arXiv:hep-th/0305209 · Journal of Physics A Mathematical and General

Centre of quantum group in roots of unity and the restriction of integrable models

Alexander A. Belavin

Published Dec 15, 2003• 2 citations

Abstract

We show the connection between the extended center of the quantum group in roots of unity and the restriction of the $XXZ$ model. We also give explicit expressions for operators that respect the restriction and act on the state space of the restricted models. The formulas for these operators are verified by explicit calculation for third-degree roots; they are conjectured to hold in the general case.

DOI: 10.1088/0305-4470/37/2/003

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arXiv:hep-th/0305209 · Journal of Physics A Mathematical and General

Centre of quantum group in roots of unity and the restriction of integrable models

Alexander A. Belavin

Published Dec 15, 2003• 2 citations

arXiv HTML PDF

Abstract

We show the connection between the extended center of the quantum group in roots of unity and the restriction of the

XXZ

model. We also give explicit expressions for operators that respect the restriction and act on the state space of the restricted models. The formulas for these operators are verified by explicit calculation for third-degree roots; they are conjectured to hold in the general case.

DOI: 10.1088/0305-4470/37/2/003

Related papers

Loop symmetry of integrable vertex models at roots of unity

Korff, C, Barry M. McCoy

It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$ . These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\\hat g)$ into account. Here $\\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\\hat{g})$ . For the simplest case $\\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity.It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$. These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\\hat g)$ into account. Here $\\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\\hat{g})$. For the simplest case $\\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity.

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arXiv:hep-th/9308008Jun 1, 19930 citations

Solving Baxter's TQ-equation via representation theory

Christian Korff

arXiv:math-ph/0411034Nov 9, 20047 citations

We show the connection between the extended center of the quantum group in roots of unity and the restriction of the

XXZ

It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter

(q+q^{-1})/2

is evaluated at roots of unity

q^{N}=1

. These new symmetries have been linked to an

U(A^{(1)}_1)

invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups

U_q(\\hat g)

into account. Here

\\hat g

is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of

U(\\hat{g})

. For the simplest case

\\hat g=A_1^{(1)}

the discussion is also extended to even primitive roots of unity.

Quantum Group Theory in

τ^{(2)}

-model, Duality of

τ^{(2)}

-model and XXZ-model with Cyclic

{\bf U_q(sl_2)}

-representation for

{\bf q^n =1}

, and Chiral Potts Model

We identify the quantum group ${\Large\textsl{U}}_\textsl{w}(sl_2)$ in the

L

-operator of

τ^{(2)}

-model for a generic $\textsl{w}$ as a subalgebra of

U_{\sf q} (sl_2)

with $\textsl{w} = {\sf q}^{-2}$. In the roots of unity case, ${\sf q}=q, \textsl{w} = ω$ with

q^{\bf n} = ω^N = 1

, the eigenvalues and eigenvectors of XXZ-model with the

U_q (sl_2)

-cyclic representation are determined by the

τ^{(2)}

-model with the induced ${\Large\textsl{U}}_ω(sl_2)$-cyclic representation, which is decomposed as a finite sum of

τ^{(2)}

-models in non-superintegrable inhomogeneous

N

-state chiral Potts model. Through the theory of chiral Potts model, the

Q

-operator of XXZ-model can be identified with the related chiral Potts transfer matrices, with special features appeared in the

{\bf n}=2N

, e.g.

N

even, case. We also establish the duality of

τ^{(2)}

-models related to cyclic representations of

U_q (sl_2)

, analogous to the

τ^{(2)}

-duality in chiral Potts model; and identify the model dual to the XXZ model with

U_q (sl_2)

-cyclic representation.

At roots of unity the

N

-state integrable chiral Potts model and the six-vertex model descend from each other with the

τ_2

model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different

q

-Pochhammer symbols, one construction only working well for

N

odd, the other equally well for all

N

. We also address the generalization based on the sl

(m,n)

vertex model.

q

is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact

SU(2)_q

invariance on the lattice. The algebra of the Yang-Baxter (YB) and

SU(2)_q

generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the

SU(2)_q

generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of

SU(2)_q

. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(

p

) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of

SU(2)_q

, at

q=e^{iπ/p}

, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J