## University of Sydney Algebra Seminar

# Alex Weekes (University of Toronto)

## Wednesday 13th November, 12:05-12:55pm, Carslaw 173

### Shift of argument algebras and the cactus group

Shift of argument algebras are maximal Poisson-commutative algebras of S(g), where g is a semi-simple Lie algebra. Their lifts to U(g) yield some interesting examples as limits - in particular the Gelfand-Tsetlin subaglebras in type A. These lifts naturally act on any weight space in any fixed representation of g, giving us a corresponding covering of the moduli space of shift of argument algebras. The fundamental group of this moduli space is the pure cactus group, which therefore acts on the fibers of the covering. We provide a conjectural link with the natural cactus group action on crystals.