# Yasushi Ikeda

## Commutativity of Second-Order Quasiderivations in General Linear Lie Algebras

In this talk I speak about the quantum analogue of the theorem of A. Mishchenko and A. Fomenko, which allows one to construct commutative subalgebras in the universal enveloping algebras. We replace the derivation of the symmetric algebra $S\mathit{gl}(d,\mathbb{C})$ (used in the original theorem) with the quasiderivation of the universal enveloping algebra $U\mathit{gl}(d,\mathbb{C})$, proposed by Gurevich, Pyatov, and Saponov. In my paper I derived an explicit formula for such quasiderivations and used it to prove the quantum analogue of the theorem at the first order. In this talk I proceed with the calculations for the second order: we show that complex combinatorial formulas play an important role here. We believe that understanding of the representation theory underlying these formulas can lead to a general proof of the quantum analogue.