Quantum superalgebras and the free-fermionic Yang-Baxter equation

Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

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.

Quantum group and quantum symmetry

This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.

Integrable massive QFT and conformal invariant models follow from lattice integrable models in suitable scaling limits. There are Yang-Baxter algebras (YBA) associated with all these two-dimensional models. These YBA allow one to construct the exact solution (spectrum, S-matrix, form-factors,…) for this class of theories. Braid groups and quantum groups are derived as limiting cases of YBA when θ (spectral parameter) goes to ±∞.KeywordsHopf AlgebraQuantum GroupBraid GroupScaling LimitConformal WeightThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Verma and simple modules for quantum groups at non-abelian groups

The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize [Formula: see text] to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N. In the continuum scaling limit, in sectors with magnetization Sz, we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q-Narayana numbers or, equivalently, skew q-binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L0. This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.

We use statistical mechanics--variants of the six-vertex model in the plane studied by means of the Yang---Baxter equation--to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

We introduce the notion of integrable modules over $\imath $ quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\imath $ quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath $ quantum group of finite type with specific parameters coincides with Bao-Song’s coordinate ring of the $\imath $ quantum group.

We survey the connections between the six-vertex (square ice) model of 2D statistical mechanics and random matrix theory. We highlight the same universal probability distributions appearing on both sides, and also indicate related open questions and conjectures. We present full proofs of two asymptotic theorems for the six-vertex model: in the first one the Gaussian Unitary Ensemble (GUE) and GUE–corners process appear; the second one leads to the Tracy–Widom distribution F 2 F_2 . While both results are not new, we found shorter transparent proofs for this text. On our way we introduce the key tools in the study of the six-vertex model, including the Yang–Baxter equation and the Izergin–Korepin formula.

Faddeev, Reshetikhin and Takhtadzhyan showed, some time ago [1], how to construct a nice algebraic structure out of a given matrix R ∈ End(C n ⊗ C n ) satisfying the Yang-Baxter equation. They referred to these algebraic structures as the algebras of “Functions over the Quantum Group”: they are in fact bialgebras which may have an antipode, i.e. may be Hopf algebras, when some extra conditions are satisfied.

State sum invariants of 3-manifolds and quantum 6 j-symbols

The quantum inverse scattering method is a scheme for solving integrable models in dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra , at least for . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.

Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed.