COMMUTATIVE FINITE-DIMENSIONAL ALGEBRAS SATISFYING x(x(xy))=0 ARE NILPOTENT

We investigate the structure of commutative non-associative algebras satisfying the identity x(x(xy)) = 0. Recently, Correa and Hentzel proved that every commutative algebra satisfying above identity over a field of characteristic not equal 2 is solvable. We prove that every commutative finite-dimensional algebra u over a field F of characteristic not equal 2, 3 which satisfies the identity x(x(xy)) = 0 is nilpotent. Furthermore, we obtain new identities and properties for this class of algebras.

Finite-dimensional non-associative algebras and codimension growth

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. (C) 2010 Elsevier Inc. All rights reserved.

POLYNOMIAL IDENTITIES OF FINITE DIMENSIONAL SIMPLE ALGEBRAS

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.

Some classes of semisimple group (and loop) algebras over finite fields

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components We apply our work to obtain similar information about the loop algebras of mdecomposable RA loops and to produce negative answers to the isomorphism problem over various fields (C) 2010 Elsevier Inc All rights reserved

The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant`s Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W-algebras. (C) 2009 Elsevier Inc. All rights reserved.

Coherent state quantization of paragrassmann algebras

By using a coherent state quantization of paragrassmann variables, operators are constructed in finite Hilbert spaces. We thus obtain in a straightforward way a matrix representation of the paragrassmann algebra. This algebra of finite matrices realizes a deformed Weyl-Heisenberg algebra. The study of mean values in coherent states of some of these operators leads to interesting conclusions.

Hyperbolic unit groups and quaternion algebras

We classify the quadratic extensions K = Q[root d] and the finite groups G for which the group ring o(K)[G] of G over the ring o(K) of integers of K has the property that the group U(1)(o(K)[G]) of units of augmentation 1 is hyperbolic. We also construct units in the Z-order H(o(K)) of the quaternion algebra H(K) = (-1, -1/K), when it is a division algebra.

Hyperbolicity of semigroup algebras

Let A be a finite-dimensional Q-algebra and Gamma subset of A a Z-order. We classify those A with the property that Z(2) negated right arrow U(Gamma) and refer to this as the hyperbolic property. We apply this in case A = K S is a semigroup algebra, with K = Q or K = Q(root-d). A complete classification is given when KS is semi-simple and also when S is a non-semi-simple semigroup. (c) 2008 Elsevier Inc. All rights reserved.

Tilt-critical algebras of tame type

Let A be a finite dimensional k-algebra over an algebraically closed field. Assume A=kQ/I where Q is a quiver without oriented cycles. We say that A is tilt-critical if it is not tilted but every proper convex subcategory of A is tilted. We describe the tilt-critical algebras which are strongly simply connected and tame.

Some examples of nil Lie algebras

Generalizing Petrogradsky`s construction, we give examples of infinite-dimensional nil Lie algebras of finite Gelfand-Kirillov dimension over any field of positive characteristic.

The bar-radical of a class of almost alternative baric algebras

In this article we prove that, if (U, ) is a finite dimensional baric algebra of (gamma, delta) type over a field F of characteristic not equal 2,3,5 such that gamma(2) - delta(2) + delta = 1 and 0,1, then rad(U) = R(U)boolean AND(bar(U))(2), where R(U) is the nilradical (maximal nil ideal) of U.

Filtered Multiplicative Bases of Restricted Enveloping Algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p.

Representation type of Jordan algebras

The problem of classification of Jordan bit-nodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0. (c) 2010 Elsevier Inc. All rights reserved.

HYPERBOLICITY OF SEMIGROUP ALGEBRAS II

In 1996, Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a Z-order with hyperbolic unit group. In this paper, we complete this classification and give an easy proof that deals with all finite semigroups.

Contravariantly finite subcategories closed under predecessors

Let A be an Artin algebra and mod A be the category of finitely generated right A-modules. We prove that an additive full subcategory C of mod A closed under predecessors is contravariantly finite if and only if its right Ext-orthogonal is covariantly finite, or if and only if the Ext-injectives in C define a cotilting module (over the support algebra of C) or, equivalently, if and only if C is the support of the representable functors given by the Ext-injectives. (C) 2009 Elsevier Inc. All rights reserved.