Algebra Seminar: Quasi-hereditary skew group algebras

Participate on site or via Zoom link (meeting ID: 645 5572 6999, for passcode: please contact the organizer)

Abstract: Let A be a finite-dimensional algebra and let G be a finite group acting on A. In this setting, a natural object of study is the skew group algebra A*G. In this talk, we will relate quasi- hereditary structures on A to quasi-hereditary structures on A*G. More precisely, we will show that, under a natural compatibility condition, a partial order on the set of isomorphism classes of simple A-modules induces a similar partial order for A*G. Moreover, equipped with these partial orders, A is quasi-hereditary if and only if A*G is. In [1], König, inspired by Borel subalgebras of Lie algebras, introduced the concept of an exact Borel subalgebra for a quasi-hereditary algebra.

In the last section of the talk we will show that if B is an exact Borel subalgebra of A which, as a set, is invariant under the G-action, then B*G becomes an exact Borel subalgebra of A*G.

References:

[1] Steffen Koenig. Exact Borel Subalgebras of Quasi-Hereditary Algebras, I. Mathematische

Zeitschrift, 220:399–426, 1995.

This is a seminar in our algebra seminar series.