Non-reductive lie algebras and their representations with good invariant-theoretic properties
Seminar Room 1, Newton Institute
One of the main topics of Invariant Theory is to describe finite-dimensional representations of complex reductive groups with good properties of algebras of invariants. For instance, it is known that, for simple algebraic groups, most of irreducible representations with polynomial algebras of invariants occur in connection with periodic automorphisms of semisimple Lie algebras. (These are the so-called "$\Theta$-groups" of Vinberg.)
In my talk, I will discuss several constructions of non-reductive algebras having polynomial algebras of invariants for the adjoint or coadjoint representations (e.g. contractions of semisimple algebras and iterated semi-direct products). Some of these coadjoint representations can also be understood as $\Theta$-group associated with periodic automorphisms of non-reductive Lie algebras. There are also possibilities for constructing more general representations with polynomial algebras of invariants using bi-periodic gradings and contractions.