Lieb Algebra Research Articles

The Grothendieck group of a tower of the Temperley–Lieb algebras

We provide a diagrammatic approach to compute the algebraic structures on the Grothendieck group of a tower of the Temperley–Lieb algebras.

The nil Temperley–Lieb algebra of type affine C

We introduce a type affine C analogue of the nil Temperley–Lieb algebra, in terms of generators and relations. We show that this algebra T(n), which is a quotient of the positive part of a Kac–Moody algebra of type Dn+1(2), has an easily described faithful representation as an algebra of creation and annihilation operators on particle configurations, reminiscent of the open TASEP model in statistical physics. The centre of T(n) consists of polynomials in a certain element Q, and T(n) is a free module of finite rank over its centre. We show how to localize T(n) by adjoining an inverse of Q, and prove that the resulting algebra is a full matrix ring over a ring of Laurent polynomials over a field. Although T(n) has wild representation type, over an algebraically closed field we can classify all the finite dimensional indecomposable representations of T(n) in which Q acts invertibly.

A relation for the Jones–Wenzl projector and tensor space representations of the Temperley–Lieb algebra

ABSTRACT A relation for the Jones–Wenzl projector is proven. It has the following consequence for representations of the Temperley–Lieb algebra on tensor product spaces: if such a representation is built from a Hermitian matrix T of rank r such that , then either and or . For the latter class of representations, new examples are found. This includes explicit examples for r=2,3,4 and any (with one exception) and a solution for n=r+1 with arbitrary r.

Gröbner–Shirshov bases for Temperley–Lieb algebras of types B and D

For Temperley–Lieb algebras of types [Formula: see text] and [Formula: see text], we construct their Gröbner-Shirshov bases and the corresponding standard monomials

Higher Jones Algebras and their Simple Modules

Let G be a connected reductive algebraic group over a field of positive characteristic p and denote by $\mathcal T$ the category of tilting modules for G. The higher Jones algebras are the endomorphism algebras of objects in the fusion quotient category of $\mathcal T$. We determine the simple modules and their dimensions for these semisimple algebras as well as their quantized analogues. This provides a general approach for determining various classes of simple modules for many well-studied algebras such as group algebras for symmetric groups, Brauer algebras, Temperley–Lieb algebras, Hecke algebras and BMW-algebras. We treat each of these cases in some detail and give several examples.

Borelic pairs for stratified algebras

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley–Lieb and Jones algebras. For this, we develop and apply a general theory of finite dimensional algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Auslander–Dlab–Ringel algebras admit exact Borel subalgebras.

Morphisms from Hecke algebras to graphic Temperley–Lieb algebras

In this paper we construct many surjective morphisms from the Hecke algebras HΓ of “uniform type” (infinite type mostly) to certain graphic Temperley–Lieb algebras, which provide a way to obtain representations of those Hecke algebras. When Γ is the Coxeter matrix I2(2n+1) of the Dihedral group D2n+1 we show there are altogether n such morphisms, exactly realize all n different two dimensional irreducible representations of HΓ. In the study the Chebyshev polynomials appear and play a prominent role in the computation.

Simple modules for Temperley–Lieb algebras and related algebras

Let k be an arbitrary field and let q∈k∖{0}. In this paper we use the known tilting theory for the quantum group Uq(sl2) to obtain the dimensions of simple modules for the Temperley–Lieb algebras TLn(q+q−1) and related algebras over k. Our main result is an algorithm which calculates the dimensions of simple modules for these algebras. We take advantage of the fact that TLn(q+q−1) is isomorphic to the endomorphism ring of the n'th tensor power of the natural 2-dimensional module for the quantum group for sl2. This algorithm is easy when the characteristic is 0 and more involved in positive characteristic. We point out that our results for the Temperley–Lieb algebras contain a complete description of the simple modules for the Jones quotient algebras. Moreover, we illustrate how the same results lead to corresponding information about simple modules for the BMW-algebras with special parameters and other algebras closely related with endomorphism algebras of families of tilting modules for Uq(sl2).

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.

Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones

We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley–Lieb algebra TL _k(d) , converging for all complex loop parameters d with |d| > 2\cos\big(\frac{\pi}{k+1}\big) . In particular, this yields a new formula for the structure coefficients of the Jones–Wenzl projection in TL _k(d) . The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation in terms of a certain graph structure we place on non-crossing pairings, and these coefficients turn out to have the remarkable property that they either always positive integers or always negative integers. As an application, we answer affirmatively a question of Vaughan Jones, asking whether every Temperley–Lieb diagram appears with non-zero coefficient in the expansion of each dual basis element in TL _k(d) , when d \in \mathbb R \backslash \big[-2\cos\big(\frac{\pi}{k+1}\big),2\cos\big(\frac{\pi}{k+1}\big)\big] . Specializing to Jones–Wenzl projections, this result gives a new proof of a result of Ocneanu [27], stating that every Temperley–Lieb diagram appears with non-zero coefficient in a Jones–Wenzl projection. Our methods establish a connection with the Weingarten calculus on free quantum groups, and yield as a byproduct improved asymptotics for the free orthogonal Weingarten function.

A homology theory for a special family of semi-groups

In this paper, we construct a new homology theory for semi-groups satisfying the right self-distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one-term and two-term (rack) homology theories of self-distributive algebraic structures. Finally, we propose connections between the homology theory and knot theory via Temperley–Lieb algebras.

On the length of fully commutative elements

In a Coxeter group W W , an element is fully commutative if any two of its reduced expressions can be linked by a series of commutations of adjacent letters. These elements have particularly nice combinatorial properties, and index a basis of the generalized Temperley–Lieb algebra attached to W W . We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that this sequence always satisfies a linear recurrence with constant coefficients, by showing that reduced expressions of fully commutative elements form a regular language. Then we classify those groups W W for which the sequence is ultimately periodic, extending a result of Stembridge. These results are applied to the growth of generalized Temperley–Lieb algebras.

Diagram calculus for a type affine C Temperley–Lieb algebra, II

In a previous paper, we presented an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley–Lieb algebra having a basis indexed by the fully commutative elements of the Coxeter group of type affine C. We also provided an explicit description of a basis for the diagram algebra. In this paper, we show that the diagrammatic representation is faithful and establish a correspondence between the basis diagrams and the so-called monomial basis of the Temperley–Lieb algebra of type affine C.

Restriction and induction of indecomposable modules over the Temperley–Lieb algebras

Both the original Temperley–Lieb algebras and their dilute counterparts form families of filtered algebras: and , for all . For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over (or ) is computed. To accomplish this, a thorough description of each indecomposable is given, including its projective cover and injective hull, some short exact sequences in which it appears, its socle and head, and its extension groups with irreducible modules. These data are also used to prove the completeness of the list of indecomposable modules, up to isomorphism. In fact, two completeness proofs are given—the first is based on elementary homological methods and the second uses Auslander–Reiten theory. The latter proof offers a detailed example of this algebraic tool that may be of independent interest.

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.

Yang–Baxter solution of dimers as a free-fermion six-vertex model

It is shown that Dimers is Yang–Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter . A one-to-many mapping of vertices onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for Dimers with periodic boundary conditions on the cylinder. 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 (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . At the isotropic point, the exact solution allows for the explicit counting of rotated dimer configurations on a periodic rectangular lattice. We show that the modular invariant partition function on the torus is the same as that of symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a logarithmic conformal field theory with central charge , minimal conformal weight and effective central charge .

Conformal partition functions of critical percolation from D3 thermodynamic Bethe Ansatz equations

Using the planar Temperley–Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as in the Yang–Baxter integrable family of logarithmic minimal models . We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y-system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights for with . For the periodic case, the finite-size corrections agree with the conformal weights , with . These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying q-binomial and q-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine u(1) characters given by the u(1) partition function . The u(1) operator content is for and for and . This result is compatible with the general conjecture of Pearce and Rasmussen, namely with , where the minimal partition function is and the lattice derivation fixes n2,3 = −1.

Classical link invariants from the framizations of the Iwahori–Hecke algebra and the Temperley–Lieb algebra of type A

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras [Formula: see text], which are not topologically equivalent to the Homflypt polynomial. We then present the algebra [Formula: see text] which is the appropriate Temperley–Lieb analogue of [Formula: see text], as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra [Formula: see text] and we prove an isomorphism of this algebra with a subalgebra of [Formula: see text].

A No-Go Theorem for the Continuum Limit of a Periodic Quantum Spin Chain

We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.

Graphical solutions of the Yang–Baxter Equation with cut strands

We study the representations of the braid group in a generalization of the Temperley–Lieb algebra. This algebra was introduced by Kloster and contains the diagrams of the TL algebra with cut strands. We address the question of existence of Markov traces on these representations and make some observations on the possibility of their Baxterizations.