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

Zengo Tsuboi

doi:10.1016/j.nuclphysb.2014.06.017

Abstract

We consider a class of asymptotic representations of the Borel subalgebra of the quantum affine superalgebra Uq(glˆ(M|N)). This is characterized by Drinfeld rational fractions. In particular, we consider contractions of Uq(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 Uq(glˆ(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 Uq(slˆ(2|1)) case [1] in part, and also give a cue for the operator realization of our Wronskian-like formulas on T- and Q-functions in [2,3].

Highlights

The Baxter Q-operators were introduced [4] by Baxter when he solved the 8-vertex model.Nowadays his method of Q-operators is recognized as one of the most powerful tools in quantum integrable systems
Our Q-operators in [1] are universal in the sense that they do not depend on the quantum space and can be applied for both lattice models and quantum field theoretical models as well
The Q-function in [2] is labeled by the index set I, which is a subset of the set {1, 2, . . . , M + N }. We continue these our previous works and define model independent universal Q-operators for Uq(gl(M|N )) (or Uq (sl(M|N ))) as the supertrace of the universal R-matrix for any (M, N ). This gives a cue for the operator realization of the Wronskian-like formulas in [2,3]

Summary

Introduction

The Baxter Q-operators were introduced [4] by Baxter when he solved the 8-vertex model. Q-operators as the trace of the universal R-matrix over q-oscillator representations of the Borel subalgebra of the quantum affine algebra Uq(sl(2)). We mention their extension to Uq(gl(M|N )). We consider q-oscillator realizations of these contracted algebras These induce representations of the Borel subalgebra of the quantum affine superalgebra (or q-super-Yangian) via the evaluation map. The hart of an idea is to synchronize the highest weight of the representations and automorphisms of the algebra in the limit so that one can obtain finite quantities In this way, we obtain spectral parameter dependent L-operators whose matrix elements are written in terms of the q-oscillator superalgebras. There is an evaluation map evx : Uq (sl(M|N )) → Uq (gl(M|N )): e0 → xq−(−1)p(1)e11 eM+N,1q−(−1)p(M+N)eM+N,M+N , f0 → (−1)p(M+N )x−1q(−1)p(M+N)eM+N,M+N e1,M+N q(−1)p(1)e1,1 ,

This may be achieved by adding an extra Cartan element

Representations

Representations of the contracted algebras

T- and Q-operators

Concluding remarks