Temperley-Lieb Algebra Research Articles

On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain withintegrable boundary

Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’representation of the affine Hecke algebra we give a construction for crossing tensor spacerepresentations of ordinary Hecke algebras. These representations build solutions to theYang–Baxter equation satisfying the crossing condition (that is, integrable quantum spinchains). We show that every crossing representation of the Temperley–Lieb algebra appearsin this construction, and in particular that this construction builds new representations. Weextend these to new representations of the blob algebra, which build new solutions to theboundary Yang–Baxter equation (i.e. open spin chains with integrable boundaryconditions).We prove that the open spin chain Hamiltonian derived from Sklyanin’s commutingtransfer matrix using such a solution can always be expressed as the representation of anelement of the blob algebra, and determine this element. We determine the representationtheory (irreducible content) of the new representations and hence show that all suchHamiltonians have the same spectrum up to multiplicity, for any given value of thealgebraic boundary parameter. (A corollary is that our models have the samespectrum as the open XXZ chain with nondiagonal boundary—despite differingfrom this model in having reference states.) Using these multiplicity data, andother ideas, we investigate the underlying quantum group symmetry of the newHamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.

Hochschild cohomology of algebras with homological ideals

Let $\varphi: A \to B$ be a homological epimorphism of $k$-algebras. We investigate the relationship of the Hochschild cohomologies $H^{i}(A)$ and $H^{i}(B)$ of $A$ and $B$, and show that they can be connected by a long exact sequence. In particular, if $A$ is a quasihereditary algebra and $B$ is the quotient of $A$ by a minimal heredity ideal, then the long exact sequence provides information on $H^{i}(A)$, $H^{i}(B)$ and the extension groups between costandard modules and standard modules, thus one can actually compute $H^{i}(A)$ inductively. As a consequence, we obtain the Hochschild cohomology of all nonsemisimple Temperley-Lieb algebras and representation-finite Schur algebras.

Markov traces on cyclotomic Temperley–Lieb algebras

In this note, we use generalized Tchebychev polynomials to define a trace function which satisfies certain conditions. Such a trace will be called the Markov trace. In particular, we obtain formulae for the weights of the Markov trace. As a corollary, we get a combinatorial identity. This generalizes Jones’s 1983 result on Temperley–Lieb algebras.

EXTRASPECIAL 2-GROUPS AND IMAGES OF BRAID GROUP REPRESENTATIONS

We investigate a family of (reducible) representations of the braid groups [Formula: see text] corresponding to a specific solution to the Yang–Baxter equation. The images of [Formula: see text] under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of [Formula: see text] factoring over Temperley–Lieb algebras and the corresponding link invariants.

Representation theory of towers of recollement: Theory, notes, and examples

We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley–Lieb algebras. We demonstrate the utility of our formalism by applying it to this class.

Algèbre de Temperley–Lieb de type B

We give a dual presentation, in the sense of the dual presentation of Artin groups, of the Temperley–Lieb algebra of type B. In particular, we obtain a basis of this algebra by considering the homomorphic images of the simple elements of the dual monoid. This algebra is the largest quotient of the Hecke algebra whose irreducible representations are parametrized by pairs of Young diagrams with at most one column in each component. To cite this article: C. Vincenti, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

The Temperley–Lieb algebra and its generalizations in the Potts andXXZ models

We discuss generalizations of the Temperley–Lieb algebra in the Potts andXXZ models. These can be used to describe the addition of integrable boundary terms ofdifferent types.We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodicextensions to classify different integrable boundary terms in the two-, three-, and four-statePotts models. The representations always lie at critical points where the algebras becomesnon-semisimple and possess indecomposable representations. In the one-boundary case weshow how to use representation theory to extract the Potts spectrum from anXXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models.In the two-boundary case we find that the Potts spectrum can be obtained by combining severalXXZ models with different boundary terms. As in the Temperley–Lieb case, there is a directcorrespondence between representations of the lattice algebra and those in the continuumconformal field theory.

Temperley-Lieb Immanants

We use the Temperley-Lieb algebra to define a family of totally nonnegative polynomials of the form \( \Sigma _{{\upsigma \in S_{n} }} f(\upsigma )x_{{1,\upsigma (1)}} \cdots x_{{n,\upsigma (n)}} \). The cone generated by these polynomials contains all totally nonnegative polynomials of the form \( \Delta _{{J,J' }} (x)\Delta _{{L,L' }} (x) - \Delta _{{I,I' }} (x)\Delta _{{K,K' }} (x) \), where, \( \Delta _{{I,I' }} (x), \ldots ,\Delta _{{K,K' }} (x) \) are matrix minors. We also give new conditions on the sets I,...,K′ which characterize differences of products of minors which are totally nonnegative.

Magic in the spectra of the XXZ quantum chain with boundaries at [formula omitted] and [formula omitted

We show that from the spectra of the U q ( sl ( 2 ) ) symmetric XXZ spin- 1 / 2 finite quantum chain at Δ = − 1 / 2 ( q = e π i / 3 ) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for Δ = 0 ( q = e π i / 2 ). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley–Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

Cyclotomic blob algebra and its representation theory

A class of finite dimensional associative algebras, called cyclotomic blob algebras, is introduced by means of diagrammatic generators and relations. They are in fact a generalization of Martin and Saleur's blob algebras and have as subalgebras cyclotomic Temperley–Lieb algebras defined recently by Rui and Xi. For precise formalization, an equivalent ring theoretical definition of such algebras is also provided. Based upon the diagrammatic definition, we show that cyclotomic blob algebras are both tabular in the sense of R.M. Green and cellular in the sense of Graham and Lehrer, and we thus obtain a description of all the irreducible representations and a criterion for the quasi-heredity of cyclotomic blob algebras. A branching rule for the cell modules of such algebras is also derived. As a by-product, we prove that cyclotomic Temperley–Lieb algebras are tabular as well, and partially recover their cellularity.

