This article is cited in 4 papers

On the Degenerate Multiplicity of the $\mathrm{sl}_2$ Loop Algebra for the 6V Transfer Matrix at Roots of Unity

Tetsuo Deguchi

Department of Physics, Faculty of Science, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-Ku, Tokyo 112-8610, Japan

Abstract: We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight $\bar d_k^{\pm}$, which leads to evaluation parameters $a_j$. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.

Keywords: loop algebra; the six-vertex model; roots of unity representations of quantum groups; Drinfeld polynomial.

MSC: 81R10; 81R12; 81R50; 81V70

Received: October 31, 2005; in final form February 6, 2006; Published online February 17, 2006

Language: English

DOI: 10.3842/SIGMA.2006.021

Bibliographic databases:

• 
• 
• 
• 

ArXiv: cond-mat/0602427