## University of Sydney Algebra Seminar

# Alexander Molev (University of Sydney)

## Friday 16 March, 12-1pm, Place: Carslaw 375

## Vinberg’s problem for classical Lie algebras

The symmetric algebra S(g) of a Lie algebra g is equipped with the Poisson-Lie bracket. A family of Poisson commutative subalgebras can be produced by "shifting the arguments" of g-invariants of S(g). Vinberg’s problem stated in 1990 concerns the existence of commutative subalgebras of the universal enveloping algebra U(g) which would "quantise" these subalgebras of S(g). When the Lie algebra g is simple, a general solution of Vinberg’s problem is provided by the vertex algebra theory. This leads to an explicit quantisation of the shift-of-argument subalgebras in the classical types via the symmetrisation map.