Second order quantum argument shifts in general linear Lie algebras, Yasushi Ikeda

Yasushi Ikeda (Moscow State University). ============================================= Title: Second order quantum argument shifts in general linear Lie algebras Abstract: Let g be a Lie algebra. Argument shift subalgebras in Sg is the Poisson commutative subalgebra (with respect to the Lie-Poisson structure), spanned by the iterated derivations of the central elements in Sg with respect to a constant vector field. Inspired by quantum partial derivative operators on Ugl proposed by Gurevich, Pyatov, and Saponov, I and Georgiy Sharygin showed that the quantum argument shift algebras are generated by the corresponding iterated quantum shifts of the central elements in Ugl (later Alexander Molev wrote the computations in an advanced manner). Thus the quantum argument shift operator gives us an alternative way to find generators of the quantum argument shift subalgebra. In this talk, I introduce a general formula for calculating the second order quantum argument shift of an arbitrary central element in Ugl and obtain generators of the quantum argument shift subalgebra (up to the second order). Behind this there is some interesting combinatorial observation.