Frederick M. Goodman, Hans Wenzl
We prove the following result: For a generic value of the parameter, the Temperley-Lieb category has no non-zero, proper tensor ideal. When the parameter $d$ is equal to $2\cos(π/n)$ for some $n \ge 3$, then the Temperley-Lieb category has exactly one non-zero, proper ideal, namely the ideal of negligible morphisms.
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We study two families of polynomials that play the same role, in the generalized Temperley Lieb algebra of a Coxeter group, as the Kazhdan Lusztig and R polynomials in the Hecke algebra of the group. Our results include recursions, closed formulas, and other combinatorial properties for these polynomials. We focus mainly on non branching Coxeter graphs.
Origins of the Temperley-Lieb algebra: early history
本文回顾了Temperley-Lieb代数早期历史中的基本代数与组合结果，并特别梳理了那些已广为流传但缺乏系统记载的经典结论。
We introduce a generalization of the Temperley--Lieb algebra. This generalization is defined by adding certain relations to the algebra of braids and ties. A specialization of this last algebra corresponds to one small Ramified Partition algebra, this fact is the motivation for the name of our generalization.
We develop the theory of diagrammatic Temperley-Lieb categories in order to construct examples of spherical fusion categories, which we then use to construct Turaev-Viro skein modules for n-holed disks. This is a thesis submitted for the degree of Bachelor of Science with Honours at the Australian National University.
Algebraic basics on Temperley-Lieb algebras are proved in an elementary and straightforward way with the help of tensor categories behind them.
Fiber functors on Temperley-Lieb categories are investigated with the help of classification results on non-degenerate bilinear forms. The case of unitary fiber functors is also considered.
We describe the cell structure of the affine Temperley-Lieb algebra with respect to a monomial basis. We construct a diagram calculus for this algebra.
We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley-Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and $D$. This generalizes Kauffman's diagram calculus for the Temperley-Lieb algebra.
In this thesis, I present an associative diagram algebra that is a faithful representation of a particular Temperley--Lieb algebra of type affine $C$, which has a basis indexed by the fully commutative elements of the Coxeter group of the same type. The Coxeter group of type affine $C$ contains an infinite number of fully commutative elements, and so the corresponding Temperley--Lieb algebra is of infinite rank. With the exception of type affine $A$, all other generalized Temperley--Lieb algebras with known diagrammatic representations are of finite rank. In the finite rank case, counting argu…
We give a combinatorial description of a new diagram algebra, the partial Temperley--Lieb algebra, arising as the generic centralizer algebra $\mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k})$, where $V = V(0) \oplus V(1)$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_2)$. It is a proper subalgebra of the Motzkin algebra (the $\mathbf{U}_q(\mathfrak{sl}_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur--Weyl duality for the new algebras, and describe their generic representation theory.
We introduce a new basis of the Temperley-Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple elements of the Birman-Ko-Lee braid monoid to the Temperley-Lieb algebra. The combinatorics of the new basis involve the Bruhat order restricted to noncrossing partitions. As an application we can derive properties of the coefficients of the base change matrix between Zinno's basis and the well-known diagram or Kazhdan-Lusztig basis of the Temperley-Lieb algebra. I…
In 2000, Jones found two copies of the cyclic category in the annular Temperley-Lieb category ATL. We give an abstract presentation of ATL to discuss how these two copies of the cyclic category generate ATL together with the coupling constants and the coupling relations. We then discuss modules over the annular category and homologies of such modules, the latter of which arises from the cyclic viewpoint.
Factorization of Temperley--Lieb diagrams
本文针对 Temperley--Lieb 代数中的图，提出一种高效算法，能从任意图重构其对应的约化分解，解决了从图到分解的逆向难题。
We study some non-semisimple representations of affine 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.
Limits of traces of Temperley-Lieb algebras
本文研究无限 Temperley-Lieb 代数上正极值迹的分类，并将其从一般情形推广到非半单单位根情形，从而构造出相应的 Hilbert 空间结构。
Construction of the diagrammatic version of the affine Temperley-Lieb algebra of type $\widetilde{A_N}$ as a subring of matrices over the Laurent polynomials is given. We move towards geometrical understanding of cellular structure of the Temperley-Lieb algebra. We represent its center as a coordinate ring of the certain affine algebraic variety and describe this variety constructing its desingularization.
