Yasushi Ikeda
Quantum argument shift method for the universal enveloping algebra $U\mathfrak{gl}_n$
The argument shift method is a technique for constructing Lie--Poisson maximal commutative subalgebras (argument shift subalgebras) of the symmetric algebra $S(\mathfrak g)$ of a Lie algebra $\mathfrak g$. The quantizations of these subalgebras---known as quantum argument shift subalgebras---are maximal commutative subalgebras of the universal enveloping algebra $U(\mathfrak g)$ and play a fundamental role in quantum integrable systems. Although their existence and uniqueness have been established in many cases, the argument shift procedures themselves remained unquantized. Last year, Georgy Sharygin and I defined quantized argument shift procedures for the general linear Lie algebra $\mathfrak{gl}_n$ and proved that they generate the corresponding quantum argument shift subalgebras.
