Journal of Lie Theory Volume 9 (1999) 113–123 ◯^{C}1999 Heldermann Verlag

ℤ-gradations of Lie algebras and infinitesimal generators

David A. Richter

Communicated by P. Olver

Abstract. In this paper we arrive at explicit formulae for the infinitesimal generators of the action of a complex simple Lie group G on the manifold M = G/P where P is a maximal parabolic subgroup. These formulae are obtained by assuming that local coordinates on M are furnished by the nilpotent subalgebra 𝔫 complementary to the maximal parabolic subalgebra 𝔭 corresponding to P. For the classical isogeny classes A_{r}, B_{r}, C_{r}, and D_{r}, the components of the infinitesimal generators are never worse than quartic polynomials in the coordinate functions, but for the exceptional cases, G_{2} , F_{4} , and E_{r}, higher-degree polynomials frequently occur.

1. Introduction

This note explains a table (at the end of this paper) of expressions I obtained and outlines my solution to a problem associated with infinitesimal actions: Suppose G is a complex simple Lie group and P is a maximal parabolic subgroup. The group G acts transitively on the homogeneous space M = G/P by translation of right cosets. We are interested in giving an infinitesimal version of this action explicitly. That is, we know that, given local coordinates on the manifold M, we can write down the infinitesimal generators of this action by giving a realization of the Lie algebra 𝔤 by vector fields where the isotropy subalgebra is the Lie subalgebra for P. We wish to find explicit formulae for the components of these generators. It turns out that this has a purely algebraic solution, (not surprisingly since this is a local problem), but the algebra gets somewhat intractable in some cases; in fact, new complications arise in the computations only when we want to consider the five exceptional simple Lie algebras. The algebraic solution involves the construction of various “tensor”-polynomials, denoted by F_{j}^{i}, which are constructed using the elementary Schur polynomials. It is interesting that the tensors F_{j}^{i} can be constructed in a universal way, independent of the choice of 𝔤. Thus it is the author’s belief that it may be possible to extend these results to the more general case when 𝔤 is an indecomposable Kac-Moody-Lie algebra.

I first encountered this problem when studying primitive actions in connection with quasi-exactly solvable systems, [3]. (Recall that a transitive action

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag of a Lie group G on a manifold M is primitive if the action admits no invariant foliations.) According to a result of Golubitsky [2], on the local level, if 𝔤 is simple and the isotropy subalgebra 𝔭 of the primitive action is not reductive, then 𝔭 is necessarily maximal parabolic. In the quasi-exact-solvability picture, it is useful to have infinitesimal realizations of these actions by spaces of vector fields, and thus the current problem presents itself. Using these actions, it is possible to construct, for each maximal parabolic subalgebra, a series of finite-dimensional representations of the Lie algebra 𝔤, confirming a conjecture by the authors of [3] concerning “quantization of cohomology”. For these actions, however, this is a somewhat trivial result beyond the scope of the current problem, so we will not address this here.

2. Notation and Standard Results

First we need to make some well-known [1, 6, 9] comments about simple Lie algebras and ℤ-gradations thereof. Suppose 𝔤 is a complex simple Lie algebra. As usual, 𝔤 has associated with it a Cartan subalgebra 𝔥, a set Δ ⊂ 𝔥^{∗} of roots, and a set {X_{α} : α ∈ Δ} of root vectors. Choosing a hyperplane Ω ⊂ 𝔥^{∗} not containing any roots partitions Δ into sets Δ^{+} and Δ^{−} of positive roots and negative roots respectively. There is a maximal ℤ^{+}-spanning set Σ^{+} for Δ^{+} known as the set of simple roots. The corresponding set for Δ^{−} is Σ^{−}, which happens to be the same as −Σ^{+}. The cardinality r of Σ^{+} is known as the rank of 𝔤 and this is also equal to the dimension of the Cartan subalgebra 𝔥. Given a positive root α ∈ Δ^{+}, there is a unique way to express

α = ∑ n_{i}α_{i}

as a sum of the simple roots α_{i}. The sum l(α) = ∑ n_{i} is known as the level of α, and there is a unique highest root with maximal level. Similarly, the lowest root has the lowest level, and it is the negative of the highest root.

