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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.

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.

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.

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.