Revisiting simple Lie algebras from the view of the grading group

Abstract: In this talk we discuss the role of gradings on Lie algebras and how they can be used to obtain suitable models for a wide range of structures, including simple, solvable, and nilpotent Lie algebras. We introduce the notion of a \emph{generalized group algebra}, which provides a flexible framework for describing graded algebras in full generality. Although this concept may at first appear too broad to be useful in the specific context of Lie algebras, we shall show how naturally it applies by examining several illustrative examples.

A central theme will be the use of the \emph{grading group} to gain insight into structural properties of the underlying Lie algebra. For instance, viewing $\mathfrak{so}(8)$ as a generalized group algebra sheds light on its spin representations and on the phenomenon of triality. More broadly, this perspective greatly simplifies the construction of orthonormal bases, as in the case of $\mathfrak{g}_2$.

Finally, we show that this approach renders the computation of brackets in Lie algebras obtained via \emph{graded contractions} essentially immediate. This leads to vast families of high-dimensional nonsimple Lie algebras with distinctive structural features, obtained from the exceptional simple Lie algebras through the Tits construction.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar