K. Erdmann, R. M. Green
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.
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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。
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.
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 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 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.
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…
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。
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。
In this paper, we describe the irreducible representations and give a dimension formula for the Framisation of the Temperley-Lieb algebra. We then prove that the Framisation of the Temperley-Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley-Lieb algebras. This allows us to construct a basis for it. We also study in a similar way the Complex Reflection Temperley-Lieb algebra.
We determine the representations of the Yokonuma-Temperley-Lieb algebra, which is defined as a quotient of the Yokonuma-Hecke algebra by generalising the construction of the classical Temperley-Lieb algebra.
This article examines the growth of generalized algebras of type Temperley-Lieb $ TL _ {\ Gamma, \ tau}. $ Studied them dimension or if the algebra of infinite growth.
In this paper, we will study the Dieck-Temlerley-Lieb algebras of type Bn and Cn. We compute their ranks and describe a basis for them by using some results from corresponding Brauer algebras and Temperley-Lieb algebras.
We investigate the representation theory of the valenced Temperley-Lieb algebras in mixed characteristic. These algebras, as described in characteristic zero by Flores and Peltola, arise naturally in statistical physics and conformal field theory and are a natural deformation of normal Temperley-Lieb algebras. In general characteristic, they encode the fusion rules for the category of $U_q(\mathfrak{sl}_2)$ tilting modules. We use the cellular properties of the Temperley-Lieb algebras to determine those of the valenced Temperley-Lieb algebras. Our approach is, at heart, entirely diagrammatic a…
In this paper, we define a quotient of the cyclotomic Hecke algebra of type $G(r,1,n)$ as a generalisation of the Temperley-Lieb algebras of type $A$ and $B$. We establish a graded cellular structure for the generalised Temperley-Lieb algebra and, using the technology of $KLR$ algebras, determine the corresponding decomposition matrix.
We determine the representations of the Yokonuma-Temperley-Lieb algebra, which is defined as a quotient of the Yokonuma-Hecke algebra by generalising the construction of the classical Temperley-Lieb algebra. We then deduce the dimension of this algebra, and produce an explicit basis for it.
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…
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。
It was studied the growth of Temperley-Lieb type algebras with orthogonal and commutative relations associated with 2-colored edges graphs. Depending on the structure of the graphs they can be finite-dimensional or to have linear or exponential growth.
We provide a necessary and sufficient condition for a type D Temperley-Lieb algebra ${\rm TLD}_n(δ)$ being semi-simple by studying branching rule for cell modules. As a byproduct, our result is used to study the so-called forked Temperley-Lieb algebra, which is a quotient algebra of ${\rm TLD}_n(δ)$.
We define a commuting family of operators $T_0,T_1,...,T_n$ in the Temperley--Lieb algebra $\mathcal{A}_n(x)$ of type $A_{n-1}$. Using an appropriate analogue to Murphy basis of the Iwahori--Hecke algebra of the symmetric group, we describe the eigenvalues arising from the triangular action of the said operators on the cell modules of $\mathcal{A}_n(x)$. These results are used to provide the Temperley--Lieb algebras of type $A_{n-1}$ with a semi--normal form, together with a branching law, and explicit formulae for associated Gram determinants.
We realize the Temperley-Lieb algebra by analogues of Soergel bimodules. The key point is that the monoidal structure is not given by a usual tensor product but by a slightly more complicated operation.
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…
The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the cell modules of the Temperley-Lieb algebra.
We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=\infty$) category. The End sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.
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 explicit bases of simple modules and Bernstein-Gelfand-Gelfand (BGG) resolutions of all simple modules of the (graded) Temperley-Lieb algebra of type B over a field of characteristic zero.
Oriented Temperley--Lieb algebras and combinatorial Kazhdan--Lusztig theory
该研究针对经典埃尔米特对称空间，定义了有向Temperley--Lieb代数，为这类空间的Kazhdan--Lusztig多项式存在闭合组合公式提供了解释。
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。
We study the finite-dimensional simple modules, over an algebraically closed field, of the affine Temperley--Lieb algebra corresponding to the affine Weyl group of type $A$. These turn out to be closely related to the simple modules for a certain $q$-analogue of the annular algebra of V.F.R. Jones.
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…
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。
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras over a finite field are unitary for the action of the braid groups.
Search for a basis of the Temperley-Lieb algebra, using rewriting systems
本文通过重写系统算法化地寻找Temperley-Lieb代数的基，并借助范畴论处理其定向推广版本，从而更高效地获得代数的基。
The Modular Temperley-Lieb Algebra
本文研究正特征及任意环上 Temperley–Lieb 代数的表示理论，利用图形化方法确定胞腔模中单模重数、分解数及单模维数，严格推广零特征理论。
In this paper we propose a combinatorial framework to study a diagrammatic representation of the affine Temperley-Lieb algebra of type C introduced by Ernst. In doing this, we define two procedures, a decoration algorithm on diagrams and a reduction algorithm on heaps of independent interest. Using this approach, an explicit algorithmic description of Ernst representation map is provided from which its faithfulness can be deduced. We also give a construction of the inverse map.