Having chosen a Cartan subalgebra 𝔥 and a hyperplane Ω ⊂ 𝔥^{∗} not containing any roots, there are some important subalgebras known as parabolic subalgebras. To identify these we first form the Borel subalgebra 𝔟 as the sum of 𝔥 and all of the root vectors X_{α} with α ∈ Δ^{+}. A subalgebra 𝔭 is parabolic if it contains 𝔟. It is not difficult to see that the parabolic subalgebras correspond exactly with subsets of Σ^{−}. That is, suppose Σ′ ⊂ Σ^{−}. Then there is a parabolic subalgebra which contains all of the root vectors X_{α} with α ∈ Σ′. Thus, the parabolic subalgebras comprise a lattice with 2^{r} vertices, partially ordered by inclusion. The r parabolic subalgebras directly beneath 𝔤 by the partial ordering are known as maximal parabolic subalgebras. Evidently, we can construct a maximal parabolic subalgebra by first choosing a simple root, say α_{0} ∈ Σ^{−}, and forming the parabolic subalgebra which contains the root vectors for all simple roots except α_{0}. The maximal parabolic subalgebras are interesting because each one of them induces a ℤ-gradation of 𝔤:

Theorem 2.1. Let 𝔤 be a complex simple Lie algebra and 𝔭 be a maximal parabolic subalgebra. Then 𝔤 has a decomposition

𝔤 = ⨁_{j∈ℤ} V_{j}

where (a) V_{1} ≠ 0, (b) 𝔭 is the sum of all V_{j} with j ≥ 0, (c) [V_{i}, V_{j}] ⊂ V_{i+j}, (in particular, V_{0} acts on each V_{j} via the adjoint action), (d) V_{0} = 𝔤_{0} ⊕ ℂE where E acts trivially on 𝔤_{0}, and (e) this decomposition has exactly 2n_{0} + 1 non-zero terms V_{j}, where n_{0} is the coefficient of α_{0} in the highest root.

Proof. Parts (a)-(d) are shown in [2], so we will prove only part (e). Notice that corresponding to this decomposition is a partition of the root system

Δ = ⋃_{j∈ℤ} Δ_{j},

where α ∈ Δ_{j} iff X_{α} ∈ V_{j}. Thus, showing that there are 2n_{0} + 1 non-zero terms V_{j} is equivalent to showing that there are 2n_{0} + 1 non-trivial subsets Δ_{j}. Assume the simple roots are Σ^{+} = {α_{0}, ..., α_{r}} (so that 𝔤 has rank r + 1) and that 𝔭 is the maximal parabolic subalgebra obtained by not including −α_{0} in Δ_{0}. Suppose n_{0}α_{0} + ⋯ n_{r}α_{r} is the highest root. Then for every integer j ∈ [−n_{0}, n_{0}], Δ contains a root of the form jα_{0} + α for some α ∈ Δ_{0}. This shows that each such Δ_{j} is non-trivial and if |j| > n_{0}, then Δ_{j} is trivial. Thus there are exactly 2n_{0} + 1 non-trivial subsets Δ_{j}.

Later, we will also need the following notation: We use Δ_{±} to denote the union ⋃_{j>0} Δ_{±j}. Thus, Δ_{0} ∪ Δ_{+} is the set of roots corresponding to the maximal parabolic subalgebra 𝔭 and Δ_{−} is the set of roots corresponding to the complementary nilpotent subalgebra 𝔫.

3. Infinitesimal Actions on Homogeneous Spaces

We are interested in giving a realization of 𝔤 in terms of vector fields on some manifold where 𝔭, a maximal parabolic subalgebra, is the isotropy subalgebra for the action. This means we must locally parameterize the manifold G/P with some neighborhood of the origin in ℂ^{d} for some d. There is a natural choice for this space, namely the complementary nilpotent subalgebra 𝔫. Notice that we have a local diffeomorphism [5]

𝔫 → N → G/P,

where the first arrow is the exponential map and the second is the natural projection onto right cosets σ ↦ P σ. Let v ∈ 𝔤. In order to find the corresponding infinitesimal generator of the right-translation action on G/P, we differentiate the action of the one-parameter subgroup g = exp(tv) on 𝔫 and evaluate at t = 0. We have a basis for 𝔫, namely the set of root vectors X_{α} for α ∈ Δ_{−}. Thus, we assume local coordinates are furnished by prescribing that

Z = ∑_{α∈Δ_{−}} x_{α}X_{α}

be an arbitrary point in 𝔫. The corresponding point in N is σ = exp(Z). Assume that the one-parameter subgroup translates Z to

W = Z · exp(tv) = Z + tF + o(t^{2}),

where

F = ∑_{α∈Δ_{−}} f_{α}X_{α}

is another element of 𝔫, the functions f_{α} being smooth functions in the coordinate functions x_{α}, α ∈ Δ_{−}. The corresponding infinitesimal generator is then

v̂ = ∑_{α∈Δ_{−}} f_{α} \frac{∂}{∂x_{α}} .

We know that we have a local diffeomorphism, so we can assert that a unique F exists for each v. We are now interested in determining F explicitly. Suppose the element in N corresponding to W is τ = exp(W). The group G acts on G/P by right-translation of cosets, so we must impose the condition that σg and τ correspond to the same right coset. That is,

\begin{aligned}σg ∼ τ & iff & P σg = P τ \\ & iff & σgτ^{−1} ∈ P. \end{aligned}

This is precisely the constraint we need on τ to determine F. We can now state:

Proposition 3.1. Let z(v) = ad_{Z}(v) = [Z, v]. For sufficiently small Z ∈ 𝔫^{−} and sufficiently small t, exp(Z) exp(tv) exp(−Z − tF + o(t^{2})) is an element of P iff

\frac{e^{z} − 1}{z}(F) = e^{z}(v)|_{𝔫}  (1)

and F ∈ 𝔫.

Remark 3.2. Looking at this constraint, we can see that for any v ∈ 𝔤, the functions f_{α} are always polynomials, so it may be useful to introduce a little notation. The space 𝔫 has a vector-space dual 𝔫^{∗}, which is spanned by the coordinate functions x_{α} for α ∈ Δ_{−}. Let us denote the ring Sym(𝔫^{∗}) of polynomials in the coordinate functions by ℂ [x_{α}]. Given v ∈ 𝔤, we construct the infinitesimal generator v̂ in two steps. First we determine F (v) as some element of the space ℂ [x_{α}]⊗𝔫. Then we identify the space 𝔫 with the dual of the space 𝔫^{∗} in order to write down a vector field in differential-operator form. The infinitesimal generator v̂ should therefore be considered (as usual) as lying in the space

Hom(𝔫^{∗}, ℂ [x_{α}])

of derivations on 𝔫^{∗}. Notice that we have two isomorphic vector spaces with drastically differing Lie-algebra structures. The first, ℂ [x_{α}] ⊗ 𝔫 is nilpotent, and the second Hom(𝔫^{∗}, ℂ [x_{α}]) contains a faithful image of 𝔤.

Proof. As before, set σ = exp(Z), g = exp(tv) = I + tv + o(t^{2}), and τ = exp(Z + tF + o(t^{2})). We must show that the constraint is equivalent to having σgτ^{−1} ∈ P. Notice that one may write

τ^{−1} = exp(−Z − tF + o(t^{2}))
= I − (Z + tF) + \frac{1}{2} [Z^{2} + t(F Z + ZF)] + ⋯ + o(t^{2})
= I − Z + \frac{1}{2}Z^{2} − ⋯ − tG + o(t^{2})
= exp(−Z) − tG + o(t^{2})

where, for brevity, G = F − \frac{1}{2}(F Z + ZF) + \frac{1}{6}(F Z^{2} + ZF Z + Z^{2}F) + · · ·. Now write

σgτ^{−1} = exp(Z) exp(tv) exp(−Z − tF + o(t^{2}))
= exp(Z)(I + tv) [exp(−Z) − tG] + o(t^{2})
= [exp(Z) + t exp(Z)v] [exp(−Z) − tG] + o(t^{2})
= I + t [− exp(Z)G + exp(Z)v exp(−Z)] + o(t^{2}).

Now it should be clear, since we want σgτ^{−1} ∈ P, that we must have

exp(Z)G = exp(Z)v exp(−Z)|_{𝔫} .

One will quickly see that

exp(Z)G = \frac{e^{z} − 1}{z}(F)

and

exp(Z)v exp(−Z) = e^{z}(v),

and these are the terms which appear in the proposition.

