Yuri Bahturin

Graded polynomial identities of matrices

We consider G-graded polynomial identities of the p×p matrix algebra Mp(K) over a field K of characteristic 0 graded by an arbitrary group G. We find relations between the G-graded identities of the G-graded algebra Mp(K) and the (G×H)-graded identities of the tensor product of Mp(K) and the H-graded algebra Mq(K) with a fine H-grading. We also find a basis of the G-graded identities of Mp(K) with an elementary grading such that the identity component coincides with the diagonal of Mp(K).

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.

On the generalized Lie structure of associative algebras

We study the structure of Lie algebras in the categoryHMA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA inHMA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain.

Classification of group gradings on simple Lie algebras of types A, B, C and D

Abstract For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple Lie algebras of types A n ( n ⩾ 1 ), B n ( n ⩾ 2 ), C n ( n ⩾ 3 ) and D n ( n > 4 ), in terms of numerical and group-theoretical invariants. The ground field is assumed to be algebraically closed of characteristic different from 2.

identities on associative algebras

Let R be an algebra over a field and G a finite group of automorphisms and anti-automorphisms of R. We prove that if R satisfies an essential G-polynomial identity of degree d, then the G-codimensions of R are exponentially bounded and R satisfies a polynomial identity whose degree is bounded by an explicit function of d. As a consequence we show that if R is an algebra with involution ∗ satisfying a ∗-polynomial identity of degree d, then the ∗-codimensions of R are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case R must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity. §

Group Gradings on G 2

In this article, we describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G 2 over an algebraically closed field F of characteristic zero.

Конечномерные простые градуированные алгебры@@@Finite-dimensional simple graded algebras

[1] И.Л. Кантор, “Некоторые обобщения йордановых алгебр”, Тр. сем. по векторному и тензорному анализу, 16 (1972), 407–499 MathSciNet Zentralblatt MATH [2] В. Г. Кац, “Градуированные алгебры Ли и симметрические пространства”, Функц. анализ и его прил., 2:2 (1968), 93–94 Math-Net.Ru MathSciNet Zentralblatt MATH ; англ. пер.: V.G. Kats, “Graduated Lie algebras and symmetric spaces”, Funct. Anal. Appl., 2:2 (1968), 182–183 [3] A. Elduque, “Gradings on octonions”, J. Algebra, 207:1 (1998), 342–354 MathSciNet Zentralblatt MATH [4] Yu. Bahturin, I. Shestakov, “Gradings of simple Jordan algebras and their relation to the gradings of simple associative algebras”, Comm. Algebra, 29:9 (2001), 4095–4102 MathSciNet Zentralblatt MATH [5] Yu. A. Bahturin, I. P. Shestakov, M.V. Zaicev, “Gradings on simple Jordan and Lie algebras”, J. Algebra, 283:2 (2005), 849–868 MathSciNet Zentralblatt MATH [6] Yu. A. Bahturin, M.V. Zaicev, “Group gradings on simple Lie algebras of type “A””, J. Lie Theory, 16:4 (2006), 719–742 MathSciNet Zentralblatt MATH

GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ]. *Partially supported by NSERC Grant. †Partially supported by CNPq grant 300528/99-0.

Group gradings on simple Lie algebras in positive characteristic

In this paper we describe all gradings by a finite abelian group G on the following Lie algebras over an algebraically closed field F of characteristic p ≠ 2: sl n (F) (n not divisible by p), so n (F) (n > 5, n ≠ 8) and sp n (F) (n > 6, n even).

Pi-envelopes of Lie superalgebras

In this paper we find necessary and sufficient conditions on a finitedimensional Lie superalgebra under which any associative PI-envelope of L is finite-dimensional. We also extend M. Scheunert’s result which enables one to pass from color Lie superalgebras to the ordinary ones, to the case of gradings by an arbitrary abelian group.