# A note on $q$-oscillator realizations of $U_{q}(gl(M|N))$ for Baxter $Q$-operators

## TL;DR

This paper develops explicit q-oscillator representations of a Borel subalgebra of the quantum affine superalgebra $U_q(\ ext{gl}(M|N))$ to facilitate the construction of Baxter Q-operators, using asymptotic limits and generator reduction methods.

## Contribution

It introduces a novel reduction approach on q-oscillator algebra generators to obtain realizations of contracted algebras and explicit representations for Baxter Q-operators.

## Key findings

- Derived q-oscillator realizations of contracted algebras.
- Provided explicit q-oscillator representations of a Borel subalgebra.
- Simplified the construction of Baxter Q-operators for quantum superalgebras.

## Abstract

We consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra $U_{q}(gl(M|N))$, and obtain q-oscillator realizations of the contracted algebras proposed in [arXiv:1205.1471]. Instead of factoring out the invariant subspaces, we make reduction on generators of the q-oscillator algebra, which gives a shortcut to the problem. Based on this result, we obtain explicit q-oscillator representations of a Borel subalgebra of the quantum affine superalgebra $U_{q}(\hat{gl}(M|N))$ for Baxter Q-operators.