Gelfand-Kirillov dimension in Jordan Algebras

In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.

     

The nil Temperley–Lieb algebra of type affine C

We introduce a type affine C analogue of the nil Temperley–Lieb algebra, in terms of generators and relations. We show that this algebra $T(n)$, which is a quotient of the positive part of a Kac–Moody algebra of type $D_{n+1}^{(2)}$, has an easily described faithful representation as an algebra of creation and annihilation operators on particle configurations, reminiscent of the open TASEP model in statistical physics. The centre of $T(n)$ consists of polynomials in a certain element Q, and $T(n)$ is a free module of finite rank over its centre. We show how to localize $T(n)$ by adjoining an inverse of Q, and prove that the resulting algebra is a full matrix ring over a ring of Laurent polynomials over a field. Although $T(n)$ has wild representation type, over an algebraically closed field we can classify all the finite dimensional indecomposable representations of $T(n)$ in which Q acts invertibly.     

On Gelfand-Kirillov Transcendence Degree

We study some basic properties of the Gelfand-Kirillov transcendence degree and compute the transcendence degree of various infinite-dimensional division algebras including quotient division algebras of quantized algebras related to quantum groups, 3-dimensional Artin-Schelter regular algebras and the 4-dimensional Sklyanin algebra.

     

A result on the Gelfand-Kirillov dimension of representations of classical groups

Let be the reductive dual pair . We show that if is a representation of (respectively ) obtained from duality correspondence with some representation of (respectively ), then its Gelfand-Kirillov dimension is less than or equal to
(respectively ).

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

Exponential growth of Lie algebras of finite global dimension

Let be a connected finite type graded Lie algebra. If dim and gldim , then log index . If, moreover, , then for some ,    dim where log index as

     

Two remarks on witten'squantum enveloping algebra

In this note we describe two simple tricks derived from work of M. Artin , J. Tate and M. Van den Bergh [2] and apply them to the quantum enveloping algebra discovered by E. Witten [ll] encoding duality in three dimensional Chern-Simons gauge theory with structure group SU(2). In particular, we give a more conceptual interpretation of the dichotony in the representation theory and determine all finite dimensional simple representations.     

Krull dimension of the enveloping algebra of a semisimple Lie algebra

Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .

     

Strong connectedness of the invertibles in a finite subdiagonal algebra

Suppose is a finite, subdiagonal subalgebra of a von Neumann algebra. We show that the invertible group of is strongly connected.

     

Injective dimension of Weyl algebras over affine coefficient rings

The concern of this paper is to derive formulas for the injective dimension of then- th Weyl algebraA _n (R) in casek is a field of characteristic zero andR is a commutative affinek-algebra of finite injective dimension. For the casen=1 we prove a more general result from which the above result follows. Such formulas can be viewed as generalizations of the corresponding results given by J. C. McConnell in the caseR has finite global dimension.Project supported in part by the National Natural Science Foundation for Youth     

Free orbit dimension of finite von Neumann algebras

We introduce a new free entropy invariant, which yields improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property T, property Γ, nonprime factors, and thin factors.     

Varieties of affine Kac-Moody algebras

In this paper the identities of the complex affine Kac-Moody algebras are studied. It is proved that the identities of twisted affine algebras coincide with those of the corresponding nontwisted algebras. Moreover, in the class of nontwisted affine Kac-Moody algebras, each of these algebras is uniquely defined by its identities. It is shown that the varieties of affine algebras, as well as the varieties defined by finitely generated three-step solvable Lie algebras, have exponential growth. Translated fromMatematicheskie Zametki, Vol. 62 No. 1, pp. 95–102, July 1997. Translated by A. I. Shtern     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.     

A criterion for a finite endoprimal algebra to be endodualisable

In this note the strategy for finding endoprimal algebras due to B. A. Davey and J. G. Pitkethly is further explored in the finite case. The Retraction Test Algebra Lemma is used as a tool to show that, in many quasivarieties, endoprimality is equivalent to endodualisability for finite algebras which are suitably related to finitely generated free algebras. Received September 2, 1998; accepted in final form June 3, 1999.     

An example of finite dimensional Kac algebras of Kac-Paljutkin type

An example of finite dimensional Kac algebras of Kac-Paljutkin type is given.

     

An upper bound for the representation dimension of group algebras with elementary abelian Sylow p-subgroups

Abstract

In this article, we present an upper bound on the representation dimension of the group algebra of a group with an elementary abelian Sylow p-subgroup. Specifically, if k is a field of characteristic p and G is a group with elementary abelian Sylow p-subgroup P, we prove that the representation dimension of kG is bounded above by the order of P. Key to proving this theorem is the separable equivalence between the two algebras and some nice properties of Mackey decomposition.