Gelfand-Kirillov Dimension and Local Finiteness of Symmetrizations of Associative Superalgebras

In this work we extend to superalgebras a result of Skosyrskii [Algebra and Logic, 18 (1) (1979) 49–57, Lemma 2] relating associative and Jordan structures. As an application, we show that the Gelfand-Kirillov dimension of an associative superalgebra coincides with that of its symmetrization, and that local finiteness is equivalent in associative superalgebras and in their symmetrizations. In this situation we obtain that having zero Gelfand-Kirillov dimension is equivalent to being locally finite.Partially supported by MCYT and Fondos FEDER BFM2001-1938-C02-02, and MEC and Fondos FEDER MTM2004-06580-C02-01.Partially supported by a F.P.I. Grant (Ministerio de Ciencia y Tecnología).     

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     

Some properties of isoclinism in Lie superalgebras

Abstract

Isoclinism of Lie superalgebras has been defined and studied currently. In this article, it is shown that for finite dimensional Lie superalgebras of same dimension, the notation of isoclinism and isomorphism are equivalent. Furthermore, we show that covers of finite dimensional Lie superalgebras are isomorphic using isoclinism concept.     

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.

     

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.     

D4型量子包络代数的Gelfand-Kirillov维数

本文研究了D₄ 型量子包络代数的Gelfand-Kirillov 维数的计算问题. 利用文献[1] 中给出的Gelfand-Kirillov 维数的计算方法和文献[2] 中给出的D₄ 型量子包络代数的Groebner-Shirshov 基计算了D₄型量子包络代数的Gelfand-Kirillov 维数, 得到的主要结果是D₄ 型量子包络代数的Gelfand-Kirillov 维数为28. 希望此结果为计算D_n型量子包络代数的Gelfand-Kirillov 维数提供一些思路.     

Alternative Superalgebras with DCC on Two-Sided Ideals#

We study the structure of alternative superalgebras that satisfy the descending chain condition (DCC) for two-sided ideals. The main results state that the Baer radical in an alternative superalgebra of characteristic ≠ 2, 3 with DCC on two-sided ideals is solvable and every such a semiprime superalgebra (of arbitrary characteristic) is isomorphic to a subdirect sum of an associative superalgebra with this property and a finite direct sum of simple alternative non-associative superalgebras.     

On Certain Semiprime Associative Superalgebras

In this note we emphasise the relationship between the structure of an associative superalgebra with superinvolution and the structure of the Lie substructure of skewsymmetric elements. More explicitly, we show that if A is a semiprime associative superalgebra with superinvolution and K is the Lie superalgebra of skewsymmetric elements satisfying [K ², K ²] = 0, then A is a subdirect product of orders in simple superalgebras each at most 4-dimensional over its center.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Primitive Algebras with Arbitrary Gelfand-Kirillov Dimension

We construct, for every real β ≥ 2, a primitive affine algebra with Gelfand-Kirillov dimension β. Unlike earlier constructions, there are no assumptions on the base field. In particular, this is the first construction over ℝ or L. Given a recursive sequence {v_n} of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in {v_n}. We bound the dimension of the resulting algebra in terms of the growth of {v_n}. In particular, if ⋎ν_n⋎ is less than doubly exponential, then the dimension is 2. This also answers affirmatively a conjecture of Salwa (1997, Comm. Algebra 25, 3965–3972).     

Simple Special Jordan Superalgebras with Associative Even Part

We describe the simple special unital Jordan superalgebras with associative even part A whose odd part M is an associative module over A. We prove that each of these superalgebras, not isomorphic to a superalgebra of nondegenerate bilinear superform, is isomorphically embedded into a twisted Jordan superalgebra of vector type. We exhibit a new example of a simple special Jordan superalgebra. We also describe the superalgebras such that M [A,M] 0.     

Supercentralizing Superderivations on Prime Superalgebras

ABSTRACT

Let A = A ₀ ⊕ A ₁ be an associative superalgebra over a commutative associative ring F, and let Z _s (A) be its supercenter. An F-mapping f of A into itself is called supercentralizing on a subset S of A if [x, f(x)] _s  ∈ Z _s (A) for all x ∈ S. In this article, we prove a version of Posner's theorem for supercentralizing superderivations on prime superalgebras.     

Integral Bases for the Universal Enveloping Algebras of Cartan Map Superalgebras

Let ?? be a finite dimensional complex simple Lie superalgebra of Cartan type and A be a commutative, associative algebra with unity over ?. We refer to the Lie superalgebras of the form ?? ? A as Cartan map superalgebras. In this paper, following Bagci and Chamberlin (J. of Pure and Applied Algebra 218(8), 1563–1576, 2014), we define an integral form for the universal enveloping algebra of the Cartan map superalgebras, and exhibit an explicit integral basis for this integral form.     

Jordan Superderivations

Abstract

We extend Herstein's theorem on Jordan derivations of prime rings to superalgebras. Our main result states that a prime associative superalgebra admits a proper Jordan superderivation only in the case when its even part is commutative.     

Central simple superalgebras with anti-automorphisms of order two of the first kind

By a theorem of Albert's, a central simple associative algebra has an involution of the first kind if and only if it is of order 2 in the Brauer group. Our main purpose is to develop the theory of existence of anti-automorphisms of order 2 of the first kind on finite dimensional central simple associative superalgebras over K, where K is a field of arbitrary characteristic. First we need to generalize the Skolem–Noether Theorem to the superalgebra case. Then we show which kind of finite dimensional central simple superalgebras have an anti-automorphism of order 2 of the first kind.     

Existence of Hopf subalgebras of GK-dimension two

Let H be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If H is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then H contains a Hopf subalgebra of Gelfand-Kirillov dimension two.     

Gorenstein Injective Dimension for Complexes and Iwanaga–Gorenstein Rings

The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga–Gorenstein rings. In particular, we show that an associative ring R is Iwanaga–Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the ‘Ext’ functor.     

Piecewise periodicity structure estimates in Shirshov’s height theorem

The Gelfand-Kirillov dimension of l-generated general matrices is equal to (l ? 1)n ² + 1. Due to the Amitzur-Levitsky theorem, the minimal degree of the identity of this algebra is 2n. That is why the essential height of A being an l-generated PI-algebra of degree n over every set of words is greater than (l ? 1)n ²/4 + 1. In this paper we prove that if A has a finite Gelfand-Kirillov dimension, then the number of lexicographically comparable subwords with the period (n ? 1) in each monoid of A is not greater than (l ? 2)(n ? 1). The case of subwords with the period 2 can be generalized to the proof of Shirshov’s height theorem.     

On Right Alternative Superalgebras

Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M ₂(F) as even part and a unital irreducible bimodule over M ₂(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.