebras are {l_{a},l_{a^{−1}}}, a ≠ 0, a^{−1}, and n_{3}, the Heisenberg Lie algebra, is the only nilpotent one. Hence the following subvarieties:

W_{1} = {[α] ∈ L_{3} : A_{α} ∈ S_{3}}
= {[α] ∈ L_{3} : for any v ∈ L_{α} tr ad v = 0},

which is isomorphic to ℙ(S_{3}), and

W_{2} = {[α] ∈ L_{3} : rank A_{α} ≤ 2}
= {[α] ∈ L_{3} : L_{α} has an abelian ideal of dimension 2},

that is the subvariety of the solvable Lie algebras, are the irreducible components of L_{3}. For any n, n′ ∈ ℕ let G_{n′,n} be the grassmannian of all the subspaces of ℂ^{n} of dimension n′.

3. The variety of the Lie algebras of dimension 3

We identify α with A_{α} and we set A = (a_{ij}) for any A ∈ M_{3}.

Lemma 3.1. We have W_{2} = {[A] ∈ ℙ(M_{3}) : dim(ker A ∩ ker A^{t}) ≥ 1}.

Proof. Since both subsets are stable with respect to the action of GL(3, ℂ) it is sufficient to show that if A is such that a_{j1} = 0, j = 1, 2, 3, the condition cof A ∈ S_{3} is equivalent to the condition dim(ker A ∩ ker A^{t}) ≥ 1. But in this case both these conditions are equivalent to the following one: rank A ≤ 1 or a_{1j} = 0, j = 2, 3; hence we get the claim. The result also follows from the classification of the Lie algebras of dimension 3 over ℂ.

Let

W′_{2} = {(H, [A]) ∈ ℙ^{2}(ℂ) × ℙ(M_{3}) : H ⊆ ker A ∩ ker A^{t}}

and let π, π′ be the canonical projections of W′_{2} onto ℙ^{2}(ℂ) and W_{2} respectively.

Proposition 3.2. W_{2} is irreducible, dim W_{2} = 5 and π′ is a resolution of singularities of W_{2}. The set of the singular points of W_{2} is Z = {[A] ∈ ℙ(M_{3}) : dim(ker A ∩ ker A^{t}) = 2}, that is the orbit of n_{3}, and dim Z = 2.

Proof. For i = 1, 2, 3 let U_{i} be the open subset of ℙ^{2}(ℂ) given by the condition that the i-th coordinate doesn’t vanish and let F_{i} be the subset of ℙ(M_{3}) of all [A] such that the i-th row and column of A vanish. Let G_{i} ∈ GL(3, ℂ) be such that G_{i}(e_{i}) ∈ ⟨e_{i}⟩ and let G_{i}^{1}, G_{i}^{2}, G_{i}^{3} be the columns of G_{i}. Let ϕ_{i} : U_{i} → GL(3, ℂ) be such that for any H = ⟨(x_{1}, x_{2}, x_{3})⟩ ∈ U_{i} the i-th column of ϕ_{i}(H) is G_{i}^{i}−∑_{j≠i} x_{j}(x_{i})^{−1}G_{i}^{j}, the others are equal to those of G_{i}; then ϕ_{i}(H)(H) = ⟨e_{i}⟩. If A_{i} = π^{−1}(U_{i}) the map (H, [A]) ↦ (H, [(ϕ_{i}(H)^{−1})^{t}Aϕ_{i}(H)^{−1}]) from A_{i} to U_{i} × F_{i} is an isomorphism. Hence W′_{2}, with the map π, is a vector bundle on ℙ^{2}(ℂ) with fibers isomorphic to ℙ(M_{2}). The map ([A]) ↦ (ker A ∩ ker A^{t}, [A]) from W_{2} to W′_{2} is regular except in the points of Z, where the fibers of π′ have dimension 1, and is a birational inverse of π′. Let Z′ = {(H, [A]) ∈ G_{2,3} × ℙ(M_{3}) : H ⊆ ker A ∩ ker A^{t}}. If π_{1} and π_{2} are the canonical projections of Z′ on G_{2,3} and Z respectively, π_{2} is a birational morphism and the fibers of π_{1} have only one point. Hence Z′ and Z are irreducible of dimension 2 and (π′)^{−1}(Z) is irreducible of dimension 3. Then by Theorem 2 of chap. II, §4 of [10] we get the claim.

Corollary 3.3. The set of the singular points of L_{3} is W_{1} ∩ W_{2}, that is the union of the orbits of n_{3} and l_{−1}.

For any [α] ∈ L_{n} the tangent space in [α] to L_{n} is ℙ(V_{α}), where V_{α} is the vector space of 2-cocycles in the cohomology of L_{α} as L_{α}-module ([5]). By the equations of the space of 2-cocycles of a Lie algebra we have found that the dimensions of the tangent spaces to L_{3} in n_{3} and l_{−1} are 7 and 6 respectively.

