Journal of Physics A Mathematical and General

Integrable quantum chain and the representation of the quantum group SUq(2)

Bo-Yu Hou, Kangjie Shi, Xhong-Xia Yang, Rui‐Hong Yue

Published Aug 21, 1991• 12 citations

Abstract

The authors discuss the XXZ spin chain with certain boundary terms and the representation of quantum group SU q (2). It is shown that Bethe ansatz states are highest-weight states of SU q (2). With a generic q they construct the irreducible representations of the quantum group. For q being a root of unity, they show that there are new Bethe ansatz states, which coincide with null states of SU q (2). By taking certain limits the authors can derive the state mod b), which is necessary for constructing the indecomposable but reducible representations of SU q (2) and for the completeness of the state space. In this case the Hamiltonian may not be completely diagonalized.

DOI: 10.1088/0305-4470/24/16/020

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Journal of Physics A Mathematical and General

Integrable quantum chain and the representation of the quantum group SUq(2)

Bo-Yu Hou, Kangjie Shi, Xhong-Xia Yang, Rui‐Hong Yue

Published Aug 21, 1991• 12 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/24/16/020

Related papers

INTEGRABLE QUANTUM CHAIN AND THE REPRESENTATION OF QUANTUM GROUP SUq(2)

Bo-Yu Hou, Shi Kang-Jie, YANG ZHONG-XIA +2 more

Acta Physica SinicaJan 1, 19920 citations

BETHE ANSATZ AND QUANTUM GROUPS

Hector J. de Vega

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Tetsuo Deguchi

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arXiv:1702.01482Journal of Physics A Mathematical and TheoreticalMay 31, 20175 citations

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

The

sl_2

loop algebra symmetry of the XXZ spin chain: an algorithm for the degeneracy of a regular Bethe state(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

The Hamiltonian of the XXZ spin chain has the

sl_{2}

loop algebra symmetry if the

q

parameter is given by a root of unity,

q_{0}^{2N}=1

, for an integer

N

.We review an algorithm for determining the dimensions of the degenerate eigenspace generated by a given regular Bethe state in some sectors.The proof of the algorithm has been formulated in Ref.[3] as follows: We show in some sectors that regular Bethe ansatz eigenvectors are highest weight vectors and generate irreducible representations of the

sl_{2}

loop algebra; We prove that every finite-dimensional highest weight representation of the

sl_{2}

loop algebra is irreducible; We then derive the dimensions of the highest weight representation generated by a given regular Bethe state through the Drinfeld polynomial, which is expressed explicitly in terms of the Bethe roots.

XXZ Bethe states as highest weight vectors of the

sl_2

loop algebra at roots of unity

We prove some part of the conjecture that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the

sl_2

loop algebra. Here

q

is related to the XXZ anisotropic coupling

Δ

Δ=(q+q^{-1})/2

, and it is given by a root of unity,

q^{2N}=1

, for a positive integer

N

. We show that regular XXZ Bethe states are annihilated by the generators

{\bar x}_k^{+}

's, for any

N

. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra,

{\bar h}_k

's. Here the loop algebra

U(L(sl_2))

is generated by

{\bar x}_k^{\pm}

and

{\bar h}_k

for

k \in {\bf Z}

, which are the classical analogues of the Drinfeld generators of the quantum loop algebra

U_q(L(sl_2))

. A representation of

U(L(sl_2))

is called highest weight if it is generated by a vector

Ω

which is annihilated by the generators

{\bar x}_k^{+}

's and such that

Ω

is an eigenvector of the

{\bar h}_k

's. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of

U(L(sl_2))