ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Pages 7–20 (March 1, 2000) S 1079-6762(00)00075-5

FAMILY ALGEBRAS

A. A. KIRILLOV

(Communicated by Svetlana Katok)

Abstract. A new class of associative algebras is introduced and studied. These algebras are related to simple complex Lie algebras (or root systems). Roughly speaking, they are finite dimensional approximations to the enveloping algebra U(𝔤) viewed as a module over its center.

It seems that several important questions on semisimple algebras and their representations can be formulated, studied and sometimes solved in terms of our algebras.

Here we only start this program and hope that it will be continued and developed.

0. Introduction

The aim of this paper is to introduce and study a new class of associative algebras: the so-called family algebras. We assume that the reader is acquainted with the general background of the theory of semisimple Lie algebras (see e.g. [H]).

Let 𝔤 be a simple complex Lie algebra with the canonical decomposition

𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}.

We denote by P (resp. Q) the weight (resp. root) lattice in 𝔥^{∗} and by P_{+} (resp. Q_{+}) the semigroup generated by fundamental weights ω_{1}, ω_{2}, …, ω_{l} (resp. by simple roots α_{1}, α_{2}, …, α_{l}).

For every λ ∈ P_{+} let (π_{λ}, V_{λ}) be an irreducible representation of 𝔤 with highest weight λ. We denote by d(λ) the dimension of V_{λ}.

Let λ^{∗} denote the highest weight of the dual (or contragredient) representation which acts in V_{λ}^{∗} by π_{λ^{∗}}(X) = −(π_{λ}(X))^{∗}. It is clear that d(λ) = d(λ^{∗}).

The space End V_{λ} is isomorphic to the matrix space Mat_{d(λ)}(ℂ) and has a 𝔤module structure defined by

X · A = [π_{λ}(X), A].

Recall that the symmetric algebra S(𝔤) and the enveloping algebra U(𝔤) also have (isomorphic) 𝔤-module structures.

Let G be a connected and simply connected Lie group with Lie(G) = 𝔤. The action of 𝔤 on End V_{λ}, S(𝔤) and U(𝔤) gives rise to the corresponding action of

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

Received by the editors December 31, 1999.

2000 Mathematics Subject Classification. Primary 15A30, 22E60.

Key words and phrases. Enveloping algebras, invariants, representations of semisimple Lie algebras.

G. We define two kinds of family algebras: the classical algebra C_{λ}(𝔤) and the quantum algebra Q_{λ}(𝔤) by

C_{λ}(𝔤) := (End V_{λ} ⊗ S(𝔤))^{G}, Q_{λ}(𝔤) := (End V_{λ} ⊗ U(𝔤))^{G}. (1)

We hope to apply the theory of family algebras to several important questions on semisimple algebras and their representations.

Here we only start this program and formulate some preliminary results.

1. Generalities about family algebras

1.1. Main definitions. First of all, we make the basic definition (1) more visual and practical. For a given pair (𝔤, λ) let us consider the set of all matrices A of order d(λ) with elements from S(𝔤) or from U(𝔤).

We can define two different actions of an element g ∈ G on A:

— the right action via conjugation by the matrix π(g):

A ↦ A · g := π(g)^{−1}Aπ(g);

— the left action by application of Ad(g) to all matrix elements of A:

A ↦ g · A,  where  (g · A)_{ij} = Ad(g)A_{ij}.

The statement that for any g ∈ G the results of these two actions on A coincide (i.e. A · g = g · A) is equivalent to the claim that A belongs to the family algebra.

The infinitesimal version of this condition is

[A, π(X)]_{ij} = ad X(A_{ij}). (2)

For the classical family algebras the condition (2) can be rewritten in terms of the canonical Poisson structure on 𝔤^{∗}. For this end we identify S(𝔤) with Pol(𝔤^{∗}) and consider the basic elements X_{1}, …, X_{n} of 𝔤 as coordinates on 𝔤^{∗}. By ∂^{i} we denote the partial derivative with respect to X_{i}. Then the Poisson bracket has the form

{f_{1}, f_{2}} = c_{ij}^{k} X_{k} ∂^{i}f_{1} ∂^{j}f_{2}.

The condition (2) is equivalent to

(2′)  [A, π(X)]_{ij} = {X, A_{ij}}  or simply  [A, π(X)] = {X, A}.

We shall use yet another interpretation of the definition of classical family algebras. Consider elements of C_{λ}(𝔤) as polynomial matrix-valued functions on 𝔤^{∗}. Then the condition that a function A corresponds to an element of the family algebra means

A(K(g)F) = π_{λ}(g)A(F)π_{λ}(g)^{−1}. (2′′)

The useful corollary of this definition is

Theorem S. Assume that π_{λ} has a simple spectrum (i.e. all weights have multiplicity 1). Then C_{λ}(𝔤) is commutative.

Proof. It is enough to check the commutativity of A(F) and B(F) for generic F. Since a generic element of 𝔤^{∗} ≅ 𝔤 is conjugate to an element of the Cartan subalgebra, we can assume that F ∈ 𝔥. But then (2′′) implies that the values of A(F) and B(F) are diagonal matrices, hence they commute.