4. Classification of the Lie algebras of dimension 4 over ℂ

For any (β, γ) ∈ ℂ^{2} let [[β, γ]] and [[β]] be the orbit in ℙ^{2}(ℂ) of [1, β, γ] and [1, β, 1−β] respectively with respect to the action of the group of the permutations of the coordinates of ℙ^{2}(ℂ).

Theorem 4.1. We have [α] ∈ L_{4} if and only if L_{α} is isomorphic to one and only one of the following Lie algebras (where we omit [e_{i}, e_{j}], i, j ∈ {1, …, 4}, if it is 0):

g_{[[β,γ]]} : [e_{4}, e_{1}] = e_{1}, [e_{4}, e_{2}] = βe_{2}, [e_{4}, e_{3}] = γe_{3}, β, γ ∈ ℂ;
g_{[[β]]} : [e_{2}, e_{3}] = e_{1}, [e_{4}, e_{1}] = e_{1}, [e_{4}, e_{2}] = βe_{2},
[e_{4}, e_{3}] = (1 − β)e_{3}, β ∈ ℂ;
g_{c} : [e_{4}, e_{1}] = ce_{1}, [e_{4}, e_{2}] = e_{2}, [e_{4}, e_{3}] = e_{2} + e_{3}, c ∈ ℂ;
a_{1} : [e_{2}, e_{3}] = e_{1}, [e_{4}, e_{1}] = 2e_{1}, [e_{4}, e_{2}] = e_{2}, [e_{4}, e_{3}] = e_{2} + e_{3};
a_{2} : [e_{4}, e_{1}] = e_{1}, [e_{4}, e_{2}] = e_{1} + e_{2}, [e_{4}, e_{3}] = e_{2} + e_{3};
a_{3} : [e_{3}, e_{2}] = e_{2}, [e_{4}, e_{1}] = e_{1};
a_{4} : [e_{4}, e_{1}] = e_{1}, [e_{4}, e_{2}] = −e_{2}, [e_{1}, e_{2}] = e_{4};
a_{5} : [e_{1}, e_{2}] = e_{3}, [e_{4}, e_{1}] = e_{1}, [e_{4}, e_{2}] = −e_{2};
a_{6} : [e_{4}, e_{1}] = e_{1}, [e_{4}, e_{2}] = e_{3};
a_{7} : [e_{2}, e_{3}] = e_{1};
a_{8} : [e_{2}, e_{3}] = e_{1}, [e_{4}, e_{3}] = e_{2}.

Proof. Let L be a Lie algebra over ℂ of dimension 4. Let H be a Cartan subalgebra of L, h ∈ H be such that H = L_{0}(ad h) = {v ∈ L : ∃ n ∈ ℕ : (ad h)^{n}v = 0} and ad h, if not nilpotent, has the eigenvalue 1, H′ be a subspace of L such that H ⊕ H′ = L, [h, H′] = H′. Let dim H = 1. Then H′ = [L, L]. Let {x, y, z} be a basis of H′ such that the matrix of ad_{H′} h with respect to it is in Jordan canonical form. From the Jacobi’s relations between h and the pairs of elements of {x, y, z}, when ad_{H′} h is represented by a diagonal matrix with diagonal entries 1, β, γ respectively, β, γ ≠ 0, we get

(β + 1)[x, y] = [h, [x, y]], (γ + 1)[x, z] = [h, [x, z]], (β + γ)[y, z] = [h, [y, z]],

hence either H′ is abelian or, permuting x, y, z and multiplying them by a scalar if necessary, β + γ = 1 and H′ is a Heisenberg Lie algebra with x = [y, z]. We get the Lie algebras g_{[[β,γ]]}, β, γ ≠ 0, and g_{[[β]]}, β ≠ 0, 1, respectively. If ad_{H′} h is represented by two Jordan blocks, the first one of order 2 and eigenvalue 1, the second one of eigenvalue c ≠ 0, we get

(c + 1)[z, x] = [h, [z, x]], [z, x] + (c + 1)[z, y] = [h, [z, y]], 2[x, y] = [h, [x, y]],

hence [z, x] = [z, y] = 0 and either H′ is abelian or c = 2 and H′ is a Heisenberg Lie algebra, with (multiplying x and y by a scalar) [x, y] = z. We get the Lie algebras g_{c}, c ≠ 0, and a_{1} respectively. If ad_{H′} h is represented by only one Jordan block we get

2[x, y] = [h, [x, y]], 2[x, z] = [h, [x, z]] − [x, y], 2[y, z] = [h, [y, z]] − [x, z],

hence H′ is abelian and we get the Lie algebra a_{2}. Let dim H = 2. Then, since H is abelian, ad_{L} H is abelian and H′ = [H, H′] = [H, L]. Let {x, y} be a basis of H′ such that the matrix of ad_{H′} h with respect to {x, y} is in Jordan canonical form. We have to require

