Affine Temperley-Lieb Algebra Research Articles

Topological Defects in Lattice Models and Affine Temperley–Lieb Algebra

This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley-Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as non-unitary models like percolation or self-avoiding walks. Our approach is essentially algebraic and focusses on the defects from two points of view: the "crossed channel" where the defect is seen as an operator acting on the Hilbert space of the models, and the "direct channel" where it corresponds to a modification of the basic Hamiltonian with some sort of impurity. Algebraic characterizations and constructions are proposed in both points of view. In the crossed channel, this leads us to new results about the center of the affine Temperley-Lieb algebra; in particular we find there a special basis with non-negative integer structure constants that are interpreted as fusion rules of defects. In the direct channel, meanwhile, this leads to the introduction of fusion products and fusion quotients, with interesting algebraic properties that allow to describe representations content of the lattice model with a defect, and to describe its spectrum.

Spin Chains as Modules over the Affine Temperley–Lieb Algebra

The affine Temperley-Lieb algebra $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$ is an infinite-dimensional algebra parametrized by a number $\beta \in \mathbb{C}$ and an integer $N\in \mathbb{N}$. It naturally acts on $(\mathbb{C}^2)^{\otimes N}$ to produce a family of representations labeled by an additional parameter $z\in\mathbb C^\times$. The structure of these representations, which were first introduced by Pasquier and Saleur in their study of spin chains, is here made explicit. They share their composition factors with the cellular $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-modules of Graham and Lehrer, but differ from the latter representations by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin as well as new maps that intertwine various $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-actions on the XXZ chain and generalize applications studied by Deguchi $\textit{et al}$ and after by Morin-Duchesne and Saint-Aubin.

Geometric algebra and algebraic geometry of loop and Potts models

We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

On fully commutative elements of type [formula omitted] and [formula omitted

We define a tower of injections of B˜-type (resp. D˜-type) Coxeter groups W(B˜n) (resp. W(D˜n)). Let Wc(B˜n) (resp. Wc(D˜n)) be the set of fully commutative elements in W(B˜n) (resp. W(D˜n)), we classify the elements of this set by giving a normal form for them. We define a B˜-type tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. We use this normal form to define two injections from Wc(B˜n−1) into Wc(B˜n). We then define the tower of affine Temperley-Lieb algebras of type B˜ and use the injections above to prove the faithfulness of this tower. We follow the same track for D˜-type objects.

On the computation of fusion over the affine Temperley–Lieb algebra

Fusion product originates in the algebraization of the operator product expansion in conformal field theory. Read and Saleur (2007) introduced an analogue of fusion for modules over associative algebras, for example those appearing in the description of 2d lattice models. The article extends their definition for modules over the affine Temperley–Lieb algebra TLna.Since the regular Temperley–Lieb algebra TLn is a subalgebra of the affine TLna, there is a natural pair of adjoint induction-restriction functors (↑ar,↓ra). The existence of an algebra morphism ϕ:TLna→TLn provides a second pair of adjoint functors (⇑ar,⇓ar). Two fusion products between TLa-modules are proposed and studied. They are expressed in terms of these four functors. The action of these functors is computed on the standard, cell and irreducible TLna-modules. As a byproduct, the Peirce decomposition of TLna(q+q−1), when q is not a root of unity, is given as direct sum of the induction ↑raSn,k of standard TLn-modules to TLna-modules. Examples of fusion products of various pairs of affine modules are given.

On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

The relationship between bulk and boundary properties is one of the founding features of (rational) conformal field theory (CFT). Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley–Lieb algebra, and use the concept of ‘braid translation’, which is a natural way, in physical terms, to ‘close’ an open spin chain by adding an interaction between the first and last spins using braiding to ‘bring’ them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain—the ‘ordinary’ Temperley–Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic Temperley–Lieb algebra. We show that for the restricted solid-on-solid models, which are known to be described by minimal unitary CFTs (with central charge ) in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed (identical) boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine Temperley–Lieb algebra. We then turn to the same kind of analysis for two models whose continuum limits are logarithmic CFTs (LCFTs)—the alternating and spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant (in that case, indecomposable but not irreducible) modules of the Temperley–Lieb algebra does not give rise to the modules known to be present in the periodic chains. In the case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit. The situation is similar for the case. The problem of relating bulk and boundary lattice models for LCFTs thus remains open.

Ring theoretical properties of affine cellular algebras

As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine Hecke algebras of type A and affine versions of diagram algebras like affine Temperley–Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, Khovanov–Lauda–Rouquier algebras and various other classes of algebras. In this paper we will study ring theoretical properties of affine cellular algebras. We show that any affine cellular algebra A satisfies a polynomial identity. Furthermore, we show that A can be embedded into its asymptotic algebra if the occurring commutative affine k-algebras Bj are reduced and the determinants of the swich matrices are non-zero divisors. As a consequence, we show that the Gelfand–Kirillov dimension of A is less than or equal to the largest Krull dimension of the algebras Bj and that equality holds, in case all affine cell ideals are idempotent or if the Krull dimension of the algebras Bj is less than or equal to 1. Special emphasis is given to the question when an affine cell ideal is idempotent, generated by an idempotent or finitely generated.

Tower of fully commutative elements of type [formula omitted] and applications

Let Wc(A˜n) be the set of fully commutative elements in the affine Coxeter group W(A˜n) of type A˜. We classify the elements of Wc(A˜n) and give a normal form for them. We give a first application of this normal form to fully commutative affine braids. We then use this normal form to define two injections from Wc(A˜n−1) into Wc(A˜n) and examine their properties. We finally consider the tower of affine Temperley–Lieb algebras of type A˜ and use the injections above to prove the injectivity of this tower.

Markov trace on a tower of affine Temperley–Lieb algebras of type Ã

We define a tower of affine Temperley–Lieb algebras of type Ã. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov–Ocneanu–Jones trace on the tower of Temperley–Lieb algebras of type A. We define an invariant of special kind of links as an application of this trace.

The periodic sℓ(2|1) alternating spin chain and its continuum limit as a bulk logarithmic conformal field theory at c = 0

The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $\mathbb{CP}^{1|1} = \mathrm{U}(2|1) / (\mathrm{U}(1) \times \mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this continuum limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the continuum limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras. We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.

Commuting Families in Hecke and Temperley-Lieb Algebras

AbstractWe define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

Representations of the odd affine Temperley-Lieb algebra

AbstractWe define the odd affine Temperley-Lieb algebra (OATLA) to be the category defined by planar diagrams in an annulus with an odd number of marked points on each boundary connected in pairs by disjoint strings and a modulus δ. This algebra is the odd part of the annularization of the Temperley-Lieb planar algebra. A positivity result is proved, which allows us to completely characterize the Hilbert space representations of OATLA when the parameter δ is of the form 2cosπN. The results finish the project of describing the irreducible Hilbert representations of the affine Temperley-Lieb algebra, which naturally arises in the study of subfactors.

Diagram algebras, Hecke algebras and decomposition numbers at roots of unity

We prove that the cell modules of the affine Temperley–Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley–Lieb algebra of type B, which provides a connection between Temperley–Lieb algebras on n and n−1 strings. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan–Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein–Zelevinsky on the “generic case” and of Jones on link invariants.

On representations of affine Temperley–Lieb algebras, II

We study some non-semisimple representations of afine Temperley-Lieb algebras and related cellular algebras. In particular, we classify extensions between simple standard modules. Moreover, we construct a completion which is an infinite dimensional cellular algebra.

On the Affine Temperley-Lieb Algebras

The paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.