We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.
This article concerns a generalization of the Temperley-Lieb algebra, important in applications to conformal field theory. We call this algebra the valenced Temperley-Lieb algebra. We prove salient facts concerning this algebra and its representation theory, which are both of independent interest and used in our subsequent work, where we uniquely and explicitly characterize the monodromy invariant correlation functions of certain conformal field theories.
We construct a representation of the Temperley-Lieb algebra from a multiplicity-free semisimple monoidal Abelian category ${\cal C}$, with two simple objects $λ$ and $ν$ such that $λ\otimesν$ is simple and Hom$_{\cal C}(λ\otimes λ, ν)$ is not empty. A self-contained manual to tensor categories is also provided as well as a summary of the best known example of the construction: Schur-Weyl duality for $U_q(sl_2))$.
We 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 $U_h\mathfrak{gl}_n$. We also give the explicit specializations of these results to the finite Tempe…
We show that there is a unique Markov trace on the tower of Temperley--Lieb type quotients of Hecke algebras of Coxeter type $E_n$ (for all $n \geq 6$). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan--Lusztig polynomials.
In this article we consider conditions under which projection operators in multiplicity free semi-simple tensor categories satisfy Temperley-Lieb like relations. This is then used as a stepping stone to prove sufficient conditions for obtaining a representation of the Birman-Murakami-Wenzl algebra from a braided multiplicity free semi-simple tensor category. The results are found by utalising the data of the categories. There is considerable overlap with the results found in arXiv:1607.08908, where proofs are shown by manipulating diagrams.
We define a triangular change of basis in which the form is diagonal and explicitly compute the diagonal entries of this matrix as products of quotients of Chebyshev polynomials, corroborating the determinant computation of Ko and Smolinsky. The proof employs a recursive method for defining the required orthogonal basis elements in the Temperley-Lieb algebra, similar in spirit to Jones' and Wenzl's recursive formula for a family of projectors in the Temperley-Lieb algebra. In short, we define a partial order on the non-crossing chord diagram basis and find the orthogonal basis using a recursiv…
In this paper, we focus on a class of quantum channels which are covariant for symmetries from free orthogonal quantum groups $O_N^+$. These quantum channels are called $O_N^+$-Temperley-Lieb channels, and their information-theoretic properties such as Holevo information and coherent information were analyzed in [BCLY20], but their additivity questions remained open. The main result of this paper is to approximate $O_N^+$-Temperley-Lieb quantum channels by much simpler ones in terms Bures distance. As applications, we study strong additivity questions for $O_N^+$-Temperley-Lieb quantum channel…
Diagram calculus for a type affine $C$ Temperley--Lieb algebra, II
本文证明了仿射 $C$ 型 Temperley--Lieb 代数的图解表示是忠实的，并建立了基图解与单项式基的对应关系。
Idempotents in the Temperley-Lieb Monoid and Other Categories
本文研究由恒等态射分解产生的代数与范畴中的幂等元，利用平面曲线组合学将 Temperley-Lieb 幺半群中的幂等元与迷宫联系起来，并推广至 Brauer 幺半群及 Tangle 范畴。
We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for $TL_{n}$ that generalizes to give extensions of $TL_{\infty}$ representations. Finally we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.
The Temperley--Lieb algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, Penrose showed that this algebra can be realized in terms of certain diagrams. Then in 1987, Jones showed that the Temperley--Lieb algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$. This realization of the Temperley--Lieb algebra as a Hecke algebra quotient was later generalized to the case of an arbitrary Coxeter group by Graham. Cellular algebras were introduced by…
We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank- n Hadamard type representations of the Temperley-Lieb algebra $TL_N(\sqrt n)$. They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined from the "Master matrix" associated to the chosen realization of Temperley-Lieb algebra
We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we sho…
The general solutions of the reflection equation associated with Temperley-Lieb $R$-matrices are constructed. Their parametrization is defined and the Hamiltonians of corresponding integrable spin systems are given.