[h, [x, y]] = [x, [h, y]] + [y, [x, h]] = (tr ad_{H′} h)[x, y],

hence either [x, y] = 0 or for any v ∈ H ad_{H′} v has the eigenvalues 1, −1 and dim ad H ≤ 1. If dim ad H = 2 and there exist in H elements v such that ad_{H′} v has two different eigenvalues, we may choose w, z ∈ H such that with respect to the basis {x, y} ad_{H′} w and ad_{H′} z are represented by two diagonal matrices with diagonal entries 1, 0 and 0, 1 respectively, hence we get the Lie algebra a_{3}. If dim ad H = 2 but for any v ∈ H ad_{H′} v has only one eigenvalue we may choose h, z ∈ H such that with respect to the basis {x, y} ad_{H′} h and ad_{H′} z are represented respectively by the identity matrix and by the nilpotent Jordan block of order 2, hence we get the Lie algebra g_{[[0]]}. If dim ad H = 1 let z ∈ H \ {0} be such that ad z = 0. If the Jordan form of ad_{H′} h is diagonal and [x, y] = 0 we get the Lie algebras g_{[[0,γ]]}, γ ∈ ℂ \ {0}. If the Jordan form of ad_{H′} h is diagonal and [x, y] ∉ ⟨z⟩ we may assume h = [x, y] and we get the Lie algebra a_{4}. If the Jordan form of ad_{H′} h is diagonal and [x, y] ∈ ⟨z⟩ \ {0} we may assume [x, y] = z getting the Lie algebra a_{5}. If the Jordan form of ad_{H′} h has only one Jordan block we get the Lie algebra g_{0}. Let dim H = 3. If H is abelian, since dim ad H = 1 there exist y, z ∈ H linearly independent such that ad y = ad z = 0, hence we get the Lie algebra g_{[[0,0]]}. If H is a Heisenberg Lie algebra, since the subset of all v ∈ H such that H = L_{0}(ad v) is open in H, we may assume H = ⟨h, y, z⟩ with [h, y] = z, [h, x] = x, x ∉ H. Since ad h and ad z commute, [z, x] ∈ ⟨x⟩. Since ad y and ad z commute, if we had [z, x] ≠ 0 we would have [y, x] ∈ ⟨x⟩ and then, since ad_{H} y and ad_{H} h commute, ad y and ad h would commute; but this holds if and only if ad z = 0. Hence [z, x] = 0 and [y, x] ∈ ⟨x⟩. Since dim[H, x] = 1 we may choose y such that [y, x] = 0; we get the Lie algebra a_{6}. Let dim H = 4, that is L nilpotent. If L isn’t abelian there exists x ≠ 0 such that x ∈ Z(L)∩[L, L]. If x = [y, z], since H′′ = ⟨x, y, z⟩ is a nilpotent subalgebra, dim H′′ = 3 and H′′ is a Heisenberg Lie algebra. Since L is nilpotent [h, H′′] ⊆ H′′ for any h ∈ L . Since [h, x] = 0 it is possible to choose h, x, y, z such that h ∉ H′′, the matrix of ad_{H′′} h with respect to the basis {x, y, z} is in Jordan canonical form and [h, y] = 0 (in fact if [h, y] = x then [h + z, y] = 0). We get the Lie algebras a_{7} and a_{8}.

5. The variety of the Lie algebras of dimension 4

For any Lie algebra L let Z(L) be the center of L.

Proposition 5.1. L_{4} is the union of the following closed subsets:

C_{1} = {[α] ∈ L_{4} : L_{α} has an abelian ideal of dimension 3},
C_{2} = {[α] ∈ L_{4} : L_{α} has a nilpotent ideal J_{α} of dimension 3 such
that \frac{1}{2} tr ad v is eigenvalue of ad_{J_{α}} v for any v ∈ L_{α}\},
C_{3} = {[α] ∈ L_{4} : dim[L_{α}, L_{α}] ≤ 2, ad_{[L_{α},L_{α}]} L_{α} is abelian},
C_{4} = {[α] ∈ L_{4} : Z(L_{α}) ≠ {0}, tr ad v = 0 for any v ∈ L_{α}}

and C_{i} ⊈ ⋃_{j≠i} C_{j} for i, j = 1, …, 4.

Proof. Since by Theorem 4.1 each one of these subsets is the union of the orbits of the following Lie algebras:

C_{1} : g_{[[β,γ]]}, g_{c}, a_{2}, a_{6}, a_{7}, a_{8}
C_{2} : g_{[[γ+1,γ]]}, g_{[[β]]}, g_{0}, g_{2}, a_{1}, a_{5}, a_{7}, a_{8}
C_{3} : g_{[[0,γ]]}, g_{[[0]]}, g_{0}, a_{3}, a_{6}, a_{7}, a_{8}
C_{4} : g_{[[0,−1]]}, a_{4}, a_{5}, a_{7}, a_{8}

where β, γ, c ∈ ℂ, we get the claim.

