Finite-dimensional Semisimple Lie Algebra Research Articles

On the Global Dimension of Certain Primitive Factors of the Enveloping Algebra of a Semi-Simple Lie Algebra

Let g be a finite-dimensional semi-simple Lie algebra over an algebraically closed field k of characteristic zero and letg = n+rj + n~ be a triangular decomposition of g. Denote by U = U(Q) the enveloping algebra of g. For Xe\)*, denote by Dk the primitive factor ring £//t/ ^W> where M(X) is the Verma module of highest weight X—p (where p is the half-sum of the positive roots). The main aim of this paper is to prove Theorem 3.9, which states that if A is regular, then gldim Dx < dim . n + -In{X), where n{X) is a non-negative integer less than or equal to dimfcn+. In [12] Levasseur computed the injective dimension of D^ in terms of the Gelfand-Kirillov dimension of L{X), the unique simple quotient of M(X). For X regular, Theorem 3.9 implies that the global dimension of D^ is finite and hence must be equal to the injective dimension of Dk. It can be shown that this figure coincides with the bound given in Theorem 3.9 (the authors would like to thank Levasseur for pointing this out). On the other hand, if X is not regular, Joseph and Stafford [11] have shown that gldim D^= oo. Special cases of this result have already appeared in the literature. In particular Stafford [17] computed the global dimension of Dk for all A el)* in the case when g = si (2, C). On the other hand Roos [15] computed gldim D^ for general g when X satisfies some transcendental but generic conditions. This paper is divided into two distinct parts. In Section 2 we prove a result analogous to a theorem of Roos relating the weak global dimension of a ring R to that of certain sets of torsion-theoretic localisations. If R is a commutative ring and {Si)tei i a s e t °f localisations of R such that © S{ is faithfully flat as an /^-module, then it is well known that wgldim R = sup{wgldim St}. To what extent this result is true for non-commutative rings is unknown. Roos and others have proved various results under the assumption that wgldim R is finite (see for instance [3, 7, 13,14]). We prove here a slightly more general sort of result than Roos's which does not require this assumption. However, it is necessary to impose certain additional assumptions on the localisations St. The special case of this result needed for the second part is the following. Let R be a prime Noetherian ring and let St, i = 1, ..., n be a finite collection of rings lying between R and its quotient ring. If © St is a faithfully flat right /^-module and St ®R S} = S} ®R St as /?-/?-bimodules for all pairs (i,j), then

Homomorphisms Between Generalized Verma Modules

Let $\mathfrak {g}$ be a finite-dimensional complex semisimple Lie algebra and $\mathfrak {p}$ a parabolic subalgebra. The first result is a necessary and sufficient condition, in the spirit of the Bernstein-Gelfand-Gelfand theorem on Verma modules, for Lepowsky’s "standard map" between two generalized Verma modules for $\mathfrak {g}$ to be zero. The main result gives a complete description of all homomorphisms between the generalized Verma modules induced from one-dimensional $\mathfrak {p}$-modules, in the "hermitian symmetric" situation.

Homomorphisms between generalized Verma modules

Let g be a finite-dimensional complex semisimple Lie algebra and p a parabolic subalgebra. The first result is a necessary and sufficient condition, in the spirit of the Bernstein-Gelfand-Gelfand theorem on Verma modules, for Lepowsky's standard map between two generalized Verma modules for g to be zero. The main result gives a complete description of all homomorphisms between the generalized Verma modules induced from one-dimensional p-modules, in the hermitian symmetric situation.

On the Harishchandra homomorphism for infinite-dimensional Lie algebras

A natural generalization of the finite-dimensional semisimple Lie algebras to the infinite-dimensional case is the Kac-Moody Lie algebra G(A) or more generally, the contragredient Lie algebra associated to an arbitrary square matrix A [9a, 111. The category a introduced in [la, b] to study the representations of finite-dimensional semisimple Lie algebras was extended [9b] to the case of Kac-Moody Lie algebras. In [lo], Kac and Kazhdan describe the irreducible subquotients of Verma modules. Using this a decomposition of the category B was obtained in [3]. In both [3] and [lo] the center 3 of the universal enveloping algebra plays no role, since there is no analog to the theory of central characters in the infinite-dimensional setup. Let G = G(A) be the Kac-Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A, and let H = H(A) be the abelian diagonalizable subalgebra of G(A). In this article we introduce alalgebra % (&i(U(G), a)) of operators on the category P and a surjectio2 /I of % onto U(H). The Weyl group W acts on U(H) and the restriction of /3 to the cznter 3 of (li maps into the invariants U(EQW. In Section 3, we prove that /3 Ij is injective and that 3 injects into 3, i: 3 4 3. If G(A) is infinite dimensional, then the existence of the Casimir operator [5, 9b] implies that i(j) f 3, and hence it follows that the Harishchandra homomorphism is never surjective. In Section 4, we deal with the case of affme Kac-Moody Lie algebras. Using a theorem of Hochschild and Mostow [6], we deduce that U(H)W is the polynomial algebra on two generators. This disproves the conjecture in [3] that the image of the Harishchandra homomorphism separ$es the Weyl group orbits.* Further, we show that the restriction of p to 3 is an isomorphism onto U(H)W.

