Ellen Kirkman

A q-analog for the virasoro algebra

(1994). A q-analog for the virasoro algebra. Communications in Algebra: Vol. 22, No. 10, pp. 3755-3774.

Noetherian down-up algebras

Down-up algebras A = A(α, β, γ) were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that β 6= 0 is equivalent to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3.

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.

q-Analogs of harmonic oscillators and related rings

A ringHq which is aq-analog of the universal enveloping algebra of the Heisenberg Lie algebraU(h) is constructed, and its ring theoretic properties are studied. It is shown thatHq has a factor ringAq which is a simple domain with properties that are compared to the Weyl algebra. A secondq-analogHq ofU(h) is constructed, andHq is shown to be a primitive ring.

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

Hopf down–up algebras

and hence we know it has a graded Hopf structure, but wedo not know if it has a Hopf structure.In the third section we show that a theorem of De Concini and Procesi providesa furthertechnique for proving that an algebra is not a Hopf algebra. We use this result to showthat under certain conditions a localization of a down–up algebra is not a Hopf algebra.Localizations of down–up algebras were considered by Jordan [13], and the algebras heconsiders include the algebra defined by Woronowicz [17].Throughout this paper let

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.

Algebras with large homological dimensions

An example is given of a semiprimary ring with infinite finitistic dimension. The construction shows that the global dimensions of finite dimensional algebras of finite global dimension cannot be bounded by a function of only Loewy length and the number of nonisomorphic simple modules. The (right) finitistic global dimension rFPD(A) of a ring A is the supremum of the projective dimensions of the right A-modules of finite projective dimension; we denote the supremum of the projective dimensions of the right finitely generated A-modules of finite projective dimension by rfPD(A). When A is a semiprimary ring with (radA)2 = 0 it is easy to show that rFPD(A) is finite; in this note we present a semiprimary graded ring A with (radA)4 = 0 and rfPD(A) = ox. It is a long standing open question (see [B]) whether rfPD(A) (or rFPD(A) ) is finite for all finite dimensional algebras A; there has been some recent progress on the question (see [Zi, GKK, IZ, GZ-H]) including a proof [GZ-H] of the fact that rfPD(A) < x when A is a right Artinian ring with (rad A)3 = 0. The example in this note shows that the Finitistic Dimension Conjecture is not true for semiprimary rings; it, of course, remains open for finite dimensional algebras and Artinian rings. A related question of current interest is to find bounds on the global dimension of a finite dimensional algebra A of finite global dimension. Schofield [S] proved that there exists an integer-valued function f, such that if A is a finite dimensional k-algebra with vector space dimension [A: k] = n and with finite global dimension, then gldim(A) < f (n); the nature of this function f is unknown, but in all known examples of algebras with finite global dimension, the global dimension of A does not exceed the vector space dimension of A. Examples of finite dimensional algebras of arbitrarily large finite global dimension can be produced by increasing either the number of isomorphism classes of simples or the Loewy length. It has been shown [G] that finite dimensional algebras with exactly two isomorphism classes of simple right modules can have Received by the editors June 28, 1989 and, in revised form, October 25, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 6A46, 1 6A5 1, 1 6A60.

On the discriminant of twisted tensor products

Abstract We provide formulas for computing the discriminant of noncommutative algebras over central subalgebras in the case of Ore extensions and skew group extensions. The formulas follow from a more general result regarding the discriminants of certain twisted tensor products. We employ our formulas to compute automorphism groups for examples in each case.