Now we recall some properties of Lie algebras 𝔤 which possess an Ad(G)-invariant symmetric bilinear form (· , ·). For a simple Lie algebra 𝔤 such a form always exists and is unique up to a scalar factor. We will usually use the form

(X, Y) = tr (π(X)π(Y)), (3)

where π is a non-trivial irreducible representation of 𝔤 of minimal dimension.

The Ad(G)-invariant form allows one to identify adjoint and coadjoint modules and thus identify S(𝔤) not only with the polynomial algebra P (𝔤^{∗}) but also with the algebra S(𝔤^{∗}), which we regard as the algebra of differential operators on 𝔤^{∗} with constant coefficients. We denote by Δ(P) the differential operator on 𝔤^{∗} associated with P ∈ S(g).

In particular, if {X_{i}} and {X^{i}} are any dual bases in 𝔤 with respect to the chosen Ad-invariant bilinear form on 𝔤, then we have

(4)  Δ(X^{i}) = ∂^{i}  and  X_{i}Δ(X^{i}) = E (the Euler operator).

The form (· , ·) extends to a non-degenerate Ad(G)-invariant form on S(𝔤) ≅ P (𝔤^{∗}) (also denoted by parentheses) via

(P, Q) = Δ(P)Q |_{X=0} . (5)

One can check that if {X_{i}}_{i=1,2,...,n=dim 𝔤} is an orthonormal basis in 𝔤, then the monomials

\frac{X^{k}}{\sqrt{k!}} := \frac{X_{1}^{k_{1}} ⋯ X_{n}^{k_{n}}}{\sqrt{(k_{1})! ⋯ (k_{n})!}}

form an orthonormal basis in S(𝔤).

The bilinear form (5) extends further to the space Mat_{d(λ)}(ℂ) ⊗ S(𝔤) (which can be viewed as the space of matrix-valued polynomials on 𝔤^{∗}). For decomposable elements this extension is given by

(A_{0} ⊗ P, B_{0} ⊗ Q) := tr(A_{0}B_{0})(P, Q) for A_{0}, B_{0} ∈ Mat_{d(λ}(ℂ), P, Q ∈ S(𝔤). (5′)

Note also that the extended form has the following useful property:

(AB, C) = (A, Δ(B)C). (6)

In other words, the matrix differential operator Δ(B) acting from the left is adjoint to the operator of right multiplication by B.

Lemma 1. Let P ∈ S(𝔤) be an invariant polynomial (i.e. we assume that P belongs to I(𝔤) = S(𝔤)^{G}). Then the matrix

M_{P} := π_{λ}(X_{i}) ⊗ ∂^{i}P (7)

belongs to C_{λ}(𝔤).

Proof. Direct verification of (2′).

We call M_{P} an M-type element of C_{λ}(𝔤).

Note that for P = C := \frac{1}{2}X_{i}X^{i} we get a special element

M := M_{C} = π_{λ}(X_{i}) ⊗ X^{i}, (8)

which belongs to C_{λ}(𝔤) (resp. to Q_{λ}(𝔤)) if we interpret X^{i} as an element of S(𝔤) (resp. of U(𝔤)).

The remarkable fact is Theorem M. The M-type elements belong to the center of C_{λ}(𝔤).

Proof. For A ∈ C_{λ}(𝔤), P ∈ I(𝔤) we have

[M_{P} , A] = [π_{λ}(X_{i}), A] ∂^{i}P =^{(2′),(4)} {A, X_{i}} ∂^{i}P = {A, P } = 0.

Below we use this theorem to show that many important classical family algebras are commutative.

1.2. The structure of the family algebras. Let us look more attentively at the 𝔤-module structure of End V_{λ} ≅ Mat_{d(λ)}(ℂ). It splits into irreducible components:

Mat_{d(λ)}(ℂ) = ⨁_{i} W_{i}, (9)

where W_{i} is a simple 𝔤-module with highest weight μ_{i}.

We shall call representations (π_{μ_{i}}, W_{i}) the children of π_{λ} (resp. the dominant weights μ_{i} will be called the children of λ).

In the case where all μ_{i} are different, we shall say that there are no twins.

Suppose that V_{λ} admits a G-invariant bilinear form. In other words, assume that π_{λ} belongs to orthogonal or symplectic type.

In the orthogonal case for an appropriate basis in V_{λ} all matrices π_{λ}(X), X ∈ 𝔤, are skew-symmetric. Then Mat_{d(λ)}(ℂ) as a 𝔤-module splits into symmetric and antisymmetric parts. We call boys those children that belong to antisymmetric part, and girls those that are in the symmetric part.

In the symplectic case, d(λ) = 2n is even and there is a π_{λ}(G)-invariant skewsymmetric form in V_{λ}. We can choose a basis such that this form is given by the matrix

J = (\begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix}) .

In this basis all matrices π_{λ}(X), X ∈ 𝔤, are J-symmetric, i.e. have the form JS, where S is symmetric.

The 𝔤-module Mat_{d(λ)}(ℂ) again splits into two parts: J-symmetric and Jantisymmetric. Here we also call girls those children that are in the J-symmetric part, and boys those in the J-antisymmetric part.

