Quantum Argument--Shift Subalgebras via Quantized Shift Operators

The argument--shift method constructs maximal Poisson-commutative subalgebras of the symmetric algebra $S(\mathfrak g)$ of a Lie algebra $\mathfrak g$ with respect to the Lie--Poisson bracket. Their quantizations---known as quantum argument--shift subalgebras---form maximal commutative subalgebras of the universal enveloping algebra $U(\mathfrak g)$ and play a fundamental role in quantum integrable systems. Although existence and uniqueness of these quantizations have been established in many cases, the underlying argument--shift procedures, realized as derivations of $S(\mathfrak g)$, had not previously been quantized. Recently, Georgy Sharygin and I defined quantized argument--shift procedures for $\mathfrak{gl}_n$ and proved that they generate the associated quantum argument--shift subalgebras.