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Journal of Mathematical Physics

Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices

Thomas Curtright, G. I. Ghandour, COSMAS K. ZACHOS

Published Mar 1, 1991• 49 citations
Publisher

Abstract

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.

DOI
10.1063/1.529410

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Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.
Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group U q [ s u ( 2 ) ] , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.
Relations are exploited among Clebsch–Gordan (CG) and Racah coefficients in the algebra suq(2), known as a deformation of su(2). These are used to show that the Yang–Baxter (YB) relation for the IRF (interaction round a face) model results from one of the symmetry relations for the 9-j symbol specific to suq(2), and that in an asymptotic limit this YB relation becomes the YB relation for the two-dimensional vertex model. The Racah coefficient, which has a particularly simple dependence on q, is efficiently used such that an asymptotic limit of the Racah coefficient is the CG coefficient and another limit gives the factorized S matrix of the vertex model.
We have constructed series of the spectral parameter dependent solutions to the Yang–Baxter equations defined on the tensor product of reducible representations with a symmetry of quantum (super)algebra. These series are produced as descendant solutions from the s l q ( 2 ) -invariant Hecke type R r r ( u ) -matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra o s p q ( 1 | 2 ) are obtained precisely. For the homogeneous solutions R r 2 − 1 r 2 − 1 there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators R r R ( u ) , extended Lax operators of Hecke type, also are suggested.
Explicit R-matrices of quantum groups are routinely transcribed from the literature, yet whether they satisfy an identity depends on basis order, tensor placement, the coproduct and, for superalgebras, the Koszul sign rule. quantum-group is a Python package that builds such matrices exactly with SymPy and returns every identity as an exact residual matrix, whose nonzero entries locate the convention that disagrees. It covers U_q(sl_2) modules, tensor products and the fundamental R-matrix, and the graded Yang–Baxter equation for GL_q(2|1). Applied to the thesis it was written for, it exposed an R-matrix that satisfies the Yang–Baxter equation but intertwines the opposite coproduct. Userguide of quantum-group 1.1.1 (software: https://doi.org/10.5281/zenodo.23039671; source: https://github.com/TerekliTahaBerk/quantum-groups). Prepared for submission to SciPost Physics Codebases.
Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\mathfrak {sl}_2sl2​ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.