Note, that in both cases there is a distinguished girl—the trivial representation (π_{0}, ℂ · 1) or (π_{0}, ℂ · J).

We also observe that the adjoint representation Ad is a common child to all nontrivial representations π_{λ}; the corresponding subspace is spanned by the matrix elements of M (see (8) above). For representations of orthogonal type it is a boy, while for those of symplectic type it is a girl. We shall see below that in some cases there are twins of type Ad.

The algebra Mat_{d(λ)} ⊗ S(𝔤^{∗}) of matrix differential operators on 𝔤^{∗} acts on the algebra Mat_{d(λ)} ⊗ S(𝔤) of matrix polynomials on 𝔤^{∗} according to the rule

(A_{0} ⊗ D) · (B_{0} ⊗ P) = A_{0}B_{0} ⊗ D(P).

Since Δ is a G-equivariant map (it is an isomorphism of 𝔤-modules), we conclude that the subalgebra of G-invariant matrix operators

D_{λ}(𝔤) := (Mat_{d(λ)} ⊗ S(𝔤^{∗}))^{G}

coincides with (1 ⊗ Δ) C_{λ}(𝔤).

The action of this subalgebra preserves the subalgebra C_{λ}(𝔤) ⊂ Mat_{d(λ)} ⊗ S(𝔤). For our goals the most important examples of elements of D_{λ}(𝔤) are

D_{P} := Δ(M_{P}) = ∑_{i} π_{λ}(X_{i}) ⊗ Δ(∂^{i}P),

and in particular

D := Δ(M) = ∑_{i} π_{λ}(X_{i}) ⊗ ∂^{i}.

1.3. Generalized exponents. Here we recall some remarkable results mostly due to B. Kostant [K].

The first result describes the structure of 𝔤-module S(𝔤). Let I(g) = S(𝔤)^{G}. We identify I(g) with the algebra P (𝔤^{∗})^{G} of G-invariant polynomials on 𝔤^{∗}.

Let I_{+}(g) be the augmentation ideal in I(𝔤) (i.e. the ideal of polynomials vanishing at the origin). Denote by J(𝔤) the ideal in S(𝔤) generated by I_{+}(g). The space H(𝔤) of harmonic polynomials on 𝔤^{∗} is defined as the orthogonal complement to J(𝔤) in S(𝔤).

According to (6), H(𝔤) can also be defined as the space of solutions h to the system of equations

Δ(P)h = 0, P ∈ I_{+}(g). (10)

(Of course, it is enough to consider the generators of I_{+}(𝔤) in the role of P.)

Theorem 1 (Kostant). a) There is an isomorphism of graded 𝔤-modules:

S(𝔤) ≅ I(𝔤) ⊗ H(𝔤). (11)

b) Each irreducible representation π_{λ} has finite multiplicity in H(𝔤).

More precisely, if s = m_{λ}(0) is the multiplicity of the zero weight in V_{λ}, then there exist numbers e_{1}(λ), …, e_{s}(λ) (not necessarily distinct) such that π_{λ} occurs in the homogeneous components H^{e_{1}(λ)}(𝔤), …, H^{e_{s}(λ)}(𝔤).

The numbers e_{1}(λ), …, e_{s}(λ) are called the generalized exponents related to the representation π_{λ}.

Since H(𝔤) is a self-dual 𝔤-module, the generalized exponents are the same for λ and λ^{∗}.

The ordinary exponents correspond to the adjoint representation for which λ = ψ, the highest root of 𝔤, and s = l, the rank of 𝔤.

The multiplicity m_{λ}(μ) of the weight μ in V_{λ} is often denoted by K_{λμ} and called Kostka number. It has the remarkable q-analog: the so-called Kostka polynomial K_{λμ}(q), which can be expressed in terms of the q-analog of the Kostant formula for ordinary multiplicities:

K_{λμ}(q) = ∑_{w∈W} (−1)^{l(w)}Q(w(λ + ρ) − (μ + ρ) | q), (12)

where Q is the q-analog of the Kostant partition function and is defined by

∑_{ν∈Q_{+}} Q(ν | q)e^{ν} = ∏_{α>0} (1 − qe^{α})^{−1}. (13)

Theorem 2 (Hesselink). The polynomials K_{λμ}(q) for μ = 0 coincide with generating functions for generalized exponents:

K_{λ0}(q) = ∑_{i=1}^{K_{λ0}} q^{e_{i}(λ)}. (14)

We observe that though the Hesselink result looks very natural and elegant, its practical use is rather restricted. The reason is that the explicit formula for generalized exponents which follows from Hesselink result is too involved. For Lie algebras of rank ≥ 3 it is practically uncomputable.

It is very interesting to compare this approach with the so-called Gelfand-Tsetlin patterns, which label a basis in V_{λ}. These patterns have the form

m_{1,1}  m_{1,2}  . . . . . . . . .  m_{1,n−1}  m_{1,n}
m_{2,1}  m_{2,2} … m_{2,n−2}  m_{2,n−1}
…  …  …
m_{n−1,1}  m_{n−1,2}
m_{n,1}

with integers m_{ij} satisfying

m_{i,j} ≥ m_{i+1,j} ≥ m_{i,j+1}.