4. The Algebraic Solution

The constraint (1) developed in the preceeding section shows that there is a welldefined function F : 𝔤 → ℂ [x_{α}] ⊗ 𝔫 which, after we identify 𝔫 with differentiation operators, imbeds 𝔤 as a set of local vector fields. We now wish to learn more about the nature of this function F. It turns out that F is much easier to study if we restrict it to the subspaces V_{j} in our ℤ-gradation of 𝔤. To see this, we will need a bit of algebra.

First, let A be the vector space of finite linear combinations of the symbols {z_{i} : i = 1, 2, 3, ...} and let T (A) = ⨂ A be the tensor algebra generated by A. We define a Lie bracket on T (A) by writing

[u, v] = u ⊗ v − v ⊗ u,

for u, v ∈ T (A). We can construct a representation of T (A) on ℂ [x_{α}] ⊗ 𝔤 as follows. Notice that we have projections π_{i} : ℂ [x_{α}] ⊗ 𝔤 → ℂ [x_{α}] ⊗ V_{i}. Suppose we set Z_{i} = π_{−i}(Z) where Z is defined as before. Evidently we have

Z_{i} = ∑_{α∈Δ_{−i}} x_{α}X_{α}.

For a generator z_{i} in A and a vector v ∈ ℂ [x_{α}] ⊗ 𝔤, we set

z_{i}(v) = [Z_{i}, v] ,

and for a decomposable tensor v_{1} ⊗ ⋯ ⊗ v_{n} ∈ T (A), set

v_{1} ⊗ ⋯ ⊗ v_{n}(v) = v_{1}(⋯ (v_{n}(v)) ⋯).

Linearly extending this gives our action of T (A) on ℂ [x_{α}] ⊗ 𝔤. By construction, this action preserves Lie brackets. Obviously, since z_{i} maps ℂ [x_{α}]⊗V_{j} → ℂ [x_{α}]⊗ V_{j−i}, this action restricts to the subspace ℂ [x_{α}]⊗𝔫. Finally, the Lie algebra T (A) has a ℤ-gradation

T (A) = ∑_{i>0} T^{i}(A),

where T^{i}(A) is spanned by all decomposable tensors whose subscripts sum to i. Notice that if u lies in T^{i}(A), then u(ℂ [x_{α}] ⊗ V_{j}) ⊂ ℂ [x_{α}] ⊗ V_{j−i}. Let us use ρ^{i} to denote the natural projections T (A) → T^{i}(A).

For later convenience, we will need elements R_{i}, S_{i}, and T_{i} of T (A) defined as follows. Write z = ∑_{i>0} z_{i} and e^{z} = 1 + z + \frac{1}{2}z ⊗ z + ⋯, and set

R_{i} = S_{i} = T_{i} = 0 if i < 0,

and S_{i} = ρ^{i} (e^{z}) ,

R_{i} = ρ^{i} (\frac{z}{e^{z} − 1}) ,

and

T_{i} = ρ^{i} (\frac{e^{z} − 1}{z})

for i ≥ 0. Notice that if we imbed the ring of polynomials Sym(A) in the tensor algebra T (A) in the natural way, then the S_{i} defined here are identified with the usual Schur polynomials defined by

∑_{i∈ℤ} S_{i}(z)x^{i} = exp ∑_{i>0} z_{i}x^{i}.

Suppose v ∈ 𝔤 and recall the algebraic criterion that F (v) be an infinitesimal generator:

\frac{e^{z} − 1}{z}(F (v)) = e^{z}(v)|_{𝔫} .

Given an arbitrary v ∈ 𝔤, it is not so easy to find the element F ∈ ℂ [x_{α}] ⊗ 𝔫 which satisfies this constraint. However, under this algebraic framework, we do have the following:

Theorem 4.1. If v ∈ V_{j}, then there is an element F_{j} ∈ T (A) such that

(\frac{e^{z} − 1}{z} ⊗ F_{j}) (v) = e^{z}(v)|_{𝔫} ,

and F_{j}(v) satisfies the constraint (1). The element F_{j} ∈ T (A) is given by

F_{j} = ∑_{i>0} F_{j}^{i},

where

F_{j}^{i} = ∑_{k=0}^{i−1} R_{k} ⊗ S_{i+j−k}.  (2)

Proof. Suppose v ∈ V_{j}. First of all, notice that since v ∈ V_{j}, we have F_{j}(v) ∈ ℂ [x_{α}] ⊗ 𝔫. We can also show that F_{j}(v) is the solution to (1): Notice that R_{0} = T_{0} = 1, so we can write