For any i = 1, …, 4 let A_{i} = {J ∈ G_{3,4} : e_{i} ∉ J} and let {i_{1}, i_{2}, i_{3}} = {1, …, 4} \ {i}, i_{1} < i_{2} < i_{3}. If J ∈ A_{i} let J = ⟨e_{i_{1}}^{J}, e_{i_{2}}^{J}, e_{i_{3}}^{J}⟩, where, with respect to the basis {e_{i_{1}}, e_{i_{2}}, e_{i_{3}}, e_{i}}, for j = 1, 2, 3 the j -th coordinate of e_{i_{j}}^{J} is 1 and for k ∈ {1, 2, 3}, k ≠ j the k-th coordinate of e_{i_{j}}^{J} is 0. Let

C′_{1} = {(J, [α]) ∈ G_{3,4} × C_{1} : J is an abelian ideal of L_{α}}

and let p_{1}, p′_{1} be the canonical projections of C′_{1} onto G_{3,4} and C_{1} respectively.

Proposition 5.2. C_{1} is irreducible, dim C_{1} = 11 and p′_{1} is a resolution of singularities of C_{1}. The set of the singular points of C_{1} is Z_{1} = {[α] ∈ C_{1} : L_{α} is nilpotent and dim[L_{α}, L_{α}] ≤ 1}, that is the orbit of a_{7}, and dim Z_{1} = 5.

Proof. Let A′_{i} := (p_{1})^{−1}(A_{i}), i = 1, …, 4. The map ξ_{i} : A_{i} ×ℙ(M_{3}) → A′_{i} defined by ξ_{i}(J, [A]) = (J, [α]) where [α] is such that in L_{α} ad_{J} e_{i} is represented by A with respect to the basis {e_{i_{1}}^{J}, e_{i_{2}}^{J}, e_{i_{3}}^{J}} is an isomorphism, hence C′_{1}, with the map p_{1}, is a vector bundle and dim C′_{1} = 11. The map p′_{1} is birational and (p′_{1})^{−1} is regular in the open subset of C_{1} of all [α] such that L_{α} is not nilpotent or there exists x ∈ L_{α} such that dim[x, L_{α}] ≥ 2 (we set (p′_{1})^{−1}([α]) = (J, [α]) where J is the subspace of all the nilpotent elements x of L_{α} such that dim[x, L_{α}] ≤ 1). It isn’t regular in the points of Z_{1} = {[α] ∈ C_{1} : L_{α} is nilpotent and dim[L_{α}, L_{α}] ≤ 1}, that is the orbit of a_{7}, since the fibers of p′_{1} on the elements of Z_{1} have dimension 1. The variety Z′_{1} := (p′_{1})^{−1}(Z_{1}), with the map p_{1}|_{Z}′_{1} , is a bundle on G_{3,4} whose fibers are isomorphic to ℙ(N′_{3}), where N′_{3} is the variety of all the nilpotent 3 × 3 matrices over ℂ of rank less or equal 1; hence it is irreducible of dimension 6, which shows the claim.

Let

C′_{2} = {(J, [α]) ∈ G_{3,4} × C_{2} : J is a nilpotent ideal of L_{α} and
for any v ∈ L_{α} \frac{1}{2} tr ad v is eigenvalue of ad_{J} v\}.

Lemma 5.3. If (J, [α]) ∈ C′_{2} and v ∈ L_{α} then [J, J] is contained in the eigenspace of ad_{J} v corresponding to \frac{1}{2} tr ad v.

Proof. Let y ≠ 0 belong to the previous eigenspace but [J, J] ⊈ ⟨y⟩. Then we may choose a basis {y, x, z} of J such that [J, J] ⊆ ⟨x⟩. Since [x, v] ∈ ⟨x⟩ (in fact 0 = [x, [y, v]] = [y, [x, v]], hence [x, v] ∈ ⟨x, y⟩, in the same way [x, v] ∈ ⟨x, z⟩), there exist a, b, c, d ∈ ℂ such that [v, y] = ay, [v, x] = bx, [v, z] = (a−b)z+cx+dy, hence by the condition [y, [z, v]] = [z, [y, v]] + [v, [z, y]] we get a = b.

Let

\begin{aligned}S′ & = & \{(H, [(A, B)]) ∈ ℙ^{2}(ℂ) × ℙ(S_{3} × M_{3}) : Im A ⊆ H, \\ & & H ⊆ ker (B − (\frac{1}{2} tr B)I_{3})\}; \end{aligned}

let S be the image of the canonical projection of S′ on ℙ(S_{3} × M_{3}) and let s, s′ be the canonical projections of S′ on ℙ^{2}(ℂ) and S respectively.

Lemma 5.4. S is irreducible, dim S = 8 and s′ is a resolution of singularities of S. The set of the singular points of S is

Ŝ = \{[(A, B)] ∈ S : A = 0, dim ker (B − ( \frac{1}{2} tr B)I_{3}) ≥ 2\},

which is irreducible of dimension 4.

Proof. The variety S′ with the map s is a vector bundle on ℙ^{2}(ℂ) with fibers of dimension 6. The map s′ is birational and (s′)^{−1} is regular in the open subset of all [(A, B)] such that A ≠ 0 or dim ker (B − (\frac{1}{2} tr B)I_{3}) = 1. It isn’t regular in the points of Ŝ, where the generic fiber of s′ has dimension 1, and Ŝ′ := (s′)^{−1}(Ŝ) is irreducible of dimension 5 (the fiber of s|Ŝ′ in H is birational to \{(V, [B]) ∈ G_{2,3} × ℙ(M_{3}) : H ⊂ V ⊆ ker (B − (\frac{1}{2} tr B)I_{3})\}, hence has dimension 3), which shows the claim.