The weight of a vector v_{M} corresponding to a pattern M is equal to

m_{n,1}, m_{n−1,1} + m_{n−1,2} − m_{n,1}, …, ∑_{j} m_{1j} − ∑_{j} m_{2,j}.

One can show that each vector v_{M} of zero weight contributes to K_{λμ}(q) a summand of the form q^{c(M)}, where c(M) is a certain combinatorial function (the so-called charge) of the pattern M.

For the case 𝔤 = sl(4) a pattern of zero weight looks like

m_{1}  m_{2}  m_{3}  m_{4} (m_{1} + m_{2} + m_{3} + m_{4} = 0)
l_{1}  l_{2}  l_{3}  (l_{1} + l_{2} + l_{3} = 0)
p  − p

0

I succeeded to compute the final expression only in several particular cases. For example, for the pattern of the form

m  0  0  − m
l  0  − l
p  − p

we have c(M) = m + l + p, K_{λ0} = \frac{(m+1)(m+2)}{2} = \binom{m+2}{2}, and K_{λ0}(q) is equal to the sum of all monomials from the following triangle table:

q^{m}  q^{m+1} … q^{2m−1}  q^{2m}
q^{m+2}  …  q^{2m+1}
. . . . . . . . .
q^{3m−2} q^{3m−1}
q^{3m}

which leads to the expression

K_{λ0}(q) = q^{m} [\begin{matrix}m + 2 \\ 2 \end{matrix}]_{q} = q^{m} · \frac{q^{m+2} − 1)(q^{m+1} − 1)}{(q − 1)(q^{2} − 1)}

= q^{m} + q^{m+1} + 2q^{m+2} + 2q^{m+3} + 3q^{m+4} + 3q^{m+5} + ⋯

+ [\frac{m}{2}] q^{2m} + ⋯ + 2q^{3m−3} + 2q^{3m−2} + q^{3m−1} + q^{3m}.

Remark 1. The general formula for c(M) in the case of sl(4) has been recently found by my student R. Masenten. It looks like

c(M) = p + |l_{2}| + max (m_{1} + l_{1} + l_{2}, −l_{2} − l_{3} − m_{4}, m_{1} + m_{2} + p). (15)

It is a challenging problem to simplify, explain and generalize this cumbersome formula.

Remark 2. The ordinary multiplicity of the zero weight in a unirrep π_{λ} of the compact form K ≅ SU(n) of G can be written as follows. Let M(K) be the algebra of (signed) measures on K with convolution product. Pick up a maximal torus T ⊂ K and denote by δ_{T} the element of M(K) given by the normalized Haar measure on T.

Then π_{λ}(δ_{T}) is the projector to V_{λ}^{0} and

K_{λ0}(q) = tr π_{λ}(δ_{T}).

Of course, we can replace the chosen torus T in this formula by any other (they are all conjugate in K). We can also take the average over all possible tori. Then we get a measure which is absolutely continuous with respect to Haar measure and has density

f(u) = \frac{n!}{∏_{i≠j} |λ_{i} − λ_{j}|},

where λ_{i} are eigenvalues of u.

It is natural to ask if there exists a simple graded version f(u | q) of f(g) such that

K_{λ0}(q) = tr π(f(u | q)).

For the case K = SU(2) we can put

f(u | q) = \frac{1 + q}{1 − q · tr(u^{2}) + q^{2}} . (16)

1.4. Relation between family algebras and generalized exponents. Now let λ ∈ P_{+} be a dominant weight such that π_{λ} has k children with no twins. We denote by W_{1}, …, W_{k} the corresponding irreducible subspaces in Mat_{d(λ)}(ℂ) and by μ_{1}, …, μ_{k} the highest weights of corresponding representations.

We denote by p_{i} the projection to W_{i} in Mat_{d(λ)}(ℂ) and use the same notation for its extension to C_{λ}(𝔤^{∗}) (which is the restriction of p_{i} ⊗ 1).

Then the subspace C_{λ}(𝔤^{∗})_{i} := p_{i}(C_{λ}(𝔤^{∗})) = (W_{i} ⊗ S(𝔤))^{G} = (W_{i} ⊗ I(𝔤) ⊗ H(𝔤))^{G} is a free I(𝔤)-module of rank K_{μ_{i}0}. In fact, it has the grading inherited from S(𝔤) and its Poincaré series is given by

ch_{C_{λ}(𝔤^{∗})_{i}}(q) = ch_{I(𝔤)}(q) · ch_{H(𝔤)_{μ^{∗}_{i}}} (q) = \frac{K_{μ_{i}0}(q)}{∏_{k=1}^{l}(1 − q^{d_{k}})},

where d_{k}, 1 ≤ k ≤ l, are the degrees of the generators of I(𝔤). (It is well known that d_{k} = e_{k} + 1, where e_{k} = e_{k}(ψ) are ordinary exponents.)

So, we get the following result:

Theorem 1. The algebra C_{λ}(𝔤) is a free I(𝔤)-module of rank m = ∑_{i=1}^{k} m_{μ_{i}}(0).

