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arXiv:1101.4084

Infinitely many commuting operators for the elliptic quantum group $U_{q,p}(\hat{sl_N})$

Takeo Kojima

Published Jan 21, 2011• 0 citations
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Abstract

We construct two classes of infinitely many commuting operators associated with the elliptic quantum group $U_{q,p}(\hat{sl_N})$. We call one of them the integral of motion ${\cal G}_m$, $(m \in {\mathbb N})$ and the other the boundary transfer matrix $T_B(z)$, $(z \in {\mathbb C})$. The integral of motion ${\cal G}_m$ is related to elliptic deformation of the $N$-th KdV theory. The boundary transfer matrix $T_B(z)$ is related to the boundary $U_{q,p}(\hat{sl_N})$ face model. We diagonalize the boundary transfer matrix $T_B(z)$ by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion ${\cal G}_m$ is open problem even for the simplest case $U_{q,p}(\hat{sl_2})$.

DOI
10.48550/arxiv.1101.4084

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Infinitely many commuting operators for the elliptic quantum group Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​)
We construct two classes of infinitely many commuting operators associated with the elliptic quantum group Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​). We call one of them the integral of motion Gm{\cal G}_mGm​, (m∈N)(m \in {\mathbb N})(m∈N) and the other the boundary transfer matrix TB(z)T_B(z)TB​(z), (z∈C)(z \in {\mathbb C})(z∈C). The integral of motion Gm{\cal G}_mGm​ is related to elliptic deformation of the NNN-th KdV theory. The boundary transfer matrix TB(z)T_B(z)TB​(z) is related to the boundary Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​) face model. We diagonalize the boundary transfer matrix TB(z)T_B(z)TB​(z) by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion Gm{\cal G}_mGm​ is open problem even for the simplest case Uq,p(sl2^)U_{q,p}(\hat{sl_2})Uq,p​(sl2​^​).
We diagonalize infinitely many commuting operators T B ( z ). We call these operators T B ( z ) the boundary transfer matrix associated with the quantum group and the elliptic quantum group. The boundary transfer matrix is related to the solvable model with a boundary. When we diagonalize the boundary transfer matrix, we can calculate the correlation functions for the solvable model with a boundary. We review the free field approach to diagonalization of the boundary transfer matrix T B ( z ) associated with U q ( A (2) 2 ) and . We construct the free field realizations of the eigenvectors of the boundary transfer matrix TB ( z ). This paper includes new unpublished formula of the eigenvector for U q ( A (2) 2 ). It is thought that this diagonalization method can be extended to more general quantum group U q ( g ) and elliptic quantum group U q,p ( g ).
Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2)Eτ,η​(sl2​)
To each representation of the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2)Eτ,η​(sl2​) is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a qqq-deformation of Hermite's solution of the Lam\'e equation.
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group E , ( sl 2 ) by difference operators. One model of statistical mechanics to which this method applies is the interaction-round-a-face model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher-order theta functions.