∑_{k=0}^{i−1} T_{k} ⊗ F_{j}^{i−k} = S_{i+j},

simply inverting the linear system (2). Now, notice that for i > 0,

\begin{aligned}π_{−i} (\frac{e^{z}−1}{z}(F_{j}(v))) & = & (ρ^{i+j} (\frac{e^{z}−1}{z} ⊗ F_{j})) (v) \\ & = & (∑_{k=0}^{i−1} T_{k} ⊗ F_{j}^{i−k}) (v), \end{aligned}

and, because v ∈ V_{j},

π_{−i}(e^{z}(v)|_{𝔫}) = π_{−i}(e^{z}(v))
= (ρ^{i+j}(e^{z}))(v)
= S_{i+j}(v).

Since π_{−i} (\frac{e^{z}−1}{z}(F_{j}(v))) and π_{−i}(e^{z}(v)|_{𝔫}) agree for all i > 0, we have

\frac{e^{z} − 1}{z}(F_{j}(v)) = e^{z}(v)|_{𝔫} .

This theorem gives us an algorithm for obtaining the infinitesimal generators of the right-multiplication action of G on G/P where P is a maximal parabolic subgroup. Start with a complex simple Lie algebra 𝔤 and a maximal parabolic subalgebra 𝔭. Then for each piece V_{j} in the induced ℤ-gradation, apply the appropriate tensor F_{j} to each element v of a spanning set for V_{j}. The number of different tensors F_{j} we need to do this is identical to the number of non-zero terms V_{j} in our ℤ-gradation, so this is a task reasonable enough to ask, say, Mathematica to perform. Also, if we know more about the structure of 𝔤, we can simplify our computations somewhat.

As a first specialization, consider the case when the choice of 𝔭 induces a gradation with at most five non-zero terms. For example, this always occurs when 𝔤 lies in one of the classical families {𝔞, 𝔟, 𝔠, 𝔡} of complex simple Lie algebras. For these cases, (a) z_{i} always acts trivially when i > 2, and (b) the actions of z_{1} and z_{2} commute (since [Z_{1}, Z_{2}] = 0). Condition (a) says that when writing down our solutions, we can set z_{i} = 0 when i > 2 and condition (b) says that we don’t have to pay attention to the order in which z_{1} and z_{2} are written; that is, z_{1} ⊗ z_{2} acts in a manner identical to that of z_{2} ⊗ z_{1}, so we write z_{1} ⊗ z_{2} = z_{2} ⊗ z_{1} = z_{1}z_{2}. Thus we have

Proposition 4.2. If 𝔤 is a complex simple Lie algebra and 𝔭 is a maximal parabolic subalgebra inducing a ℤ-gradation with at most five non-zero terms V_{j}, then the formulae for the infinitesimal generators F_{j}(v) for v ∈ V_{j} are given in the following table:

\begin{matrix}j & F_{j}(v) \\ −2 & v \\ −1 & (1 + \frac{1}{2}z_{1})(v) \\ 0 & (z_{1} + z_{2})(v) \\ 1 & (z_{2} + \frac{1}{2}z_{1}^{2} + \frac{1}{2}z_{1}z_{2} − \frac{1}{12}z_{1}^{3})(v) \\ 2 & (z_{1}z_{2} + \frac{1}{6}z_{1}^{3} + \frac{1}{2}z_{2}^{2} − \frac{1}{24}z_{1}^{4})(v) \end{matrix}

(The solutions as elements F_{j}^{i} of T (A) may be found in the table at the end of this paper, if the reader is interested.)

Frequently the ℤ-gradation has exactly three non-zero terms. In this case, z_{2} acts trivially and z = z_{1} + z_{2} + ⋯ acts just like z_{1}. Thus, restricting the preceeding case, we have

Proposition 4.3. Suppose 𝔤 = V_{−1} ⊕V_{0} ⊕V_{1} is a ℤ-gradation and 𝔭 = V_{0} ⊕V_{1} is maximal, parabolic. Then (a) if v ∈ V_{−1} then F (v) = v, (b) if v ∈ V_{0}, then F (v) = z(v), and (c) if v ∈ V_{1}, then F (v) = \frac{1}{2}z^{2}(v).

