Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl1
dc.contributor.author | Feigin, B. | |
dc.contributor.author | Jimbo, M. | |
dc.contributor.author | Miwa, T. | |
dc.contributor.author | Mukhin, Evgeny | |
dc.contributor.department | Mathematical Sciences, School of Science | en_US |
dc.date.accessioned | 2018-03-23T19:24:14Z | |
dc.date.available | 2018-03-23T19:24:14Z | |
dc.date.issued | 2017-11 | |
dc.description.abstract | We study highest weight representations of the Borel subalgebra of the quantum toroidal gl1 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 gl1 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. | en_US |
dc.eprint.version | Author's manuscript | en_US |
dc.identifier.citation | Feigin, B., Jimbo, M., Miwa, T., & Mukhin, E. (2017). Finite Type Modules and Bethe Ansatz for Quantum Toroidal $${\mathfrak{gl}_1}$$. Communications in Mathematical Physics, 356(1), 285–327. https://doi.org/10.1007/s00220-017-2984-9 | en_US |
dc.identifier.uri | https://hdl.handle.net/1805/15700 | |
dc.language.iso | en | en_US |
dc.publisher | Springer | en_US |
dc.relation.isversionof | 10.1007/s00220-017-2984-9 | en_US |
dc.relation.journal | Communications in Mathematical Physics | en_US |
dc.rights | Publisher Policy | en_US |
dc.source | ArXiv | en_US |
dc.subject | q-character theory | en_US |
dc.subject | finite type modules | en_US |
dc.subject | Bethe ansatz | en_US |
dc.title | Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl1 | en_US |
dc.type | Article | en_US |