Generalized Temperley Lieb Algebra Research Articles

The Kauffman bracket skein module of the lens spaces via unoriented braids

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces [Formula: see text], KBSM([Formula: see text]), for [Formula: see text]. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, [Formula: see text], which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, [Formula: see text], for knots and links in ST, via a unique Markov trace constructed on [Formula: see text]. The universal invariant [Formula: see text] is equivalent to the KBSM(ST). For passing now to the KBSM([Formula: see text]), we impose on [Formula: see text] relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in [Formula: see text] but not in ST, and which reflect the surgery description of [Formula: see text], obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM([Formula: see text]). We first present the solution for the case [Formula: see text], which corresponds to obtaining a new basis, [Formula: see text], for KBSM([Formula: see text]) with [Formula: see text] elements. We note that the basis [Formula: see text] is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case [Formula: see text], we first show how the new basis [Formula: see text] of KBSM([Formula: see text]) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case [Formula: see text]. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.

On the annihilator ideal in the bt-algebra of tensor space

We study the representation theory of the braids and ties algebra, or the bt -algebra, E n ( q ) . Using the cellular basis { m s t } for E n ( q ) obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules M ( λ ) and M ( Λ ) for E n ( q ) . We show that the tensor product module V ⊗ n for E n ( q ) is a direct sum of M ( λ ) 's. We introduce the dual cellular basis { n s t } for E n ( q ) and study its action on M ( λ ) and M ( Λ ) . We show that the annihilator ideal I in E n ( q ) of V ⊗ n enjoys a nice compatibility property with respect to { n s t } . We finally study the quotient algebra E n ( q ) / I , showing in particular that it is a simultaneous generalization of Härterich's ‘generalized Temperley-Lieb algebra’ and Juyumaya's ‘partition Temperley-Lieb algebra’.

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.

A cellular basis for the generalized Temperley–Lieb algebra and Mahler measure

Just as the Temperley–Lieb algebra is a good tool to compute the Jones polynomial, the Kauffman bracket skein algebra of a disk with 2k colored points on the boundary, each with color n, is a good tool to compute the nth colored Jones polynomial. We show that the colored skein algebra is a cellular algebra and find a set of separating Jucys–Murphy elements. This is done by explicitly providing the cellular basis and the JM-elements. Having done this, several results of Mathas on such algebras are considered, including the construction of pairwise non-isomorphic irreducible submodules and their corresponding primitive idempotents. These idempotents are then used to define recursive elements of the colored skein algebra. Recursive elements are of particular interest as they have been used to relate geometric properties of link diagrams to the Mahler measure of the Jones polynomial. In particular, a single proof is given for the result of Champanerkar and Kofman, that the Mahler measure of the Jones and colored Jones polynomial converges under twisting on any number of strands.

Fully commutative elements in finite and affine Coxeter groups

An element of a Coxeter group $$W$$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular in the finite case. They index naturally a basis of the generalized Temperley–Lieb algebra. In this work we deal with any finite or affine Coxeter group $$W$$ , and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrresponding growth sequence is ultimately periodic in each type. When the sequence is infinite, this implies that the associated Temperley–Lieb algebra has linear growth.

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D n is shown to be semisimple and free of rank (2 n + 1)n!! − (2 n−1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n − 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ℤ[δ±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D n is a subalgebra of the BMW algebra of the same type.

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

In this paper, we study the quantum sl(<TEX>$n$</TEX>) representation category using the web space. Specially, we extend sl(<TEX>$n$</TEX>) web space for <TEX>$n{\geq}4$</TEX> as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial <TEX>$P_n(q)$</TEX> specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(<TEX>$n$</TEX>). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (<TEX>$n=0$</TEX>) and Jones polynomial (<TEX>$n=2$</TEX>) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

Diagram calculus for a type affine [formula omitted] Temperley–Lieb algebra, I

In this paper, we present 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 (in the sense of Stembridge) of the Coxeter group of type affine C. Moreover, we provide an explicit description of a basis for the diagram algebra. In the sequel to this paper, we show that this diagrammatic representation is faithful. The results of this paper and its sequel will be used to construct a Jones-type trace on the Hecke algebra of type affine C, allowing us to non-recursively compute leading coefficients of certain Kazhdan–Lusztig polynomials.

The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E_ n for n = 6, 7, 8 are shown to be semisimple and free over the integral domain Z[δ^(±1),l^(±1),m]/(m1−δ)(l−l^(−1)) of ranks 1,440,585; 139,613,625; and 53,328, 069,225. We also show they are cellular over suitable rings. The Brauer algebra of type E_n is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring Z[δ^(±1)]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. The generalized Temperley-Lieb algebra of type E_n turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type E_n share many structural properties with the classical ones (of type A_n) and those of type D_n .

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs $$ {\tilde A_n} $$ , $$ {\tilde D_n} $$ , $$ {\tilde E_6} $$ , or $$ {\tilde E_7} $$ . An algebra $$ T{L_{\Gamma, \tau }} $$ has exponential growth if and only if the graph Γ coincides with none of the graphs $$ {A_n} $$ , $$ {D_n} $$ , $$ {E_n} $$ , $$ {\tilde A_n} $$ , $$ {\tilde D_n} $$ , $$ {\tilde E_6} $$ , and $$ {\tilde E_7} $$ .

TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE Dn

A generalization of the Kauffman tangle algebra is given for Coxeter type D n. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A n - 1. The proof involves a diagrammatic version of the Brauer algebra of type D n of which the generalized Temperley–Lieb algebra of type D n is a subalgebra.

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 ∂.

Decorated Tangles and Canonical Bases

We study the combinatorics of fully commutative elements in Coxeter groups of type H n for any n > 2. Using the results, we construct certain canonical bases for non-simply-laced generalized Temperley–Lieb algebras and show how to relate them to morphisms in the category of decorated tangles.

Murphy bases of generalized Temperley-Lieb algebras

In this article we show how to use some results of G. E. Murphy on the so-called standard basis of Hecke-Algebras of Type A to derive a similar basis for generalized Temperley-Lieb algebras. This standard basis is compared to the usual diagrammatic basis of the original Temperley-Lieb algebra used in knot theory and statistical physics.

On the algebraic approach to cubic lattice Potts models

We consider Diagram algebras, $\Dg(G)$ (generalized Temperley-Lieb algebras) defined for a large class of graphs $G$, including those of relevance for cubic lattice Potts models, and study their structure for generic $Q$. We find that these algebras are too large to play the precisely analogous role in three dimensions to that played by the Temperley-Lieb algebras for generic $Q$ in the planar case. We outline measures to extract the quotient algebra that would illuminate the physics of three dimensional Potts models.

Algebras in higher-dimensional statistical mechanics - the exceptional partition (mean field) algebras

We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models (in the continuous $Q$ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is non-semi-simple iff $Q$ is a non-negative integer less than $n$. We determine the dimension of the key irreducible representation in every specialization.