James Kuzmanovich

Rigidity of graded regular algebras

. We prove a graded version of Alev-Polos rigidity theorem: the homogenization of the universal enveloping algebra of a semisimple Lie algebra and the Rees ring of the Weyl algebras A n (k) cannot be isomorphic to their fixed subring under any finite group action. We also show the same result for other classes of graded regular algebras including the Sklyanin algebras.

Finite Groups of Matrices Whose Entries Are Integers

Finite groups of matrices appear early as examples in a first course in abstract algebra, and most of the time these examples are given with integral entries. While these groups provide a setting in which to illustrate new concepts and to pose problems, they also have surprising and beautiful properties. For example, Minkowski proved the unexpected result that GL(n, Z), the group of n x n matrices having inverses whose entries are also integers, has only finitely many isomorphism classes of finite subgroups. As a consequence, there are only finitely many possible orders for elements of GL(n, Z); fortunately, the possible orders can be determined using linear algebra. In general, the number of possible orders increases as n increases, but even here we have the surprising result that no new possible orders are obtained when going from GL(2k, Z) to GL(2k + 1, 7). This paper is an exposition of these and other related results and questions. Although finite groups of integral matrices have a long history, most of the beautiful theorems concerning them do not appear in texts and are not known to many algebraists (for example, the first named author). While still an area of active research (see [7], [12], [23], and [24]), major parts are accessible to students; indeed, this paper has its origin in a term paper written by the second author for a beginning modem algebra course that used [9] as a text. One of our goals for this paper is to be a source of problems, readings, and projects for other students; the topic seems ideal since it synthesizes results from number theory, algebra, and linear algebra, gives a natural and somewhat gentle introduction to deeper subjects not yet encountered by a beginning algebra student, has historical features, and could be a subject of continuing student research. With this goal in mind, the authors have included exercises and have tried to organize the paper so that parts can be read independently, or given to students in outline form. Our second goal is to give the reader examples of current research in the area.

Primitivity of Noetherian Down-Up Algebras

These algebras arose in the study of the representations of differential partially ordered sets. Examples of down-up algebras include enveloping algebras of many three dimensional Lie algebras and their quantizations. Any down-up algebra A(&, 0, y) with not both a and ,d equal to 0 is isomorphic to a Witten 7-parameter deformation of U ( d 2 ) (see [B], [Wl], [W2]). We assume throughout that A is Noetherian. In [KMP] it was shown that A is Noetherian, if and only if /3 # 0, if and only if A is a domain, and if and only if C[du,ud] is a conlmutative polynomial ring in two indeterminants. As in [BR] consider the roots rl and 7-2 of the polynomial equation x2 a x ,d = 0. Since P = -rlrz, we may assume that ri # 0 for i = 1 , 2 . Answering a question of Jacobson, Ooms [0] gave conditions on a finite dimensional Lie algebra C over a field k of characteristic zero such that its

Finitistic dimensions of noetherian rings

The global dimension gldim(R) of a Noetherian ring R, while a useful invariant when finite, is often infinite. Several homological dimensions have been introduced to handle rings of infinite global dimension. These dimensions include the injective dimension of the ring R as a left R-module, injdim,(R), and the tinitistic global dimensions, lfPD(R) and lFPD(R). The tinitistic dimensions are defined as follows: IFPD(R) is the supremum of the projective dimensions pd(M) of left R-modules M of finite projective dimension, and lfPD(R) is the supremum of the projective dimensions pd(M) of left R-modules M which are finitely generated and have finite projective dimension. For a Noetherian ring R, when gldim(R) is finite, all these dimensions are equal (on the right and the left). Clearly, lfPD(R) < lFPD(R), and Bass [B2, Proposition 4.31 has shown that for left Noetherian rings lFPD(R) < injdim,(R). We will note some further relationships between the dimensions in Proposition 2.1. More is known about these dimensions when R is a commutative Noetherian ring. Auslander and Buchsbaum [AuBu, Theorem 1.61 showed that fPD(R) = codim(R) Q Kdim(R) (where codim(R) is the least upper bound on the lengths of R-sequences). Bass [B2, Proposition 5.43 showed

Minimal prime ideals in enveloping algebras of Lie superalgebras

Let g be a finite dimensional Lie superalgebra over a field of characteristic zero. Let U(g) be the enveloping algebra of g. We show that when g = b(n), then U(g) is not semiprime, but it has a unique minimal prime ideal; it follows then that when g is classically simple, U(g) has a unique minimal prime ideal. We further show that when g is a finite dimensional nilpotent Lie superalgebra, then U(g) has a unique minimal prime ideal. Throughout let g = g0 ⊕ g1 be a finite dimensional Lie superalgebra over a field k of characteristic zero (see [S] or [K] for general definitions). Let U = U(g) be the enveloping algebra of g. When g is a Lie algebra U(g) is a domain, but when g is a Lie superalgebra U(g) need not be semiprime. In this paper we will present some classes of Lie superalgebras for which we can prove that U(g) has a unique minimal prime ideal; we begin by noting that we know of no example of a finite dimensional Lie superalgebra g for which U(g) does not have a unique minimal prime ideal. A. Bell gave a sufficient condition [Be, Theorem 1.5] for U to be a prime ring (so that 0 is the unique minimal prime ideal). Let ([yi, yj ]) be the m ×m matrix of brackets of basis elements of g1, and let d(g) be the determinant of that matrix, considered as a matrix with elements in the symmetric algebra S(g1). If d(g) 6= 0, then U is prime. Using this criterion, Bell showed [Be, Corollary 3.6] that all the classically simple, finite dimensional Lie superalgebras g (except for one class b(n)) (called P (n− 1) in the notation of [K]) have d(g) 6= 0, and hence U is prime. Since d(b(n)) = 0, Bell’s criterion could not determine whether U(b(n)) was prime. Behr [B] had shown U(b(2)) was not semiprime, but b(n) is classically simple only for n ≥ 3. Bell [Be, Section 1.6] had further shown that when g0 is supercentral in g, then U has a unique minimal prime ideal, and he raised the question of whether U always has a unique minimal prime ideal. Here, in Section 2, we will show that U(b(n)) is not semiprime for all n, but that U(b(n)) always has a unique minimal prime ideal. Hence the enveloping algebra U(g) of any classically simple Lie superalgebra g has a unique minimal prime ideal. This result has been used by Letzter and Musson in [LM]. Wilson [W] has shown that W (n), a class of simple Lie superalgebras of Cartan type, has a prime enveloping algebra when n is even and n ≥ 4. Received by the editors August 12, 1994 and, in revised form, December 13, 1994. 1991 Mathematics Subject Classification. Primary 16S30; Secondary 16D30, 17B35, 17A70.

Down-Up Algebras From Trees

It is shown that the global dimension of any n-ary down-up algebra A n = A(n,α, β,γ) is less than or equal to n + 2, and when γ i = 0 for all i (A n is graded by total degree in the generators), then the global dimension of A n is n + 2. Furthermore, a sufficient condition for A n to be prime is given; when γ i = 0 for all i this condition is also necessary. An example is given to show that the condition is not always necessary.