Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.     

Relatively Free Algebras with the Identity x 3 = 0 #

A basis for a relatively free associative algebra with the identity x ³ = 0 over a field of an arbitrary characteristic is found. As an application, a minimal generating system for the 3 × 3 matrix invariant algebra is determined.     

A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings

The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module.     

Soluble Block-transitive Automorphism Groups of Design

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Krull–Gabriel Dimension of 1-domestic String Algebras

We classify the indecomposable pure injective modules over a wide class of -domestic string algebras and calculate the Krull–Gabriel dimension of these algebras.This paper was written during the visit of the second author to the University of Manchester supported by EPSRC grant GR/R44942/01.     

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let be a linear representation of a finite group over a field of characteristic 0. Further, let R be the corresponding algebra of invariants, and let P (t) be its Hilbert–Poincaré series. Then the series P (t) represents a rational function (t)/(t). If R is a complete intersection, then (t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where is an almost regular (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.     

Automorphism Group of Toeplitz Algebras on Certain Pseudoconvex Domains

in this paper, we characterize the automorphism groups of Toeplitz algebras oncertain strongly pseudoconvex domains of Cn, and obtain a generalized BDF theorem for aspecial kind of essential normal operator tupies.     

Gelfand–Kirillov Dimensions of the Z~2-graded Oscillator Representations of sl(n)

We find an exact formula of Gelfand–Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand–Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.     

Ringel–Hall Algebras of Quotient Algebras of Path Algebras

We determine the generating relations for Ringel–Hall algebras associated with quotient algebras of path algebras of Dynkin and tame quivers, and investigate their connection with composition subalgebras.     

A generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.     

Structures of Wey1 Groups of Some Kac-Moody Algebras

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Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group,π_e(G) denotes the set of orders of elements in G.If Ω is a subsetof the set of natural numbers,h(Ω) stands for the number of isomorphism classes of finite groups withthe stone set Ω of element orders.We say that G is k-distinguishable if h(π_e(G))=k<∞,otherwiseG is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.Itis shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M~cL and O'N,then Aut(M) is characterizable by its element orders.It is also proved that if M is isomorphic toM_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.