We study the two-boundary Temperley--Lieb $O(n)$ loop model on Kazhdan--Lusztig bases of type A and B. We obtain explicit expressions of the ground state of the two-boundary Temperley--Lieb Hamiltonian by means of a coideal subalgebra of $U_q(\mathfrak{sl}_2)$. This ground state possesses a positive integral property. We conjecture that some components of the ground state are directly related to an enumeration of binary or permutation matrices.
A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U_q with a representation ring equivalent to the representation ring of the sl_2 Lie algebra. This algebra U_q is the symmetry algebra of the corresponding open spin chain.
We consider nonquasiclassical solutions to the quantum Yang-Baxter equation and the corresponding quantum cogroups $\Fun(SL(S))$ constructed earlier by one of the authors . We give a criterion of the existence of a dual quasitriangular structure in the algebra $\Fun(SL(S))$ and describe a large class of such objects related to the Temperley-Lieb algebra satisfying this criterion. We show also that this dual quasitriangular structure is in some sense nondegenerate.
Invariants of 4-Dimensional 2-Handlebodies from the Temperley-Lieb Category in Positive Characteristic
本文研究了在正特征下，基于Temperley-Lieb范畴的四维2柄体不变量，探讨其在特定条件下的消失性质及对exotic平滑结构的检测潜力。
The Temperley-Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum sl(2) on the n-th tensor power of the 2-dimensional irreducible module. We define and study a quotient of the Birman-Wenzl-Murakami algebra, which plays an analogous role for the 3-dimensional representation of quantum sl(2). In the course of the discussion we prove some general results about the radical of a cellular algebra, which may be of independent interest.
Temperley-Lieb Crystals
本文针对 Temperley-Lieb 子式推广 Littlewood-Richardson 法则，引入洗牌 tableau，并利用 Stembridge 公理证明其构成 A 型 Kashiwara 晶体，从而给出相关 Schur 系数的组合解释。
The orientifold Temperley--Lieb algebra
本文通过构造取向性鸟笼代数的商，研究了2边界温伯格李代数和辛blob代数的梯度结构，并提出了计算其梯度分解矩阵的算法。
We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the…
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 $D_{n+1}^{(2)}$, 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 adjoinin…
We present in detail a recent conjecture on self-adjoint representations of the Temperley-Lieb algebra for particular values on the unit circle. The formulation in terms of graphical calculus is emphasized and discussed for several examples. The role of PT (parity and time reversal) invariance is highlighted as it might prove important for generalising the construction to other cases.
Modules of the Temperley-Lieb algebra at zero
本文研究了Temperley-Lieb代数在零处的模结构，并通过构造精确序列范畴化了Jones多项式在t=-1处的评估，同时推导了对称群在特征二下的表示论结果。
We study a class of quantum channels arising from the representation theory of compact quantum groups that we call Temperley-Lieb quantum channels. These channels simultaneously extend those introduced in [BC18], [AN14], and [LS14]. (Quantum) Symmetries in quantum information theory arise naturally from many points of view, providing an important source of new examples of quantum phenomena, and also serve as useful tools to simplify or solve important problems. This work provides new applications of quantum symmetries in quantum information theory. Among others, we study entropies and capaciti…
The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-We…
Classifying Module Categories for Generalized Temperley-Lieb-Jones *-2-Categories
本文提出通过加权双向公平和平衡图对广义Temperley-Lieb-Jones *-2-范畴的单位ary模块进行分类。