Remark 3. Assume that all weights of the representation π_{λ} are simple (i.e. have multiplicity 1). Then the zero weight subspace in Mat_{d(λ)}(ℂ) ≅ End V_{λ} is the subspace of diagonal matrices so that m = d(λ). I do not know the meaning of the corresponding q-analog.

Theorem 2. The algebra C_{λ}(𝔤) is commutative if and only if all weights of the representation π_{λ} have multiplicity 1.

It follows from the following general result:

Theorem 3. Let K(𝔤) be the field generated by the zero weight subalgebra S(𝔤)^{0} of S(𝔤), let μ_{i} be children of λ, and δ(μ_{i}) their multiplicities. Then the algebra K_{λ}(𝔤) := C_{λ}(𝔤) ⊗_{S(𝔤)^{0}} K(𝔤) is isomorphic to the direct sum of matrix algebras:

K_{λ}(𝔤) ≅ ⨁_{i} Mat_{δ(μ_{i})}(K(𝔤)). (17)

Sketch of the proof. We shall regard elements of K_{λ}(𝔤) as rational matrix-valued functions on 𝔤^{∗} with values in Mat_{λ}(ℂ). The condition (2′′) implies that this function is uniquely determined by its restriction to 𝔥^{∗} ⊂ 𝔤^{∗}.

Conversely, this restriction can be any rational function on 𝔥^{∗} with values in the zero weight subspace Mat_{λ}(ℂ)^{0}. (It follows from the birational isomorphism G ≅ G/B × H.) But the latter is exactly the direct sum ⨁_{i} Mat_{δ(μ_{i})}(ℂ).

2. Family algebras for standard representations of classical simple Lie algebras

Here we illustrate the general theory described above by examples related to the simplest representations of classical simple Lie algebras. It can be viewed as a beautiful exercise in linear algebra which ties together many classical results.

One of interesting consequences of the results listed below is the fact that all family algebras in question are commutative. It is not true for the quantum algebras and for some classical algebras related to other representations.

2.1. The case 𝔤 = A_{n} ≅ 𝔰𝔩(n + 1, ℂ). As usual it is slightly more convenient to consider 𝔤 together with the bigger algebra 𝔤̃ = 𝔤𝔩(n + 1, ℂ).

Denote by {E_{ij}}_{1≤i,j≤n+1} the standard basis in 𝔤̃, so that π(E_{ij}) is a matrix unit with the only non-zero entry in the i-th row and j-th column.

Let π = π_{ω_{1}} be the standard representation of 𝔤̃ with the highest weight

ω_{1}(E_{ii}) = δ_{i,1}.

The irreducible subspaces of Mat_{n+1}(ℂ) are W_{0} and W_{1}, consisting respectively of scalar and traceless matrices.

It follows that π has two children: the trivial representation π_{0} and the representation π_{ψ} with the highest weight

ψ(E_{ii}) = δ_{i,1} − δ_{i,n+1}.

Note, that the restriction of π_{ψ} on 𝔤 is the adjoint representation of 𝔤. The element M̃ ∈ C_{ψ}(𝔤̃) has the form

M̃ = ∑_{i,j=0}^{n} π(E_{ij}) ⊗ E_{ji} = (\begin{matrix}E_{00} & … & E_{n0} \\ … & … & … \\ E_{0n} & … & E_{nn} \end{matrix}) .

The element M ∈ C_{π}(𝔤) is just the traceless part of M̃. For example, for n = 2 we have

M = (\begin{matrix}\frac{1}{3}(2H_{α} + H_{β}) & X_{−α} & X_{−γ} \\ X_{α} & \frac{1}{3}(H_{β} − H_{α}) & X_{−β} \\ X_{γ} & X_{β} & −\frac{1}{3}(H_{α} + 2H_{β}) \end{matrix})

where we use the standard notation: X_{α} = E_{01}, X_{β} = E_{12}, X_{γ} = E_{02}, X_{−α} = E_{10}, X_{−β} = E_{10}, X_{−γ} = E_{20}, H_{α} = E_{00} − E_{11}, H_{β} = E_{11} − E_{22}.

Theorem A. a) The algebra I(𝔤) coincides with ℂ [tr (M^{k}), 2 ≤ k ≤ n + 1] and C_{π}(𝔤) is a free I(𝔤)-module of rank n + 1 generated by elements M^{k}, 0 ≤ k ≤ n.

b) The algebra I(𝔤̃) coincides with ℂ [tr (M̃^{k})], 1 ≤ k ≤ n + 1, and C_{π}(𝔤̃) is a free I(𝔤̃)-module of rank n + 1 generated by M̃^{k}, 0 ≤ k ≤ n.

Sketch of the proof. We start with part b). By the very definition p_{0}(C_{π}(𝔤̃)) is isomorphic to I(𝔤̃): it consists of scalar matrices with elements from I(𝔤̃). So, the elements I_{k} = tr(M̃^{k}) belong to I(𝔤̃). On the other hand, it is well known that the polynomials I_{1}, …, I_{n+1} freely generate I(𝔤̃) (the main theorem on symmetric functions).

