Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl1

• Published in: Communications in mathematical physics Vol. 356; no. 1; pp. 285 - 327
• Main Authors: Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-017-2984-9

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl 1 algebra with finite-dimensional weight spaces. In particular, we develop the q -character theory for such modules. We introduce and study the subcategory of ‘finite type’ modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current ψ + ( z ) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T V , W ( u ; p ) analogous to those of the six vertex model. In our setting T V , W ( u ; p ) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl 1 with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules V the corresponding transfer matrices, Q ( u ; p ) and T ( u ; p ) , are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q ( u ; p ). Then we show that the eigenvalues of T V , W ( u ; p ) are given by an appropriate substitution of eigenvalues of Q ( u ; p ) into the q -character of V .