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Our main goal is to extend one of classical Razmyslov’s Theorem saying that any two simple finite-dimensional $$\Omega $$ -algebras over an algebraically closed field, satisfying the same polynomial identities, are isomorphic. We suggest a method that allows one to reduce problems about identities of algebras with additional structure to the identities of $$\Omega $$ -algebras. For the convenience of the reader, we start with a full detailed proof of Razmyslov’s Theorem. Then we describe our method and its consequences for the identities of graded algebras, algebras with involution, and several others.

We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence., Comment: 24 pages

We survey a vast array of known results and techniques in the area of polynomial identities in pointed Hopf algebras. Some new results are proven in the setting of Hopf algebras that appeared in papers of D. Radford and N. Andruskiewitsch - H.-J. Schneider., 20 pages

We study actions of pointed Hopf algebras on matrix algebras. Our approach is based on known facts about group gradings of matrix algebras [Y. Bahturin, S. Sehgal and M. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2001) 667–698] and other sources.

Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G . In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B . We also prove that every finite dimensional simple real algebra with a G -grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras Q(n), n ≥ 2, over an algebraically closed field of characteristic different from 2 (and not div...

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field [Formula: see text] can be reduced to the following three classifications, for each finite Galois extension [Formula: see text] of [Formula: see text]: (1) finite-dimensional central division algebras over [Formula: see text], up to isomorphism; (2) twisted group algebras of finite groups over [Formula: see text], up to graded-isomorphism; (3) [Formula: see text]-forms of certain graded matrix algebras with coefficients in [Formula: see text] where [Formula: see text] is as in (1) and [Formula: see text] is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.

Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.

For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field)., 35 pages

We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.

For a given abelian group G, we classify the isomorphism classes of Ggradings on the simple restricted Lie algebras of types W.mI1/ and S.mI1/ for m 2, in terms of numerical and group-theoretical invariants. Our main tool is automorphism group schemes, which we determine for the simple restricted Lie algebras of types S.mI1/ and H.mI1/. The ground field is assumed to be algebraically closed of characteristic p > 3.

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.

We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero.

In this paper we apply the method of functional identities to the study of group gradings by an abelian group G on simple Jordan algebras, under very mild restrictions on the grading group or the base field of coefficients.

We consider Lie superautomorphisms of prime associative superalgebras. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.

In this paper we describe all gradings by an abelian group G on the simple Lie algebra p s l n ( F ) where F is an algebraically closed field of characteristic p different from 2 and dividing n .

Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of the paper Olshanskii, A. Yu.; Osin, D.V., Large groups and their periodic quotients, Proc. Amer. Math. Soc., 136 (2008), 753 - 759., 17 pages

In this paper we describe all gradings by a finite abelian group G G on the following Lie algebras over an algebraically closed field F F of characteristic p ≠ 2 p\neq 2 : s l n ( F ) \mathfrak {sl}_n(F) ( n n not divisible by p p ), s o n ( F ) \mathfrak {so}_n(F) ( n ≥ 5 n\geq 5 , n ≠ 8 n\neq 8 ) and s p n ( F ) \mathfrak {sp}_n(F) ( n ≥ 6 n\geq 6 , n n even).

We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application, we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which can serve as counterexamples for various Burnside-type questions.

We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.

The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Zaĭtsev, S.K. Segal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the structure of graded simple rings from the gradings by groups to the gradings by cancellative semigroups.

In this paper we describe all group gradings by a finite Abelian group G of several types of simple Jordan and Lie algebras over an algebraically closed field F of characteristic zero.

In this paper we extend a well-known theorem of M. Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

In this paper we find necessary and sufficient conditions on a finite-dimensional Lie superalgebra under which any associative PI-envelope of L 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.

We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank. In case of rank 0, we describe conditions to obtain non trivial $\Z_k$-gradings., Comment: 21 pages

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2., Comment: Several typographical errors have been corrected

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals here is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra.

We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong "infiniteness" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional.

In this paper we describe graded automorphisms and antiautomorphisms of finite order on matrix algebras endowed with a group gradings by a finite abelian group over an arbitrary algebraically closed field of charcteristic different from 2.

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.

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).

We consider a certain decomposition of the matrix algebra M"n(F), where F is a field. The commutation relations of that decomposition yield an n^2xn^2 matrix M^M^"^n^(^F^), which determines the multilinear polynomial identities of M"n(F). Thus if char(F)=0, the matrix M^M^"^n^(^F^) determines the polynomial identities of M"n(F). We show that M^M^"^n^(^F^) is very close to the tensor product of two nxn Vandermonde matrices. In particular this allows us to evaluate the determinant of M^M^"^n^(^F^).

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z 2 -graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274–296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M n ( F ) over the field F is regular, which is closely related to M n ( F ) being Z n -graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A ⊗ B theorem for Z 2 -graded PI algebras.

In this paper we apply the method of functional identities to the study of group gradings by an abelian group G on simple Lie algebras, under very mild restrictions on the grading group or the base field of coefficients.