Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators

TL;DR

This paper develops explicit solutions to the graded Yang-Baxter equation using q-oscillator superalgebras, based on asymptotic representations of quantum affine superalgebras, advancing the understanding of Baxter Q-operators.

Contribution

It introduces new asymptotic representations and explicit q-oscillator solutions for the graded Yang-Baxter equation, extending to contracted algebras and universal Q-operators.

Findings

Explicit solutions of graded Yang-Baxter equation in terms of q-oscillators

Construction of universal Q-operators as supertrace of universal R-matrix

Extension of representations to contracted algebras and relation to T- and Q-functions

Abstract

We consider a class of asymptotic representations of the Borel subalgebra of the quantum affine superalgebra U_q(gl(M|N)^). This is characterized by Drinfeld rational fractions. In particular, we consider contractions of U_q(gl(M|N)) in the FRT formulation and obtain explicit solutions of the graded Yang-Baxter equation in terms of q-oscillator superalgebras. These solutions correspond to L-operators for Baxter Q-operators. We also discuss an extension of these representations to the ones for contracted algebras of U_q(gl(M|N)^) by considering the action of renormalized generators of the other side of the Borel subalgebra. We define model independent universal Q-operators as the supertrace of the universal R-matrix and write universal T-operators in terms of these Q-operators based on shift operators on the supercharacters. These include our previous work on U_q(sl(2|1)^) case [arXiv:0805.4274] in part, and also give a cue for the operator realization of our Wronskian-like formulas on T-and Q-functions in [arXiv:0906.2039, arXiv:1109.5524].