Let p_{2} and p′_{2} be the canonical projections of C′_{2} on G_{3,4} and C_{2} respectively.

Lemma 5.5. C′_{2}, with the map p_{2}, is a bundle on G_{3,4} with fibers isomorphic to S.

Proof. Let U_{i} = (p_{2})^{−1}(A_{i}), i = 1, …, 4. For any (J, [α]) ∈ C′_{2} let α_{J} ∈ Hom(J ∧ J, J) be defined by α_{J}(v ∧ v′) = α|_{J∧J}(v ∧ v′) for any v, v′ ∈ J. The map ν_{i} : A_{i} × S → U_{i} such that ν_{i}(J, [(A, B)]) = (J, [α]) where α is such that the matrix of α_{J} with respect to the bases {e_{i_{2}}^{J} ∧e_{i_{3}}^{J}, e_{i_{3}}^{J} ∧e_{i_{1}}^{J}, e_{i_{1}}^{J} ∧e_{i_{2}}^{J}} and {e_{i_{1}}^{J}, e_{i_{2}}^{J}, e_{i_{3}}^{J}} is A and in L_{α} the matrix of ad_{J} e_{i} with respect to the basis {e_{i_{1}}^{J}, e_{i_{2}}^{J}, e_{i_{3}}^{J}} is B is an isomorphism, which shows the claim.

For any i = 1, …, 4 and J ∈ A_{i} let B_{i}^{J} = {e_{i_{1}}^{J}, e_{i_{2}}^{J}, e_{i_{3}}^{J}, e_{i}}. Let J ∈ A_{i} ∩ A_{i′} and let G_{J} be the matrix whose columns are the coordinates of the elements of B_{i}^{J} with respect to B_{i′}^{J}. Let δ : S_{3} × M_{3} → M_{4×6} be the isomorphism such that, by regarding δ((A, B)) as a block matrix, we have

δ((A, B)) = ( \begin{matrix}A & B \\ 0 & 0 \end{matrix} ) .

Then, by using the notations of the proof of Lemma 5.5, we have that the automorphism (ν_{i′})^{−1} ∘ ν_{i} of (A_{i} ∩ A_{i′}) × S is given by

(ν_{i′})^{−1} ∘ ν_{i}(J, [(A, B)]) = (J, [δ^{−1}(G_{J}δ((A, B))Ĝ_{J})]).

Let C′′_{2} be the vector bundle on G_{3,4} which is the union of open subsets U′_{i}, i = 1, …, 4, with isomorphisms ν′_{i} : A_{i} × S′ → U′_{i} such that

(ν′_{i′})^{−1} ∘ ν′_{i}(J, (H, [(A, B)])) = (J, (H_{J}, [δ^{−1}(G_{J}δ((A, B))Ĝ_{J})]))

where if H = [h_{1}, h_{2}, h_{3}] then H_{J} = [h_{1}^{J}, h_{2}^{J}, h_{3}^{J}] is such that (h_{1}^{J}, h_{2}^{J}, h_{3}^{J}, 0) = G_{J}^{−1}(h_{1}, h_{2}, h_{3}, 0). Let p′′ : C′′_{2} → C′_{2} be the morphism such that p′′(U′_{i}) = U_{i} and, if p′′_{i} is p′′|_{U}′_{i} as map onto U_{i}, we have ν_{i} ∘ (id_{A_{i}} ×s′) = p′′_{i} ∘ ν′_{i} for any i = 1, …, 4. Then p′′ is a resolution of singularities of C′_{2}.

Proposition 5.6. C_{2} is irreducible, dim C_{2} = 11 and p′_{2} ∘ p′′ is a resolution of singularities of C_{2}. The set of the singular points of C_{2} is Z_{2} = Ẑ_{2} ∪ Z̃_{2} where Ẑ_{2} = {[α] ∈ C_{2} : L_{α} is nilpotent} and Z̃_{2} = {[α] ∈ C_{2} : L_{α} has an abelian ideal of dimension 3 and for any v ∈ L_{α} dim Im(ad v − ( \frac{1}{2} tr ad v) id) ≤ 1\}. We have that Ẑ_{2} is irreducible of dimension 8 and is the union of the orbits of a_{7} and a_{8}; Z̃_{2} is irreducible of dimension 7 and is the union of the orbits of g_{[[0,1]]}, g_{0} and a_{7}.

