TY - JOUR
T1 - Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl1
AU - Feigin, B.
AU - Jimbo, M.
AU - Miwa, T.
AU - Mukhin, E.
N1 - Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.
PY - 2017/11/1
Y1 - 2017/11/1
N2 - 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 TV,W(u; p) analogous to those of the six vertex model. In our setting TV,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 TV,W(u; p) are given by an appropriate substitution of eigenvalues of Q(u; p) into the q-character of V.
AB - 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 TV,W(u; p) analogous to those of the six vertex model. In our setting TV,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 TV,W(u; p) are given by an appropriate substitution of eigenvalues of Q(u; p) into the q-character of V.
UR - http://www.scopus.com/inward/record.url?scp=85027970512&partnerID=8YFLogxK
U2 - 10.1007/s00220-017-2984-9
DO - 10.1007/s00220-017-2984-9
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AN - SCOPUS:85027970512
SN - 0010-3616
VL - 356
SP - 285
EP - 327
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 1
ER -