Related papers: Gradings on semisimple algebras
We classify group gradings on the simple Lie algebras of types $G_2$ and $D_4$ over the field of real numbers (or any real closed field): fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism.
The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the…
This paper presents a survey of the results and ideas behind the classification of the fine gradings, up to equivalence, on the simple finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It provides an…
We classify up to equivalence the gradings on Hurwitz superalgebras and on symmetric composition superalgebras, over any field. Also, classifications up to isomorphism are given in case the field is algebraically closed. By grading, here we…
First we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an…
We classify, up to equivalences, all abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras. Any grading on a null-filiform Leibniz algebra is toral but there are non-toral gradings on one-parametric filiform…
Let A,B be finite dimensional G-graded algebras over an algebraically closed field K with char(K)=0, where G is an abelian group, and let Id_G(A) be the set of graded identities of A (res. Id_G(B)). We show that if A,B are G-simple then…
We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular…
We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a…
We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric…
In this paper we describe all gradings by abelian groups without elements of order p, where p > 2 is the characteristic of the base field, on the simple graded Cartan type Lie algebras.
All gradings by abelian groups are classified on the following algebras over an algebraically closed field of characteristic not 2: the simple Lie algebra of type $G_2$ (characteristic not 3), the exceptional simple Jordan algebra, and the…
We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these…
Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…
Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite…
This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with…
This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras…
We establish correspondances between factorisations of finite abelian groups (direct factors, unitary factors, non isomorphic subgroup classes) and factorisations of integer matrices. We then study counting functions associated to these…
We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading…
We generalize the notion of semi-normalized classes of systems of differential equations, study properties of such classes and extend the algebraic method of group classification to them. In particular, we prove the important theorems on…