Proof. The map p′_{2} is birational and the subset of C_{2} in which (p′_{2})^{−1} isn’t regular is Ẑ_{2}, since (p′_{2})^{−1}([α]) = (J, [α]) where J is the subspace of L_{α} of all the nilpotent elements and the generic fiber of p′_{2} on Ẑ_{2} has dimension 1. Let Ẑ′_{2} := (p′_{2})^{−1}(Ẑ_{2}). If we set \overline{S} = \{[(A, B)] ∈ S : B is nilpotent\} we have that \overline{S} is irreducible and dim \overline{S} = 6 (in fact, by Lemma 5.4, (s′)^{−1}(\overline{S}) has these properties). Since the fibers of p_{2}|Ẑ′_{2} are isomorphic to \overline{S} we get that Ẑ′_{2} is irreducible of dimension 9 and Ẑ_{2} is irreducible of dimension 8, hence by Theorem 2 of chap. II, §4 of [10] the points of Ẑ_{2} are singular for C_{2}. By Lemma 5.4 and Lemma 5.5 Z̃_{2} is irreducible of dimension 7 and the points of Z̃_{2} \ Ẑ_{2} are singular for C_{2}, hence we get the claim.

For any n ∈ ℕ let C_{n} = {(A, B) ∈ M_{n} × M_{n} : [A, B] = 0}. If (x_{0}, …, x_{7}) are coordinates of ℂ^{8}, we set

A = ( \begin{matrix}x_{0} & x_{2} \\ x_{4} & x_{0} + x_{6} \end{matrix} ) , B = ( \begin{matrix}x_{1} & x_{3} \\ x_{5} & x_{1} + x_{7} \end{matrix} )

and we regard C_{2} as a subvariety of ℂ^{8}. Let V′ = {(x_{0}, …, x_{7}) ∈ C_{2} : (x_{2}, …, x_{7}) ≠ (0, …, 0)}; then the map u : V′ → ℙ^{5}(ℂ) such that u((x_{0}, …, x_{7})) = [x_{2}, …, x_{7}] is a morphism. Let V = u(V′), let:

W = {((x_{0}, …, x_{7}), [z_{2}, …, z_{7}]) ∈ C_{2} × V : x_{i}z_{j} = z_{i}x_{j}, i, j = 2, …, 7}

and let r be the canonical projection of W on C_{2}.

Lemma 5.7. C_{2} is irreducible, dim C_{2} = 6 and

Y = {(A, B) ∈ C_{2} : A, B ∈ ⟨I_{2}⟩}

is the set of the singular points of C_{2}. The variety W is irreducible and r is a resolution of singularities of C_{2}.

Proof. For any n ∈ ℕ C_{n} is irreducible of dimension n^{2} + n ([7], [4]). If X = (x_{ij}), Y = (y_{ij}) are the coordinates of M_{n} × M_{n} and (A, B) ∈ C_{n} then [A, X] + [B, Y] = 0 are equations of the tangent space to C_{n} in (A, B). Hence the points (A, B) such that A or B is regular, that is has centralizer of minimum dimension n, are non singular for C_{n}, which shows the first claim. Since V and C_{2} have the same equations, V is an irreducible nonsingular variety of dimension 3. The map r is birational, since for any (x_{0}, …, x_{7}) ∈ C_{2} such that (x_{2}, …, x_{7}) ≠ (0, …, 0) we may set r^{−1}((x_{0}, …, x_{7})) = ((x_{0}, …, x_{7}), [x_{2}, …, x_{7}]); if (x_{2}, …, x_{7}) = (0, …, 0) we have r^{−1}({(x_{0}, …, x_{7})}) = {(x_{0}, …, x_{7})} × V. Since for any x_{0}, x_{1}, t ∈ ℂ and [z_{2}, …, z_{7}] ∈ V we have that ((x_{0}, x_{1}, tz_{2}, …, tz_{7}), [z_{2}, …, z_{7}]) ∈ W, W is irreducible. Since V has the same equations as C_{2} the tangent space to W in a point such that (x_{2}, …, x_{7}) = (0, …, 0) has the same dimension as in a point of W \r^{−1}(Y), hence we get the claim.

Let

\begin{aligned}G′ & = & \{([y_{1}, y_{2}, x_{0}, …, x_{7}], [z_{2}, …, z_{7}]) ∈ ℙ^{9}(ℂ) × ℙ^{5}(ℂ) : \\ & & : ((x_{0}, …, x_{7}), [z_{2}, …, z_{7}]) ∈ W\}; \end{aligned}

let G be the image of the canonical projection of G′ onto ℙ^{9}(ℂ) and let r′ be the canonical projection of G′ on G.

Corollary 5.8. The map r′ is a resolution of singularities of G. Let

C′_{3} = {(W, [α]) ∈ G_{2,4} × C_{3} : [L_{α}, L_{α}] ⊆ W, ad_{W} L_{α} is abelian},

and let p_{3}, p′_{3} be the canonical projections of C′_{3} on G_{2,4} and C_{3} respectively. For any i, j ∈ {1, …, 4}, i < j let A_{ij} = {W ∈ G_{2,4} : W ∩ ⟨e_{i}, e_{j}⟩ = {0}}. Let {i_{0}, j_{0}} = {1, …, 4} \ {i, j}, i_{0} < j_{0}; if W ∈ A_{ij} let W = ⟨e_{i_{0}}^{W} , e_{j_{0}}^{W} ⟩ where the first two coordinates of e_{i_{0}}^{W} and e_{j_{0}}^{W} with respect to the basis {e_{i_{0}}, e_{j_{0}}, e_{i}, e_{j}} are 1, 0 and 0, 1 respectively.