On the semi-simplicity of cyclotomic Temperley-Lieb algebras

In [7], a class of associative algebras called cyclotomic Temperley-Lieb algebras over a commutative ring was introduced. In this note, we provide a necessary and sufficient condition for a cyclotomic Temperley-Lieb algebra to be semi-simple.

One-boundary Temperley–Lieb algebras in the XXZ and loop models

We give an exact spectral equivalence between the quantum group invariant XXZ chainwith arbitrary left boundary term and the same XXZ chain with purely diagonal boundaryterms.This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arisingfrom different representations of the one-boundary Temperley–Lieb algebra. For a system of sizeL these representationsare all of dimension 2L and, for generic points of the algebra, equivalent. However, at exceptional points they canpossess different indecomposable structures.We study a centralizer of the one-boundary Temperley–Lieb algebra in the ‘non-diagonal’spin- representation and find its eigenvalues and eigenvectors. In the exceptional cases thiscentralizer becomes indecomposable. We show how to get a truncated space of ‘good’states. The indecomposable part of the centralizer leads to degeneracies in the threementioned Hamiltonians.

Quantum automorphism groups of small metric spaces

To any finite metric space $X$ we associate the universal Hopf $\c^*$-algebra $H$ coacting on $X$. We prove that spaces $X$ having at most 7 points fall into one of the following classes: (1) the coaction of $H$ is not transitive; (2) $H$ is the algebra of functions on the automorphism group of $X$; (3) $X$ is a simplex and $H$ corresponds to a Temperley-Lieb algebra; (4) $X$ is a product of simplexes and $H$ corresponds to a Fuss-Catalan algebra.

Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category $\mathcal{O}$. We show that this assignment is (up to shifts) invariant under tangle isotopies and Reidemeister moves and defines therefore invariants of tangles. The occurring functors are defined via so-called projective functors. The first part of the paper deals with the indecomposability of such functors and their connection with generalised Temperley-Lieb algebras which are known to have a realisation via decorated tangles. The second part of the paper describes a categorification of the Temperley-Lieb category and proves the main conjectures of [BFK]. Moreover, we describe a functor from the category of 2-cobordisms into a category of projective functors.

G-Vertex Colored Partition Algebras as Centralizer Algebras of Direct Products#

The partition algebras P k (x) have been defined in Martin [Martin, P. (1990). Representations of graph temperley Lieb algebras. Pupl. Res. Inst. Math. Sci. 26:485–503] and Jones [Jones, V. F. R. (1993). The Potts model and the symmetric group. In: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebra, Kyuzeso. River edge, NJ: World Scientific, pp. 259–267, 1994]. We introduce a new class of algebras for every group G called “G-Vertex Colored Partition Algebras,” denoted by P k (x, G), which contain partition algebras P k (x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra P k (n, G) is the centralizer algebra of an action of the direct product S n × G on tensor powers of its permutation module. Further we show that these algebras P k (x, G) contain as subalgebras the “G-Colored Partition Algebras P k (x;G)” introduced in Bloss [Bloss, M. (2003). G-colored partition algebras as centralizer algebras of wreath products. J. Algebra 265:690–710].

Ferromagnetic Ordering of Energy Levels

We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb–Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley–Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.

The representation theory of cyclotomic Temperley-Lieb algebras

A class of associative algebras called cyclotomic Temperley-Lieb algebras is introduced in terms of generators and relations. They are closely related to the group algebras of complex reflection groups on the one hand and generalizations of the usual Temperley-Lieb algebras on the other hand. It is shown that the cyclotomic Temperley-Lieb algebras can be defined by means of labelled Temperley-Lieb diagrams and are cellular in the sense of Graham and Lehrer. One thus obtains not only a description of the irreducible representations, but also a criterion for their quasi-heredity in the sense of Cline, Parshall and Scott. The branching rule for cell modules and the determinants of Gram matrices for certain cell modules are calculated, they can be expressed in terms of generalized Tchebychev polynomials, which therefore play an important role for semisimplicity.

Acyclic Heaps of Pieces, I

Heaps of pieces were introduced by Viennot and have applications to algebraic combinatorics, theoretical computer science and statistical physics. In this paper, we show how certain combinatorial properties of heaps studied by Fan and by Stembridge are closely related to the properties of a certain linear map ∂ E associated to a heap E. We examine the relationship between ∂ E and ∂ F when F is a subheap of E. This approach allows neat statements and proofs of results on certain associative algebras (generalized Temperley–Lieb algebras) that are otherwise tricky to prove. The key to the proof is to interpret the structure constants of the aforementioned algebras in terms of the maps ∂.

Temperely-Lieb Algebras and the Four-Color Theorem

The Temperley–Lieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1,e0,e1, . . .,en, where the generators satisfy the relations $$e^{2}_{i} = 2e_{i} ,{\kern 1pt} {\kern 1pt} e_{i} e_{j} e_{i} = e_{i}$$ if |i−j|=1 and eiej=ejei if |i−j|≥2. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of Tn to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.

The bubble algebra: structure of a two-colour Temperley–Lieb Algebra

We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley–Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang–Baxter equations.