Let $k$ be an arbitrary field and let $q \in k\setminus\{0\}$. In this paper we use the known tilting theory for the quantum group $U_q(sl_2)$ to obtain the dimensions of simple modules for the Temperley-Lieb algebras $TL_n(q+q^{-1})$ 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 $TL_n(q+q^{-1})$ is isomorphic to the endomorphism ring of the $n$'th tensor power of the natural $2$-dimensional module for the quantum group for $sl_2$. This algorithm is easy when the characteri…
Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors
本文研究带彩色区域的广义Temperley-Lieb分类，确定其半单性并分解张量积的简单对象，利用双变量Chebyshev多项式和彩色Jones-Wenzl投影子证明了一个猜想。
Diagram calculus for a type affine $C$ Temperley--Lieb algebra, I
本文提出一个无限维图解代数，其基由仿射 C 型 Coxeter 群的完全交换元素索引，并给出基的显式描述，为后续构造 Jones 型迹和计算 Kazhdan--Lusztig 多项式系数奠定基础。
The Drinfeld Center of the Generic Temperley--Lieb Category
本文研究了当q非根为1时，温度利-李特尔范畴嵌入模尔张量范畴的超积，并证明其德里文中心是半单的，描述了其简单对象。
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 $\text{TL}_k(d)$, converging for all complex loop parameters $d$ with $|d| > 2\cos\big(\fracπ{k+1}\big)$. In particular, this yields a new formula for the structure coefficients of the Jones-Wenzl projection in $\text{TL}_k(d)$. 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 coeffici…
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 $n \times n$ matrix $T$ of rank $r$ such that $T^2=Q T$, then either $n^2 = Q^2 r$ and $Q^2 =1,2,3$ or $n^2 \geq 4 r$. For the latter class of representations, new examples are found. This includes explicit examples for $r=2,3,4$ and any $n \geq r$ (with one exception) and a solution for $n=r+1$ with arbitrary $r$.
The Temperley-Lieb algebra is a fundamental component of SU(2) topological quantum field theories. We construct chain complexes corresponding to minimal idempotents in the Temperley-Lieb algebra. Our results apply to the framework which determines Khovanov homology. Consequences of our work include semi-orthogonal decompositions of categorifications of Temperley-Lieb algebras and Postnikov decompositions of all Khovanov tangle invariants.
When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $TL_n(q)$ is non-semisimple for almost all $n$. Jones showed that there is a canonical symmetric bilinear form on $TL_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_\ell\in TL_{\ell-1}(q)\subseteq TL_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. In this work, we study the quotients $Q_n(\ell):=TL_n(q)/R_n(q)$, where $|q^2|=\ell$, which are precisely the algebras generated by Jones' projections. We give the dimen…
KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems
本文证明Temperley-Lieb子积系统的Cuntz-Pimsner代数具有KK-自对偶性，利用其规范KMS-态构造Fredholm模代表对偶基本类，从而显式描述该代数的K-同调，并确立$S^3$的$U(2)$-等变KK-自对偶性。
The Center of the Temperley-Lieb Algebra
该研究计算特征零域上坦珀利-利布代数$\operatorname{TL_n}(\delta)$中心维度，利用胞腔滤过、表示理论等，通过扩张压缩映射及变形论证得出维度公式，还证明中心元素在特定自同构下不变，给出平凡根基情形同余准则及Gram矩阵计算。
Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)$_k$ Wess - Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory outlined. Explicit definition of an associated linear trace operation in terms of a certain matrix element in the space of conformal blocks of such a conformal theory is presented. Further for every primary field of a rational conformal field theory, there is a subfactor of hyperfi…
Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)
本文研究 Racah 代数与 su(2) 三个不可约表示张量积中心化子的关系，证明 Racah 代数满射到中心化子，并提出核的描述猜想，在若干情形下证明该猜想并得到 Temperley-Lieb、Brauer 等代数作为其商。
The Temperley-Lieb Tower and the Weyl Algebra
本文通过图示方法定义幺半范畴 W 和 2-范畴 2Weyl，证明 2Weyl 以诱导与限制作用于 Temperley-Lieb 代数模范畴，并揭示其 Grothendieck 群与 Weyl 代数的密切联系及渐近作用。
Temperley-Lieb Categories on Non-Orientable Surfaces
本研究构造了扩展Temperley-Lieb范畴的带边正方形范畴，其态射含不可定向有界曲面嵌入曲线图，定义了扩展张量积并给出幺半生成元，为相关范畴研究提供了新的构造框架。
Frobenius algebra objects in Temperley-Lieb categories at roots of unity
本文在根为单位的Temperley-Lieb范畴中定义了Frobenius结构，并展示了其Nakayama映射的阶为2，扩展了Frobenius代数的概念。
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