A note on q-oscillator realizations of U(gl(M|N)) for Baxter Q-operators

Abstract

A note on q-oscillator realizations of U q (gl(M |N )) forBaxter Q-operators Zengo TsuboiLaboratory of physics of living matter, AbstractWe consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra U q (gl(M |N )), and obtain qoscillator realizations of the contracted algebras proposed in [1]. 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 (ĝl(M |N )) for Baxter Q-operators.1 As for the rational (q = 1) case, see [14,15,16]. There is another approach to Q-operators [17,18,19].1 over a Borel subalgebra of U q (ŝl(3)). Moreover, Hernandez and Jimbo showed [20] that the same type of q-oscillator representations can be systematically constructed by taking asymptotic limits of Kirillov-Reshetikhin modules over one of the Borel subalgebras of any non-twisted quantum affine algebra. In addition, this approach was further developed [21,22] for U q (ŝl(M|N)) case. Hernandez and Jimbo's approach is representation theoretically sophisticated, but rather abstract, and thus it is still meaningful to seek another method to obtain explicit q-oscillator realizations, which will be useful for applications to concrete problems. In this paper, we make a proposal on this for U q (ĝl(M|N)) case, where we develop, in part, the scheme proposed in our previous paper [1]. In our classification [23] of the Q-operators, there are 2 M +N kinds of Q-operators for U q (ĝl(M|N)), each of which is labeled by a subset I of {1, 2, . . . , M + N}. In the paper [1], we mainly considered Card(I) = 0, 1, M + N − 1, M + N cases 2 . In this paper, we propose q-oscillator realizations for 2 ≤ Card(I) ≤ M + N − 2 case.In general, the Kirillov-Reshetikhin modules are considered to be derived from Verma modules based on a procedure, called the BGG-resolution. This implies that one has to factor out unnecessary invariant subspaces to get the final results if one starts from Verma modules [4,13]. In this paper, we also start from Verma modules, but realize them in terms of the q-oscillator algebra based on the Heisenberg realization (q-difference realization) of U q (gl(M|N)) [24, 25] on the flag manifold (for N = 0 case, [26]) from the very beginning, and then consider reduction on generators of the q-oscillator algebra, from which we obtain various q-oscillator realizations of U q (gl(M|N)) that interpolate the full Verma module and the simplest q-oscillator realization, namely, the q-Holstein-Primakoff type realization (cf. [27]). By taking limits of them, we obtain q-oscillator realizations of contracted algebras 3 U q (gl(M|N; I)) for U q (gl(M|N)) [1], and those of the q-super-Yangian Y q (gl(M|N)) via an evaluation map. A merit to consider reduction on the q-oscillator algebra lies in the fact that we do not have to factor out invariant subspaces, and thereby ar...

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Transfer matrices of rational spin chains via novel BGG-type resolutions

“…The factorisation formula of the Lax matrices entering the definition of the infinite-dimensional transfer matrices into the product of two degenerate Lax matrices used to construct Baxter Q-operators was initially proposed in [BLZ] for U q ( sl 2 ), see also [KT]. The degenerate Lax matrices (solutions of (1.1) with the degenerate coefficient of the leading term, which are no longer connected to quantum groups) employed in this factorisation have a long history, going back to [IK, KSS] and their relation to Q-operators [AF, BLZ, RW, Kor, BLMS], and for higher rank cases [BHK,BT,BGKNR,BFLMS,Ts2], while cases beyond A-type were first found in [Fr, CGY, FT].…”

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