It is clear that for any C ∈ I(𝔤̃) the matrix D(C) belongs to C_{π}(𝔤̃). On the other hand, for any A ∈ C_{π}(𝔤̃) the polynomial tr (AM) belongs to I(𝔤̃). The maps α : C ↦ D(C) and β : A ↦ tr (AM) are almost reciprocal: β ∘ α = deg C for homogeneous C, hence both α and β are invertible. This proves b).

Part a) easily follows from b).

Corollary. The ordinary exponents of 𝔤 are 1, 2, …, n.

2.2. The case 𝔤 = B_{n} ≅ 𝔰𝔬(2n + 1, ℂ). Here the basic representation π = π_{ω_{1}} has dimension d(ω_{1}) = 2n + 1 and is orthogonal. So the space Mat_{2n+1}(ℂ) as a 𝔤-module splits into symmetric and antisymmetric parts. Actually, in this case there are two girls and one boy.

Namely, the symmetric part has dimension (2n+1)(n+1) and contains the trivial subrepresentation W_{0} acting on the space of scalar matrices. The complementary subspace W_{2} of traceless symmetric matrices is irreducible and has the highest weight 2ω_{1}.

The antisymmetric part W_{1} is irreducible and isomorphic to the adjoint representation with highest weight ω_{2}.

The matrix M is antisymmetric and its k-th power is symmetric for even k and antisymmetric for odd k.

Theorem B. a) The algebra I(𝔤) coincides with ℂ [tr (M^{2k}), 1 ≤ k ≤ n] and C_{π}(𝔤) is a free I(𝔤)-module of rank 2n + 1 generated by M^{k}, 0 ≤ k ≤ 2n.

b) The I(𝔤)-module p_{2}(C_{π}(𝔤)) is spanned by p_{2}(M^{2k}), 1 ≤ k ≤ n, while the I(𝔤)-module p_{1}(C_{π}(𝔤)) is spanned by M^{2k−1}, 1 ≤ k ≤ n.

We omit the proofs of this and the next theorem because they follow the same scheme as above.

Corollary. The exponents of π_{2ω_{1}} are 2, 4, …, 2n and the exponents of π_{ω_{2}} (the ordinary exponents) are 1, 3, …, 2n − 1.

2.3. The case 𝔤 = C_{n} ≅ 𝔰𝔭(2n, ℂ). This time the basic representation π = π_{ω_{1}} has dimension 2n and is symplectic. The space Mat_{2n}(ℂ) splits into 1-dimensional space W_{0} = ℂ · J, the subspace W_{2} of J-symmetric matrices orthogonal to J, and the subspace W_{1} of J-antisymmetric matrices.

Thus, there are two girls: 0 and 2ω_{1}, and one boy: ω_{2}.

The matrix M is J-symmetric and its k-th power is J-symmetric for odd k and J-antisymmetric for even k.

Theorem C. a) The algebra I(𝔤) coincides with ℂ [tr (M^{2k}), 1 ≤ k ≤ n], and C_{π}(𝔤) is a free I(𝔤)-module of rank 2n generated by M^{k}, 0 ≤ k ≤ 2n − 1.

b) The I(𝔤)-module p_{2}(C_{π}(𝔤)) is spanned by p_{2}(M^{2k}), 1 ≤ k ≤ n − 1, while the I(𝔤)-module p_{1}(C_{π}(𝔤)) is spanned by M^{2k−1}, 1 ≤ k ≤ n.

Corollary. The exponents of π_{ω_{2}} are 2, 4, …, 2n − 2, and the exponents of π_{2ω_{1}} (the ordinary exponents) are 1, 3, …, 2n − 1.

2.4. The case 𝔤 = D_{n} ≅ 𝔰𝔬(2n, ℂ). The basic representation π = π_{ω_{1}} of 𝔤 has dimension d(ω_{1}) = 2n. Here again, as in the case B_{n}, there are two girls: 0 and 2ω_{1}, and one boy: ω_{2}.

More precisely, the symmetric part has dimension n(2n + 1) and contains the trivial subrepresentation W_{0} of scalar matrices and the complementary irreducible subspace W_{2} of traceless symmetric matrices with highest weight 2ω_{1}.

The antisymmetric part W_{1} is irreducible and isomorphic to the adjoint representation with highest weight ω_{2}.

The matrix M is antisymmetric and its k-th power is symmetric for even k and antisymmetric for odd k.

Theorem D. a) The algebra I(𝔤) coincides with the polynomial algebra generated by tr (M^{2k}), 1 ≤ k ≤ n − 1, and Pf (M). The algebra C_{π}(𝔤) is a free I(𝔤)-module of rank 2n generated by elements M^{k}, 0 ≤ k ≤ 2n − 1, and M^{−1} · Pf(M).

b) The I(𝔤)-module p_{2}(C_{π}(𝔤)) is spanned by p_{2}(M^{2k}), 1 ≤ k ≤ n − 1, while the I(𝔤)-module p_{1}(C_{π}(𝔤)) is spanned by M^{2k−1}, 1 ≤ k ≤ n − 1, and M^{−1} · Pf(M).

