Algebra Seminar: Molev -- Vinberg’s problem for classical Lie algebras

Alexander Molev (University of Sydney) 

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

Title: Vinberg’s problem for classical Lie algebras 

Abstract: 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.