Corner Transfer Matrices and Quantum Affine Algebras

Abstract

Let H be the corner-transfer-matrix Hamiltonian for the six-vertex model in the antiferroelectric regime. It acts on the infinite tensor product W = V ⊗ V ⊗ V ⊗ · · ·, where V is the 2-dimensional irreducible representation of the quantum affine Lie algebra U q sl (2) . We observe that H is the derivation of U q sl (2) , and conjecture that the eigenvectors of H form the level-1 vacuum representation of U q sl (2) . We report on checks in support of our conjecture.

An integrable -model: corner transfer matrices and Young skew diagrams

“…The classifications and character formulae in [32], [35] and [43] refer to the standard set of simple roots for sl(m|m ′ ) containing only one unique odd simple root. In contrast, [45] employs the simple root set characterized by (3). It turns out that all but one irreducible level-one modules relevant to the model studied below are found among the (infinitely many) modules listed in [45].…”

“…This amounts to rescaling the matrix R(q 2 w, z) by a factor z −2 which does not affect integrability. Renormalizations of that type are familiar from the vertex models related to quantum affine algebras [3]. The diagonal elements h ren h ren…”

“…The trace of the CTM of a homogeneous integrable vertex model based on U q ĝ relates to an affine Lie algebra character [2,3]. Incorporating the concept of vertex operators, a mathematical characterization of physical objects including transfer matrices and N-point correlators has been developed [4,5].…”

Quantum Loop Modules and Quantum Spin Chains

“…It is shown in [21] that the left-hand half (k > 0) of the CTM may be identified with the derivation d of U q sl 2 , acting on a standard level-1 highest weight module. There are two such modules, V (Λ 0 ) and V (Λ 1 ), which correspond to the two possible choices of boundary condition in the anti-ferromagnetic regime.…”

“…For the SOS models the relation is more technical: the Boltzmann weights are the connection matrices which intertwine vertex operators -themselves intertwiners -discovered by Frenkel and Reshetikhin [20]. An important ingredient in the use of the quantum affine algebra for the solution of these models is the fact that the eigenvectors of their CTMs may be idendified with the weight vectors of level-k infinite-dimensional modules, and their CTMs with the derivation operator d [21,22]. In this way, physical calculations are reduced to problems of representation theory.…”