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arXiv:hep-th/9303161 · Journal of Physics A Mathematical and General

Multi-colour braid-monoid algebras

Uwe Grimm, Paul A. Pearce

Published Dec 21, 1993• 36 citations
arXivHTMLPDF

Abstract

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis and Seaton(1992) as well as by Roche(1992). These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.

DOI
10.1088/0305-4470/26/24/018

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We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis and Seaton(1992) as well as by Roche(1992). These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.
The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.
A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, in the cylinder continuum limit, to a Perturbed Minimal Conformal Field Theory description is analysed and supported. b ∗
<!-- *** Custom HTML *** --> The Yang–Baxter algebras (YBA) are introduced and formulated in a general way stressing graphical methods. Their various physical applications are then exposed: lattice statistical models, integrable field theories and factorizable S-matrices. The Bethe Ansatz (BA) and its generalizations provide the explicit solutions of all these models using the appropiate YBA. The six-vertex model solution is exposed. YB algebras and their associated physical models are classified in terms of simple Lie algebras. It is exposed how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is exposed as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YB algebras. To conclude braid and quantum groups are derived from the YBA in the limit of infinite spectral parameter.
The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropiate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.
Multiple equations in math, physics, quantum information, and elsewhere are referred to as "the" Yang-Baxter equation, in spite of being a broad family of equations. Most of the equations are nonlinear matrix equations, where the unknown variable is a matrix. This is the case for the so called braided, algebraic, and generalized forms of "the" equation, which are the primary focus of this dissertation. Finding solutions to the various forms of these equations has been the subject of much research. The equations in all their forms are largely considered intractable in high dimensions, and only in dimension 2 have the solutions been fully classified. We begin with an introduction to quantum computation, with a focus on the topological model and its connection to the braid group. Next, we introduce the braided, algebraic, and generalized forms of "the" Yang-Baxter equation. We provide a full classification of diagonal solutions to each form. In particular, we show that any diagonal matrix is a solution to the algebraic form in any dimension, and each instance of the braided and generalized forms only have diagonal solutions that are scalar multiples of the identity. We exploit the relationship between the algebraic and braided forms to construct a solution in any dimension that is applicable to topological quantum computation as a universal gate. The generalized form of the equation is parameterized by three natural numbers, (d,m,l), and we show that the only invertible solutions when l ≥ m are scalar multiples of the identity. We completely classify all solutions arising from an X-shaped ansatz for five different choices of (d,m,l), and provide a complete classification of X-shaped solutions to every odd dimensional braided equation, where there are no X-shaped solutions in any dimension. We fully classify permutation solutions to each instance of the braided and algebraic equations that can be written as a product of 3 or fewer transpositions. We show that the problem of classifying all invertible upper triangular solutions to the 4-dimensional algebraic Yang-Baxter equation can be split into 48 cases, and fully classify one of the cases.