Lemma 5.9. C′_{3} with the map p_{3} is a bundle on G_{2,4} with fibers isomorphic to G.

Proof. Let U_{ij} = (p_{3})^{−1}(A_{ij}), i, j ∈ {1, …, 4}, i < j. The map η_{ij} : A_{ij} × G → U_{ij} defined by η_{ij}(W, [(y_{1}, y_{2}, A, B)]) = (W, [α]) where α is such that in L_{α} [e_{i}e_{j}] = y_{1}e_{i_{0}}^{W} + y_{2}e_{j_{0}}^{W} and ad_{W} e_{i}, ad_{W} e_{j} are represented, with respect to the basis {e_{i_{0}}^{W} , e_{j_{0}}^{W} }, respectively by A and B is an isomorphism, which shows the claim.

For any i, j ∈ {1, …, 4}, i < j, and W ∈ A_{ij} let B_{ij}^{W} = {e_{i_{0}}^{W} , e_{j_{0}}^{W} , e_{i}, e_{j}}. Let W ∈ A_{ij} ∩ A_{i′j′} and let G_{W} be the matrix whose columns are the coordinates of the elements of B_{ij}^{W} with respect to B_{i′j′}^{W}. Let ζ : ℂ^{2} × M_{2} × M_{2} → M_{4×6} be the isomorphism such that, by regarding ζ((y_{1}, y_{2}, A, B)) as a block matrix, we have:

ζ((y_{1}, y_{2}, A, B)) = ( \begin{matrix}0 & Y & A & B \\ 0 & 0 & 0 & 0 \end{matrix} ) , Y = ( \begin{matrix}y_{1} \\ y_{2} \end{matrix} ) .

Then, by using the notations of the proof of Lemma 5.9, we have that the automorphism (η_{i′j′})^{−1} ∘ η_{ij} of (A_{ij} ∩ A_{i′j′}) × G is given by

\begin{matrix}(η_{i′j′})^{−1} ∘ η_{ij} & (W, [y_{1}, y_{2}, x_{0}, …, x_{7}]) = \\ = & (W, [ζ^{−1}(G_{W} ζ((y_{1}, y_{2}, x_{0}, …, x_{7}))Ĝ_{W})]). \end{matrix}

Let \overline{u} : ℂ^{2}×V′ → V be defined by \overline{u}((y_{1}, y_{2}, x_{0}, …, x_{7})) = [x_{2}, …, x_{7}]. Let C′′_{3} be the vector bundle on G_{2,4} which is the union of open subsets U′_{ij}, i, j ∈ {1, …, 4}, i < j, with isomorphisms η′_{ij} : A_{ij} × G′ → U′_{ij}, such that

(η′_{i′j′})^{−1} ∘ η′_{ij} (W, ([y_{1}, y_{2}, x_{0}, …, x_{7}], [z_{2}, …, z_{7}])) =
= (W, ([ζ^{−1}(G_{W} ζ((y_{1}, y_{2}, x_{0}, …, x_{7}))Ĝ_{W})],
\overline{u} ∘ ζ^{−1}(G_{W} ζ((0, …, 0, z_{2}, …, z_{7}))Ĝ_{W}))

and let q′′ : C′′_{3} → C′_{3} be the morphism such that q′′(U′_{ij}) = U_{ij} and, if q′′_{ij} is q′′|_{U}′_{ij} as map onto U_{ij}, we have η_{ij} ∘(id_{A_{ij}} ×r′) = q′′_{ij} ∘η′_{ij} for any i, j ∈ {1, …, 4}, i < j. Then q′′ is a resolution of singularities of C′_{3}.

Proposition 5.10. C_{3} is irreducible, dim C_{3} = 11 and p′_{3} ∘ q′′ is a resolution of singularities of C_{3}. The set of the singular points of C_{3} is Z_{3} = {[α] ∈ C_{3} : ad_{[L_{α},L_{α}]} L_{α} ⊆ ⟨id⟩}, that is the union of the orbits of g_{[[0,0]]}, g_{[[0,1]]} and a_{7}, which is irreducible of dimension 7.

Proof. The map p′_{3} is birational and the subset of C_{3} in which (p′_{3})^{−1} isn’t regular is Ẑ_{3} := {[α] ∈ C_{3} : dim[L_{α}, L_{α}] < 2} (since (p′_{3})^{−1}([α]) = ([L_{α}, L_{α}], [α]) and the generic fiber of p′_{3} on Ẑ_{3} has dimension 2). By Theorem 4.1 we have Ẑ_{3} ⊂ Z_{3} and by Lemma 5.7 and Lemma 5.9 the points of Z_{3} \Ẑ_{3} are singular for C_{3}. If Z′_{3} := (p′_{3})^{−1}(Z_{3}), the fibers of p_{3}|_{Z}′_{3} are isomorphic to ℙ^{3}(ℂ), hence Z_{3} is irreducible of dimension 7. Since the subset of the singular points is closed this shows the claim.

