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Journal of Physics A Mathematical and General

Quantum R matrix for G2and a solvable 175-vertex model

Atsuo Kuniba

Published Apr 21, 1990• 35 citations
Publisher

Abstract

The quantum R matrix for Cartan's exceptional simple Lie algebra G 2 in its seven-dimensional (minimal) representation is presented. The R matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra U q (G 2 ). This defines a new solvable seven-state 175-vertex model on the planar square lattice whose Boltzmann weights satisfy the Yang-Baxter equation. For the equivalent face model, the local state probability is obtained in terms of a relative of a level-1 string function related to the affine Lie algebra pair B 3 (1) contains/implies G 2 (1) .

DOI
10.1088/0305-4470/23/8/011

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The quantum R matrix for Cartan's exceptional simple Lie algebra G 2 in its seven-dimensional (minimal) representation is presented. The R matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra U q (G 2 ). This defines a new solvable seven-state 175-vertex model on the planar square lattice whose Boltzmann weights satisfy the Yang-Baxter equation. For the equivalent face model, the local state probability is obtained in terms of a relative of a level-1 string function related to the affine Lie algebra pair B 3 (1) contains/implies G 2 (1) .
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We have found a family of solvable nineteen vertex model with statistical configurations invariant by the time reversal symmetry within a systematic study of the respective Yang–Baxter relation. The Boltzmann weights sit on a degree seven algebraic threefold which is shown birationally equivalent to the three-dimensional projective space. This permits to write parameterized expressions for both the transition operator and the R-matrix depending on three independent affine spectral parameters. The Hamiltonian limit tells us that the azimuthal magnetic field term is connected with the asymmetry among two types of spectral variables. The absence of magnetic field defines a physical submanifold whose geometrical properties are remarkably shown to be governed by a quartic K3 surface. This expands considerably the class of irrational manifolds that could emerge in the theory of quantum integrable models.
A bstract This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the su(3)k \mathfrak{su}{(3)}_k su(3)k​ su 3 k affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the Uq(sl(3)) {U}_q\left(\mathfrak{sl}(3)\right) Uq​(sl(3)) U q sl 3 quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of R matrix elements.