Structure of some categories of representations of infinite-dimensional lie algebras

The highest weight representations of a finite-dimensional complex semisimple Lie algebra 9 have been studied extensively and a number of interesting results have been obtained in the last few years (an extensive bibliography is given in [7]). In [2, 3] a category cY of representations of g was introduced which turns out to be the proper setting for considering various questions regarding highest weight representations. The aim of this paper is to extend the study of this notion to a Kac-Moody algebra, or, more generally, to a contragredient Lie algebra G(A) corresponding to a square matrix A over C. The study of such algebras was begun independently in [8, 14] and subsequently a number of interesting results were proved. These results show that many important properties from the finite-dimensional setup (i.e., from the context of 9) can be proved in the general setup involving G(A). In particular, the notion of category • was extended in [9] to the case of Kac-Moody algebras and used in the proof of combinatorial identities.

Lie algebras with descending chain condition

In this note we investigate Lie algebras which satisfy the descending chain condition on ideals of ideals. We show that a Lie algebra L satisfies this descending chain condition if and only if the following two conditions hold: (i) L contains a finite dimensional solvable ideal N such that every solvable ideal of L is contained in N, and (ii) L/N is a subdirect sum of a finite number of prime algebras satisfying the descending chain condition. We also show that if L is a prime algebra with this chain condition then there exists a Lie algebra B, which is either simple or the tensor product of a simple Lie algebra with a truncated polynomial algebra, such that L is isomorphic to a subalgebra of Der B containing adB. A decade ago a theory of Jordan algebras with descending chain condition on inner ideals was developed [3, Chapter IV] which emulates and connects with the theory of Artinian rings. More recently Benkart [1] studied Lie algebras with descending chain condition on inner ideals (a subspace B of a Lie algebra L is called an inner ideal of L if [B, [B, L]] £ JS). It has not been settled yet whether a Jordan algebra with DCG on inner ideals necessarily has a nilpotent radical. One of the purposes of the present paper is to show that a Lie algebra with DCC on inner ideals has a radical which is solvable and finite dimensional. This follows from the results stated in the last paragraph since any ideal of an ideal is an inner ideal and hence DCC on inner ideals implies DCC on ideals of ideals. It is known that a finite dimensional semisimple Lie algebra M of characteristi c p is not necessarily a direct sum of simple algebras, but there do not seem to be any results published which express M in terms of algebras which belong to a more restricted class than M. A second purpose of this paper is to show that M is a subdirect sum of prime algebras. Rather than finite dimensionality the assumption of DCC on ideals of ideals seems to be the most natural level of generality for this proof. The results in this paper hold for Lie algebras over a field Φ of any characteristic including 2. Suppose now that L is a Lie algebra with DCC on ideals of ideals. We begin with LEMMA 1. If G is a solvable ideal of L, then C is finite

Hilbert space methods in the theory of Lie algebras

Introduction. In the study of the complex finite-dimensional semi-simple Lie algebras a crucial role is played by the fundamental bilinear form (x, y) = Tr(ad(x)ad(y)). Since the definition is meaningless when the restriction of finite-dimensionality is removed, if any of the highly desirable properties of the form are to be retained in this case they must necessarily be given a priori. By reconsidering the finite-dimensional situation it is possible to formulate suitable conditions in a more convenient form. To see this let P be a complex finite-dimensional semi-simple Lie algebra and let Po be a compact real form for P with cr as the associated involution (conjugation). If we let x* = —a(x) and (x, y) = (x, y*) then 7 becomes a finite-dimensional Hilbert space, the mapping x into x* is a Hilbert space conjugation, and the connecting property ([x, y], z) = (y, [x*, z]) holds for all x, y, z. An P* algebra as defined here is simply a Lie algebra whose vector space is a Hilbert space such that the connecting property above holds. This paper is a study of such algebras with emphasis, of course, on the infinite-dimensional ones. For finite dimensions nothing new is obtained and it is shown here that in this case every semisimple P* algebra arises essentially from a construction like that above (see the remark after 2.5). There is an associative algebra analogue of this problem in the paper of Ambrose [l ] on TZ* algebras and some of his results are used here. Any H* algebra gives rise to an P* algebra by letting [x, y]=xy — yx and the only known examples of P* algebras are those obtained as Lie subalgebras of H* algebras. The main result of this paper is a classification of the (separable) simple P* algebras which have Cartan decompositions (see §2) and it is shown that this class coincides with the simple self-adjoint Lie subalgebras of a (separable) simple H* algebra. The results turn out to be the natural extensions of the finite-dimensional theory. Associated with each of the Lie algebras considered here there is a gener-