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arXiv:1007.3795 · Journal of Mathematical Physics

Diagonalization of infinite transfer matrix of boundary Uq,p(AN−1(1)) face model

Takeo Kojima

Published Jan 1, 2011• 5 citations
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Abstract

We study infinitely many commuting operators TB(z), which we call infinite transfer matrix of boundary Uq,p(AN−1(1)) face model. We diagonalize the infinite transfer matrix TB(z) by using free field realizations of the vertex operators of the elliptic quantum group Uq,p(AN−1(1)).
DOI
10.1063/1.3521604

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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​^​).
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