These formulae yield the familiar realization

𝔰𝔩_{2}ℂ ≅ \{\frac{∂}{∂x}, 2x\frac{∂}{∂x}, −x^{2} \frac{∂}{∂x}\} ,

the differentiated action of SL_{2}ℂ on ℂP^{1}. The cases with three non-zero terms are interesting also because these cases coincide with the classification of irreducible Hermitian symmetric spaces [5].

5. An Example

We now wish to illustrate the application of these formulae to the case when 𝔤 is the Lie algebra 𝔟_{2} and the choice of a maximal parabolic subalgebra 𝔭 induces a ℤ-gradation with exactly five non-zero terms V_{j}. Suppose 𝔤 has the basis {X_{1}, X_{2}, X_{3}, X_{4}, Y_{1}, Y_{2}, Y_{3}, Y_{4}, H_{2}, H_{4}} where H_{2} and H_{4} span a Cartan subalgebra and the remaining eight vectors are root vectors for this Cartan subalgebra. In order to perform any computations, we need to know the commutation relations among the elements of the basis. To determine these structural constants, assume the commutation relations are in accordance with choosing 𝔤 as spanned by the following set of 4-by-4 matrices:

\begin{aligned}X_{1} & = & E_{21} − E_{34}, & Y_{1} & = & E_{12} − E_{43}, \\ X_{2} & = & E_{31}, & Y_{2} & = & E_{13}, \\ X_{3} & = & E_{41} + E_{32}, & Y_{3} & = & E_{14} + E_{23}, \\ X_{4} & = & E_{42}, & Y_{4} & = & E_{24}, \\ H_{2} & = & −E_{11} + E_{33}, & H_{4} & = & −E_{22} + E_{44}. \end{aligned}

(This is quite arbitrary; all we need to know are the structural constants, so we will get by with any faithful representation of 𝔟_{2}.) Set

V_{−2} = span{Y_{2}},
V_{−1} = span{Y_{1}, Y_{3}},
V_{0} = span{H_{2}, H_{4}, X_{4}, Y_{4}},
V_{1} = span{X_{1}, X_{3}},
V_{2} = span{X_{2}}.

It is easy to check that 𝔭 = V_{0} ⊕ V_{1} ⊕ V_{2} is a maximal parabolic subalgebra and that this defines a ℤ-gradation of 𝔤.

We now wish to apply our formulae to obtain a realization of 𝔤 by vector fields. First we write

Z = Z_{1} + Z_{2}

where Z_{1} = x_{1}Y_{1} + x_{3}Y_{3} ∈ ℂ [x_{1}, x_{2}, x_{3}] ⊗ V_{1} and Z_{2} = x_{2}Y_{2} ∈ ℂ [x_{1}, x_{2}, x_{3}] ⊗ V_{2}. By definition, z_{i}(v) = [Z_{i}, v] for i = 1, 2, and, since [Z_{1}, Z_{2}] = 0, we do not need to pay any attention to the order in which the z_{i} are applied to any vector in ℂ [x_{1}, x_{2}, x_{3}] ⊗ 𝔤. Let us determine the infinitesimal generator corresponding to X_{1}. Since X_{1} ∈ V_{1}, we must apply the tensors F_{1}^{i} to obtain the infinitesimal generator. According to the table,

F (X_{1}) = F_{1}^{1}(X_{1}) + F_{1}^{2}(X_{1})
= (z_{2} + \frac{1}{2}z_{1}^{2} + \frac{1}{2}z_{1}z_{2} − \frac{1}{12}z_{1}^{3})(X_{1})
= [Z_{2}, X_{1}] + \frac{1}{2}[Z_{1}, [Z_{1}, X_{1}]] + \frac{1}{2}[Z_{1}, [Z_{2}, X_{1}]] − \frac{1}{12}[Z_{1}[Z_{1}[Z_{1}, X_{1}]]]
= −x_{1}^{2}Y_{1} − x_{1}x_{2}Y_{2} − (x_{2} + x_{1}x_{3})Y_{3}.

Thus the infinitesimal generator corresponding to X_{1} is

X̂_{1} = −x_{1}^{2} \frac{∂}{∂x_{1}} − x_{1}x_{2} \frac{∂}{∂x_{2}} − (x_{2} + x_{1}x_{3})\frac{∂}{∂x_{3}} .