Here the scheme of the proof is essentially the same with one exception. There exists an element in I(𝔤) which cannot be expressed in terms of tr M^{k}, k ∈ ℕ. It is the so-called Pfaffian Pf(M) := \sqrt{det M}.

Accordingly, the space p_{1}(C_{π}(𝔤)) contains, besides all odd powers of M, an additional element M^{−1} · Pf(M). It is a nice exercise in matrix algebra to prove that the entries of this matrix are indeed polynomials in matrix elements of M.

Corollary. The exponents of π_{2ω_{1}} are 2, 4, …, 2n − 2, and the exponents of π_{ω_{2}} (the ordinary exponents) are 1, 3, …, 2n − 3 and n − 1.

3. Other examples

3.1. The case of a 7-dimensional representation of G_{2}. In this section I use some computations performed by my student N. Rojkovskaya.

The simplest exceptional Lie algebra of type G_{2} admits a ℤ_{3}-grading such that

𝔤_{0} ≅ 𝔰𝔩(3, ℂ), 𝔤_{1} ≅ V, 𝔤_{2} ≅ V^{∗},

where V and V^{∗} are dual fundamental modules for 𝔰𝔩(3, ℂ). We realize X ∈ 𝔤_{0} as a traceless complex 3 × 3 matrix, v ∈ V as a column 3-vector with coordinates v^{1}, v^{2}, v^{3}, and f ∈ V^{∗} as a row 3-vector with coordinates f_{1}, f_{2}, f_{3}.

The 7-dimensional representation of G_{2} has the form:

X = (\begin{matrix}X & v & A(f) \\ f & 0 & −v̌ \\ B(v) & −f̌ & −X̌ \end{matrix}) . (18)

Here the check symbol denotes the transposition with respect to the second diagonal, and the following notation is used:

A(f) = \frac{1}{\sqrt{2}} (\begin{matrix}−f_{2} & f_{3} & 0 \\ f_{1} & 0 & −f_{3} \\ 0 & −f_{1} & f_{2} \end{matrix}) , B(v) = −A(v̌) = \frac{1}{\sqrt{2}} (\begin{matrix}v^{2} & −v^{1} & 0 \\ −v^{3} & 0 & v^{1} \\ 0 & v^{3} & −v^{2} \end{matrix}) .

The fact that matrices (17) form a Lie algebra is equivalent to the following simple statement.

Lemma 2. If X ∈ 𝔰𝔩(3, ℂ), v ∈ V, f ∈ V^{∗}, then a) A(−fX) = XA(f) + A(f)X̌; b)B(Xv) = −X̌B(v) − B(v)X;

c) A(f)B(v̌) = \frac{1}{2} (vf − fv · 1) .

We see that the 7-dimensional representation π = π_{ω_{2}} has 4 children: two girls and two boys.

The girls are the trivial representation and the 27-dimensional representation with the highest weight 2ω_{2}.

The boys are 14-dimensional and 7-dimensional representations with the highest weights ω_{1} and ω_{2} respectively.

We denote by p_{1}, p_{27}, p_{14} and p_{7} the corresponding projections.

Theorem G. a) The algebra I(𝔤) is generated by tr M^{k}, k = 2, 6.

b) The I(𝔤)-module p_{27}(C_{π}(𝔤)) is spanned by p_{27}(M^{2k}), 1 ≤ k ≤ 3, the I(𝔤)-module p_{14}(C_{π}(𝔤)) is spanned by p_{14}(M^{k}), k = 1, 5, the I(𝔤)-module p_{7}(C_{π}(𝔤)) is spanned by p_{7}(M^{3}).

Corollary. The generalized exponents for 2ω_{2} are 2, 4, 6; the unique generalized exponent for ω_{2} is 3 and the ordinary exponents are 1, 5.

3.2. The case of the adjoint representation of A_{n}. The adjoint representation π = π_{ω_{1}+ω_{n}} is orthogonal. Therefore, the space Mat_{n^{2}+2n}(ℂ) splits as a 𝔤-module into symmetric and antisymmetric parts.

More detailed analysis (see e.g. [OV], Table 5) shows, that there are 7 children (4 girls and 3 boys). In the table we show their dimensions and the numbers of exponents.

Highest weight boy/girl  dimension  number of exponents
2ω_{1} + 2ω_{n}  girl  \frac{n(n+1)^{2}(n+4)}{4}  \frac{n(n+1)}{2}
2ω_{1} + ω_{n−1}  boy  \frac{(n−1)n(n+2)(n+3)}{4}  \frac{n(n−1)}{2}
ω_{2} + 2ω_{n}  boy  \frac{(n−1)n(n+2)(n+3)}{4}  \frac{(n+1)(n−2)}{2}
ω_{2} + ω_{n−1}  girl  \frac{(n+1)^{2}(n^{2}−4)}{4}  \frac{n(n−1)}{2}
ω_{1} + ω_{n}  girl  n(n + 2)  n
ω_{1} + ω_{n}  boy  n(n + 2)  n
0  girl  1  1

(For small n some degenerations occur: for n = 2 there are 3 girls and 3 boys and for n = 1 there are two girls and one boy.)

