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.

Jan 1, 200113 citations

Aspects of Integrable Models

Costas John Efthimiou

Jan 1, 19940 citations

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

Alexander A. Belavin

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.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.

arXiv:hep-th/0305209Journal of Physics A Mathematical and GeneralDec 15, 20032 citations

Critical exponents for integrable models

N. M. Bogoliubov, Anatoli G. Izergin, V. E. Korepin

Nuclear Physics BDec 1, 1986201 citations

Conformal Properties of Integrable Models

M. Karowski

Springer eBooksJan 1, 19871 citation

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.

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.