Collecting the results when we apply the appropriate tensor to each element of our basis, we have

\begin{aligned}X̂_{1} & = & −x_{1}^{2}p_{1} − x_{1}x_{2}p_{2} − (x_{2} + x_{1}x_{3})p_{3} & Ŷ_{1} & = & p_{1} − x_{3}p_{2} \\ X̂_{2} & = & −x_{1}x_{2}p_{1} − x_{2}^{2}p_{2} − x_{2}x_{3}p_{3} & Ŷ_{2} & = & p_{2} \\ X̂_{3} & = & (x_{2} − x_{1}x_{3})p_{1} − x_{2}x_{3}p_{3} − x_{3}^{2}p_{3} & Ŷ_{3} & = & x_{1}p_{2} + p_{3} \\ X̂_{4} & = & x_{3}p_{1} & Ŷ_{4} & = & x_{1}p_{3} \\ Ĥ_{2} & = & x_{1}p_{1} + 2x_{2}p_{2} + x_{3}p_{3} & Ĥ_{4} & = & −x_{1}p_{1} + x_{3}p_{3} \end{aligned}

where, for brevity, we have written p_{i} in place of \frac{∂}{∂x_{i}}. (Note: Lie gave this realization of 𝔟_{2} in his classification of transitive actions on ℂ^{3} [8].)

6. Conclusion

As a concluding remark, it may be interesting to note that the associated spaces ℂ [x_{α}] carry a natural generalization of “degree”. Notice that in our example we obtained the operator

Ĥ_{2} = x_{1}p_{1} + 2x_{2}p_{2} + x_{3}p_{3}.

If we assign the monomial x_{2} a degree of 2 and the monomials x_{1}, x_{3} each degree 1, (as our ℤ-gradation suggests), then Ĥ_{2} resembles Euler’s degree operator on homogeneous polynomials:

Ĥ_{2}(f) = deg(f) · f,

where f lives in ℂ [x_{1}, x_{2}, x_{3}]. Actually, this occurs in general. Recall that V_{0} is the sum of a semisimple ideal 𝔤_{0} with a central element E. By properly scaling E, it is not hard to show that the corresponding infinitesimal generator is

Ê = ∑_{j>0} ∑_{α∈Δ_{−j}} jx_{α} \frac{∂}{∂x_{α}} ,

which, operating on ℂ [x_{α}], generalizes the usual degree operator.

Table 1.

\begin{matrix}j & F_{j}^{1} & F_{j}^{2} \\ −2 & 0 & 1 \\ −1 & 1 & \frac{1}{2}z_{1} \\ 0 & z_{1} & z_{2} \\ 1 & z_{2} + \frac{1}{2}z_{1}^{2} & \frac{1}{2}z_{1}z_{2} − \frac{1}{12}z_{1}^{3} \\ 2 & z_{1}z_{2} + \frac{1}{6}z_{1}^{3} & \frac{1}{2}z_{2}^{2} − \frac{1}{24}z_{1}^{4} \end{matrix}

References

[1] Fulton, W. and J. Harris, “Representation Theory, a First Course, ” Springer-Verlag, New York, 1991.

[2] Golubitsky, M, Primitive actions and maximal subgroups of Lie groups, J. Diff. Geom. 7 (1972), 175–191.

[3] González-López, A., N. Kamran, and P. Olver, Quasi-exact solvability, Contemp. Math. 160 (1994), 113–139.

[4] Guillemin, V. and S. Sternberg, An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc. 70 (1964) 16–47.

[5] Helgason, S., “Differential Geometry, Lie Groups, and Symmetric Spaces,” Academic Press Inc., New York, 1978.

[6] Humphreys, J. E., “Introduction to Lie Algebras and Representation Theory”,, Springer-Verlag, New York, 1972.

[7] Kac, V. G., “Infinite dimensional Lie algebras, 3rd ed.,” Cambridge University Press, Cambridge, 1990.

[8] Lie, S., “Theorie Der Transformationsgruppen,” in:, Gesammelte Abhandlungen, Vol. 6, B. G. Teubner, Leipzig, 1924.

[9] Varadarajan, V., “Lie Groups, Lie Algebras, and Their Representations,” Springer-Verlag, New York, 1984.

School of Mathematics University of Minnesota 206 Church St SE Minneapolis MN 55455 drichter@math.umn.edu

Received January 9, 1998 and in final form September 23, 1998