Graded Derived Equivalences

We consider the equivalences of derived categories of graded rings over different groups. A Morita type equivalence is established between two graded algebras with different group gradings. The results obtained here give a better understanding of the equivalences of derived categories of graded rings.

Graded-division algebras over arbitrary fields

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.

Group gradings and actions of pointed Hopf algebras

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.

Induced quotient group gradings of epsilon-strongly graded rings

Let [Formula: see text] be a group and let [Formula: see text] be a [Formula: see text]-graded ring. Given a normal subgroup [Formula: see text] of [Formula: see text], there is a naturally induced [Formula: see text]-grading of [Formula: see text]. It is well known that if [Formula: see text] is strongly [Formula: see text]-graded, then the induced [Formula: see text]-grading is strong for any [Formula: see text]. The class of epsilon-strongly graded rings was recently introduced by Nystedt et al. as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and sufficient conditions for the induced [Formula: see text]-grading of an epsilon-strongly [Formula: see text]-graded ring to be epsilon-strong. Our method involves relating different types of rings equipped with local units ([Formula: see text]-unital rings, rings with sets of local units, rings with enough idempotents) with generalized epsilon-strongly graded rings.

On H-simple not necessarily associative algebras

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.