Let

C′_{4} = {(T, [α]) ∈ ℙ^{3}(ℂ) × C_{4} : T ⊆ Z(L_{α})}

and let p′_{4} be the canonical projections of C′_{4} on C_{4}.

Proposition 5.11. C_{4} is irreducible, dim C_{4} = 11 and p′_{4} is a resolution of singularities of C_{4}. The set of the singular points of C_{4} is Z_{4} = {[α] ∈ C_{4} : dim Z(L_{α}) ≥ 2}, that is the orbit of a_{7}.

Proof. Let C′′_{4} = {(J, T, [α]) ∈ G_{3,4} × C′_{4} : J is an ideal of L_{α}} and let q_{1}, q_{2} be the canonical projections of C′′_{4} on G_{3,4} × ℙ^{3}(ℂ) and on C′_{4} respectively. If (J, T) ∈ G_{3,4} × ℙ^{3}(ℂ) is such that T ⊈ J the fiber of q_{1} in (J, T) is isomorphic to ℙ(S_{3}). If (J, T) ∈ G_{3,4} × ℙ^{3}(ℂ) is such that T ⊂ J then J is a nilpotent ideal such that [J, J] ⊂ T for any L_{α} such that (J, T, [α]) ∈ C′′_{4}, hence the fiber of q_{1} in (J, T) is also a projective subspace of dimension 5. This proves that C′_{4} is irreducible and dim C′_{4} = 11, since q_{2} is birational ((q_{2})^{−1} is regular in the open subset of all the elements (T, [α]) such that dim[L_{α}, L_{α}] = 3). For any i ∈ {1, …, 4} let A^{i} = {T ∈ ℙ^{3}(ℂ) : T ∩ ⟨e_{i_{1}}, e_{i_{2}}, e_{i_{3}}⟩ = {0}}; for any T ∈ A^{i} we set T = ⟨e^{T} ⟩ where the first coordinate of e^{T} with respect to the basis {e_{i}, e_{i_{1}}, e_{i_{2}}, e_{i_{3}}} is 1. Let

U^{i} = {(T, [α]) ∈ C′_{4} : T ∈ A^{i}, α(e_{i_{1}} ∧ e_{i_{3}}) ≠ 0},

U′′ = {[x_{1}, …, x_{8}] ∈ ℙ^{7}(ℂ) : (x_{1}, …, x_{4}) ≠ (0, …, 0)}.

The map ψ : A^{i} × U′′ → U^{i} defined by ψ(T, [x_{1}, …, x_{8}]) = (T, [α]) where α is such that α(e_{i_{1}}∧e_{i_{2}}) = x_{5}e^{T} +x_{1}e_{i_{2}}+x_{6}e_{i_{3}} , α(e_{i_{1}}∧e_{i_{3}}) = x_{2}e^{T} +x_{3}e_{i_{1}}+x_{4}e_{i_{2}}−x_{1}e_{i_{3}} , α(e_{i_{2}} ∧ e_{i_{3}}) = x_{7}e^{T} + x_{8}e_{i_{1}} − x_{3}e_{i_{2}} is an isomorphism, hence C′_{4} is nonsingular. The map p′_{4} is birational and the subset of C_{4} in which (p′_{4})^{−1} isn’t regular is Z_{4} = {[α] ∈ C_{4} : dim Z(L_{α}) ≥ 2}, that is the orbit of a_{7}, where the fibers of p′_{4} have dimension 1. By Proposition 5.2 (p′_{4})^{−1}(Z_{4}) is irreducible of dimension 6, which shows the claim.

Corollary 5.12. The varieties C_{i}, i = 1, …, 4, are the irreducible components of L_{4} and the set of the singular points of L_{4} is ⋃_{i≠j} C_{i} ∩ C_{j}, i, j = 1, …, 4, that is the union of the orbits of the following Lie algebras: g_{[[0,γ]]}, γ ∈ ℂ, g_{[[γ+1,γ]]}, γ ∈ ℂ, g_{[[0]]}, g_{c}, c = 0, 2, a_{5}, a_{6}, a_{7}, a_{8}.

Basili 407

By the equations of the space of 2-cocycles of a Lie algebra we have found that the dimension of the tangent space to L_{4} in g_{[[β,γ]]}, β = 0 or β = γ + 1, [[β, γ]] ≠ [[0, 1]], [[0, −1]], [[0, 0]], g_{[[0]]}, g_{c}, c = 0, 2, a_{5}, a_{6} is 12. It is 13 in g_{[[0,1]]}, g_{[[0,−1]]}, g_{[[0,0]]}. In a_{7} and a_{8} it is 18 and 14 respectively.

Acknowledgements

I thank Lucio Guerra for introducing me to the subject of this paper and for some useful suggestions. Moreover I thank Nicola Ciccoli for some helpful discussions.

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Roberta Basili Dipartimento di Matematica e Informatica Università di Perugia Via Vanvitelli 1 06123 Perugia Italy

Received March 19, 2001 and in final form March 26, 2002