A note on $q$-oscillator realizations of $U_q(gl(M\mathrm{\mid }N))$ for Baxter $Q$-operators

We consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra $U_q(gl(M\mathrm{\mid }N))$, 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 $U_q(\widehat{gl}(M\mathrm{\mid }N))$ for Baxter Q-operators.