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Modern Physics Letters A

Nn(n–1)/2-STATE INTERTWINER RELATED TO U q( sl (n)) ALGEBRA AT q2N=1

Rinat Kashaev, Vladimir V. Mangazeev

Published Sep 28, 1992• 1 citation
Publisher

Abstract

The Nn(n–1)/2-state R-matrix related to U q( sl (n)) algebra at q2N=1 is presented. Its matrix elements are interpreted as Boltzmann weights of an elementary box of some 2D lattice statistical model and given in terms of [Formula: see text] weights of the "minimal" sl (n) chiral Potts model. The corresponding family of transfer matrices depends on n rapidity variables living on high genus algebraic curves, the latter being defined by n moduli. The Yang-Baxter equation is conjectured to hold.

DOI
10.1142/s0217732392004201

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The Nn(n–1)/2-state R-matrix related to U q( sl (n)) algebra at q2N=1 is presented. Its matrix elements are interpreted as Boltzmann weights of an elementary box of some 2D lattice statistical model and given in terms of [Formula: see text] weights of the "minimal" sl (n) chiral Potts model. The corresponding family of transfer matrices depends on n rapidity variables living on high genus algebraic curves, the latter being defined by n moduli. The Yang-Baxter equation is conjectured to hold.
A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.
We identify the precise relationship between the five-parameter τ (2) -family in the N -state chiral Potts model and XXZ chains with U q ( sl 2 )-cyclic representation. By studying the Yang–Baxter relation of the six-vertex model, we discover a one-parameter family of L -operators in terms of the quantum group U q ( sl 2 ). When N is odd, the N -state τ (2) -model can be regarded as the XXZ chain of cyclic representations with . The symmetry algebra of the τ (2) -model is described by the quantum affine algebra via the canonical representation. In general, for an arbitrary N , we show that the XXZ chain with a U q ( sl 2 )-cyclic representation for q 2 N = 1 is equivalent to two copies of the same N -state τ (2) -model.
We adapt our previous results for the “partition function ” of the superintegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e −αH, where H is the associated hamiltonian. The spontaneous magnetization Mr can be expressed in terms of particular matrix elements of e −αH S r 1 e−βH, where S1 is a diagonal matrix. We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising model. KEY WORDS: Statistical mechanics, lattice models, transfer matrices. 1