There are two special elements in our algebra related to two covariant pairings 𝔤 × 𝔤 → 𝔤. The first is antisymmetric and given by the commutator, while the second is symmetric and given by the traceless part of the anticommutator. Below we give the precise formula for these elements.

We introduce the notation N = [\frac{dim 𝔤}{2}] = [\frac{n^{2}}{2} ] + n and enumerate the basis vectors in 𝔤 by integers from −N to N (including zero if n is even), so that the Ad-invariant form on 𝔤 look like

(X_{i}, X_{j}) = \begin{cases}1  if i + j = 0, \\ 0  otherwise. \end{cases}

For example, for n = 2 we can put

X_{±1} = X_{±α}, X_{±2} = X_{±β}, X_{±3} = X_{±γ}, X_{±4} = H_{±},

where α and β are simple roots, γ = α + β and

H_{+} = \frac{ϵH_{α} − ϵ^{2}H_{β}}{\sqrt{3}} ,  H_{−} = \frac{ϵ^{2}H_{α} − ϵH_{β}}{\sqrt{3}} ,  ϵ = e^{2πi/3}.

Then we define matrices A and S in the family algebra A := C_{ω_{1}+ω_{n}}(A_{n}) by

A_{k}^{j} = [X_{−j}, X_{k}], S_{k}^{j} = {X_{−j}, X_{k}}_{0}. (19)

Note that A is antisymmetric with respect to the second diagonal, while S is symmetric. Indeed, by the very definition we have

A_{−j}^{−k} = −A_{k}^{j}, S_{−j}^{−k} = S_{k}^{j}.

In fact, A coincides with the element M introduced above. Hence, it commutes with S.

This algebra is non-commutative for n ≥ 2. The further study of it is very interesting.

3.3. The quantum family algebras for A_{1}. Let α be the simple root of 𝔤 ≅ 𝔰𝔩(2, ℂ), and ω = \frac{1}{2}α the fundamental weight. We choose the standard basis X_{1} = E, X_{0} = \frac{1}{2}H, X_{−1} = F. Denote by C the central element X_{0}^{2}+ \frac{1}{2}(X_{1}X_{−1}+X_{−1}X_{1}). The algebra Q_{n} := Q_{nω}(𝔤) contains the element

A_{n} = ( \begin{matrix}\frac{1}{2} nH & \sqrt{1·n}F & 0 & 0 & ... & 0 & 0 \\ \sqrt{1·n}E & \frac{1}{2} (n−2)H & \sqrt{2·(n−1)}F & 0 & ... & 0 & 0 \\ 0 & \sqrt{2·(n−1)}E & \frac{1}{2} (n−4)H & \sqrt{3·(n−2)}F & ... & 0 & 0 \\ ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & 0 & ... & \frac{1}{2} (2−n)H & \sqrt{n·1}F \\ 0 & 0 & 0 & 0 & ... & \sqrt{n·1}E & − \frac{1}{2} nH \end{matrix} ) .

For n = 1 we have A_{1} = (\begin{matrix}\frac{1}{2}H & F \\ E & −\frac{1}{2}H \end{matrix}). In this case it is not difficult to check

directly that Q_{1} is a free ℂ[C]-module with generators 1 and A. The same is true in general:

Theorem Q. The quantum family algebra Q_{n} is a free ℂ[C]-module with n + 1 generators 1, A, …, A^{n}. In particular, the algebra is commutative and eventually isomorphic to the corresponding classical family algebra.

Note, however, that the defining relations for Q_{n} and C_{n} are different. For the latter algebra it is the usual Cayley identity,

A^{n+1} + ∑_{k=1}^{n+1} c_{k}A^{n+1−k} = 0, where c_{k} = (−1)^{k}tr(∧^{k}A).

The corresponding quantum identity for n = 2 looks like A^{2} − A + c_{2} · 1 = 0, while the Cayley identity is A^{2} + c_{2} · 1 = 0.

4. Some open questions

We list here some cases where the family algebras seemed to admit the full description. (The order reflects the anticipated difficulty of the problem.)

1. The classical algebras for adjoint representations of classical groups.

2. The quantum algebras for the standard representations of classical groups.

3. The quantum algebra for the 7-dimensional representations of G_{2}.

4. The classical algebra for the minimal representation of the exceptional group F_{4}.

5. The same for minimal representations of the exceptional groups E_{6}, E_{7}.

6. The classical algebra for the adjoint representation of E_{8}.

7. In general, it would be very interesting to find out which quantum family algebras are commutative and which classical algebras are spanned over I(𝔤) by powers of M or analogous elements related to other generators of I(𝔤).

References

[H] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, New York, 1972, 1980 (second edition). MR 48:2197, MR 81b:17007

[K] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327– 404. MR 28:1252 [OV] A. L. Onishchik and E. B. Vinberg, A seminar on Lie groups and algebraic groups, Nauka,

Moscow, 1988, 1995 (second edition); English translation: Series in Soviet Mathematics, Springer-Verlag, 1990, xx+328 pp. MR 92i:22014, MR 97d:22001

Department of Mathematics, The University of Pennsylvania, Philadelphia, PA 19104

E-mail address: kirillov@math.upenn.edu