Almost fine gradings on algebras and classification of gradings up to isomorphism

Abstract: Since the works of Patera-Zassenhaus (1989) and Bahturin-Sehgal-Zaicev (2001), the problem of classifying gradings by groups on various algebras has received much attention. There are typically two kinds of classification of gradings on a given algebra A: fine gradings up to equivalence or all G-gradings, for a fixed group G, up to isomorphism. These classifications are related, but it is not straightforward to pass from one to the other. In this talk, based on a recent paper with A. Elduque, we introduce a class of gradings, which we call almost fine, on a finite-dimensional algebra A over an algebraically closed field, such that every G-grading on A is obtained from an almost fine grading in an essentially unique way (which is not the case with fine gradings). For abelian groups, we give a method of obtaining all almost fine gradings if fine gradings are known. If time permits, we will illustrate this approach in the case of simple Lie algebras in characteristic 0: to any abelian group grading with nonzero identity component, we attach a (possibly nonreduced) root system Φ and construct an adapted Φ-grading.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar