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In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in [7,8].

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

This is an attempt to survey the progress made in the study of identical relations in Lie algebras during almost three decades since the publication of my book “Identical Relations in Lie Algebras”, the only monograph devoted entirely to this area. Accordingly, we assume that that reader has access to either the Russian or the English version of this book.

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 .

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.

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.

We consider a class of Lie algebras L such that L admits a grading by a finite Abelian group so that each nontrivial homogeneous component is one-dimensional. In particular, this class contains simple Lie algebras of types A, C and D where in C and D cases the rank of L is a power of 2. We give a simple construction of a family of central elements of the universal enveloping algebra U(L). We show that for the A-type Lie algebras the elements coincide with the Gelfand invariants and thus generate the center of U(L). The construction can be extended to Lie superalgebras with the additional assumption that the group grading is compatible with the parity grading.

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.

We present some results about Lie algebras, which can be written as the sum of two subalgebras in two cases: where both subalgebras are simple or both are nilpotent. In the first case we suggest new examples of simple Lie algebras admitting decomposition into the sum of simple subalgebras and give explicit realizations where the existence of such decompositions was established earlier. We single out cases where such decomposition is not possible. We also construct examples of solvable Lie algebras, which are the sums of two nilpotent subalgebras, and the derived length of the sum is greater than the sum of the nilpotent indexes of the summands.

In this article, which is the first part of a sequence of two, we discuss modern approaches to the classification of group gradings on simple and nilpotent Lie algebras. In the second article we discuss applications and related topics in differential geometry.

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.

Let P k be the symmetric polynomial of degree k i.e., the full linearization of the polynomial x k . Let G be a cancellation semigroup with 1 and R a G-graded ring with finite support of order n. We prove that if R 1 satisfies $ P_k \equiv 0 $ then R satisfies $ P_{kn} \equiv 0 $ .

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 study the structure of Lie algebras in the category H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in H MA. 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.

In this paper the main result is the rigidity of varietally free Lie algebras within the varieties where they are free. In fact, these algebras are usually free in some larger varieties, such as various types of the commutator varieties, and this is demonstrated in this paper as well. A related paper is [2] where, among some other results, we claimed the rigidity of free metabelian algebras within the respective variety of Lie algebras.