A Graphical Calculus for Induction and Restriction on Temperley-Lieb Modules
本文为 Temperley-Lieb 塔上的诱导与限制构造了一个图形 2-范畴，通过桥梁路径基证明其忠实性，并计算了分裂 Grothendieck 环。
An orthogonal realization of representations of the Temperley-Lieb algebra
本文通过在 $\mathbf{U}(\mathfrak{sl}_2)$ 的张量模 $V^{\otimes n}$ 中构造两两正交的极大向量，为 Temperley–Lieb 代数 $\text{TL}_n$ 的单模建立正交基，并阐明其与标准胞腔基的关系。
Even Temperley-Lieb algebras and the dga of planar loops
本文证明偶数链Temperley-Lieb代数同调有丰富非平凡结构，其由平面环微分分次代数（dga）控制，还提供该dga的小型模型并据此推导同调结论。
Temperley-Lieb Immanants of Ribbon Decomposition Matrices
本文证明Lusztig对偶典范基中的Temperley-Lieb immanants在带状分解矩阵上求值时是Schur正的，并猜想所有对偶典范基元素都具有此性质。
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 construct combinatorial (involutory) Gelfand models for the following diagram algebras in the case when they are semi-simple: Brauer algebra, its partial analogue, walled Brauer algebra, its partial analogue, Temperley-Lieb algebra, its partial analogue, walled Temperley-Lieb algebra, its partial analogue, partition algebra and its Temperley-Lieb analogue.
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…
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。
Orthogonal projections in ${\mathbb C}^n \otimes {\mathbb C}^n$ of rank one and rank two that give rise to unitary tensor space representations of the Temperley-Lieb algebra $TL_N(Q)$ are considered. In the rank one case, a complete classification of solutions is given. In the rank two case, solutions with $Q$ varying in the ranges $[2n/3,\infty)$ and $[n/\sqrt{2},\infty)$ are constructed for $n=3k$ and $n=4k$, $k \in {\mathbb N}$, respectively.
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。
Diagram Calculus for the Affine Temperley--Lieb Algebra of Type $D$
本文针对D型仿射Coxeter系对应的广义Temperley-Lieb代数，构造了与之同构的带装饰图无限维结合代数，并给出了与经典单项式基一一对应的显式图基。
In \cite{JS} Jensen and Su constructed 0-Schur algebras on double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters $\ut$. In particular, when all the entries of $\ut$ are 1, we have 0-Schur algerbas. When all the entries of $\ut$ are zero, we obtain a class of degenerate 0-Schur algebras. We prove that…
Cell modules for the Temperley-Lieb algebra in mixed characteristic
本文研究了温伯格-李代数在混合特征下的单元模块结构，通过图示方法描述了其子模块结构和阿尔佩因图，并探讨了二维贾茨滕过滤。
Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
本文研究了解开的仿射Temperley-Lieb代数及其Wenzl-Jones投影子，探讨了其定义关系、图示描述、维度及与仿射和斜沙漏细胞代数的关系，构造了投影子并涉及Chebyshev多项式。
Diagrammatic representations of Generalized Temperley-Lieb algebras of affine type $\widetilde{B}$ and $\widetilde{D}$
本文针对 $\widetilde{B}$ 和 $\widetilde{D}$ 型仿射 Coxeter 系统的广义 Temperley-Lieb 代数，定义了同构的无穷维装饰图代数，并给出与经典单项基双射对应的显式容许装饰图基。
The Dual Canonical Basis in the Spin Representation via the Temperley-Lieb Algebra
本文通过Temperley-Lieb代数提出双canonical基的简化构造，推导出显式公式并重新证明Khovanov的结果，同时用Hecke代数证明canonical基与双canonical基的对偶性。
Using a braid group representation based on the Temperley-Lieb algebra, we construct braid quantum gates that could generate entangled $n$-partite $D$-level qudit states. $D$ different sets of $D^n\times D^n$ unitary representation of the braid group generators are presented. With these generators the desired braid quantum gates are obtained. We show that the generalized GHZ states, which are maximally entangled states, can be obtained directly from these braid quantum gates without resorting to further local unitary transformations. We also point out an interesting observation, namely for a g…
Generalised Temperley-Lieb algebras of type $G(r,p,n)$
本文定义了与复反射群 $G(r,p,n)$ 对应的广义 Temperley-Lieb 代数 $TL_{r,p,n}$，将其刻画为 $TL_{r,1,n}$ 在自同构 $σ$ 下的不动点子代数，并证明其胞腔性、描述胞腔模与分解数，进而探讨范畴化与图式理论的可能性。
The two boundary Temperley-Lieb algebra $TL_k$ arises in the transfer matrix formulation of lattice models in Statistical Mechanics, in particular in the introduction of integrable boundary terms to the six-vertex model. In this paper, we classify and study the calibrated representations---those for which all the Murphy elements (integrals) are simultaneously diagonalizable---which, in turn, corresponds to diagonalizing the transfer matrix in the associated model. Our approach is founded upon the realization of $TL_k$ as a quotient of the type $C_k$ affine Hecke algebra $H_k$. In previous work…
Quasi-hereditary covers of Temperley-Lieb algebras and relative dominant dimension
本文研究Temperley-Lieb代数的拟遗传覆盖问题，通过证明拟遗传代数相对控制维数的两倍关系，应用于Schur代数确定其相对控制维数，得到q-模拟及Hemmer-Nakano型结果，首次完整分类相关联系并确定整系数最佳拟遗传覆盖。
We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algebra to show that all irreducible representations lie within a finite dimensional diagrammatic quotient. These representations are fully characterised by an additional parameter related to the action of th…
Both the original Temperley-Lieb algebras $\mathsf{TL}_{n}$ and their dilute counterparts $\mathsf{dTL}_{n}$ form families of filtered algebras: $\mathsf{TL}_{n}\subset \mathsf{TL}_{n+1}$ and $\mathsf{dTL}_{n}\subset\mathsf{dTL}_{n+1}$, for all $n\geq 0$. For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over $\mathsf{TL}_{n}$ (or $\mathsf{dTL}_{n}$) 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 …
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