On the supersymmetric XXX spin chains associated to $\mathfrak{gl}_{11}$
Abstract
We study the $\mathfrak{gl}_{11}$ supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation $\mathfrak{gl}_{11}$ Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and antisymmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.13360
 Bibcode:
 2019arXiv191013360L
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Representation Theory
 EPrint:
 32 pages, 1 figure