Extension of a Borel subalgebra symmetry into the sl(2) loop algebra symmetry for the twisted XXZ spin chain at roots of unity and the Onsager algebra

Abstract

We discuss a conjecture that the twisted transfer matrix of the six-vertex model at roots of unity with some discrete twist angles should have the sl(2) loop algebra symmetry. As an evidence of this conjecture, we show the following mathematical result on a subalgebra of the sl(2) loop algebra, which we call a Borel subalgebra: any given finite-dimensional highest weight representation of the Borel subalgebra is extended into that of the sl(2) loop algebra, if the parameters associated with it are nonzero. Thus, if operators commuting or anti-commuting with the twisted transfer matrix of the six-vertex model at roots of unity generate the Borel subalgebra, then they also generate the sl(2) loop algebra. The result should be useful for studying the connection of the sl(2) loop algebra symmetry to the Onsager algebra symmetry of the superintegrable chiral Potts model.

18 pages with no figure; submitted to the proceedings of the workshop RAQIS, LAPTH, Annecy, France, September 11-14, 2007

References in corpus (3)

• The L(sl_2) symmetry of the Bazhanov-Stroganov model associated with the superintegrable chiral Potts model
• The six-vertex model at roots of unity and some highest weight representations of the sl(2) loop algebra
• Generalized Drinfeld polynomials for highest weight vectors of the Borel subalgebra of the $s{l}_2$ loop algebra

Cited by in corpus (1)

• An algebraic derivation of the eigenspaces associated with an Ising-like spectrum of the superintegrable chiral Potts model