Quantum superalgebras at roots of unity and non-Abelian symmetries of integrable models

Abstract

We consider integrable vertex models whose Boltzmann weights (Rmatrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras Uq(ĝ). These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter q when periodic boundary conditions are imposed. If q is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-abelian subalgebras which are translation invariant and commute with the transfer matrix and therefore with all charges of the model. In the line of argument we introduce the restricted quantum superalgebra U res q (ĝ) and investigate its root of unity limit. We prove several new formulas involving supercommutators of arbitrary powers of the Chevalley-Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.

The twisted XXZ chain at roots of unity revisited

“…This property is common to a large class of integrable vertex models associated with trigonometric solutions to the Yang-Baxter equation and quantum affine (super)algebras. Despite the obvious modifications in the algebraic structure of the Yang-Baxter algebra we expect that the results found here can be extended to these models similar as the periodic case has been generalised to other models in [8] and [9] (albeit with different methods). Note that these are incommensurate sectors.…”

“…In particular, it avoids having first to prove translation invariance, cf. [1,8,9]. The extension to the inhomogeneous case is also discussed.…”

Auxiliary matrices for the six-vertex model atqN 1 and a geometric interpretation of its symmetries

“…In particular, it would be helpful to have a direct implementation of the group action (172) on the spin-chain. This would allow one to obtain further insight how the earlier observed loop symmetry in the commensurate sectors S z = 0 mod N [6,12,13] extends to the non-commensurate ones. It is an intriguing observation that the generators of the quantum coadjoint action are closely related with the restricted quantum group expressing the loop algebra symmetry.…”

“…Although the present discussion is limited to the six-vertex model with spin one-half, the occurrence of the loop symmetry at roots of unity is a general phenomenon. See [12,13] for generalizations to models with higher spin and higher rank.…”

“…Since all auxiliary matrices will be shown to commute with the six-vertex transfer matrix, while they in general do not commute among themselves, this group action manifests the infinite-dimensional non-abelian symmetry of the six-vertex model at q N = 1. Recall that the loop symmetry has only been established in the commensurate sectors where the total spin is a multiple of N [6,12,13]. In contrast the auxiliary matrices will be defined for all spin-sectors.…”