Motzkin algebras and the An tensor categories of bimodules

2021 ◽  
pp. 2150077
Author(s):  
Vaughan F. R. Jones ◽  
Jun Yang
Keyword(s):  
Tensor Category ◽  
Fusion Rule ◽  
Tensor Categories ◽  
Basic Construction

We discuss the structure of the Motzkin algebra [Formula: see text] by introducing a sequence of idempotents and the basic construction. We show that [Formula: see text] admits a factor trace if and only if [Formula: see text] and the higher commutants of these factors depend on [Formula: see text]. Then a family of irreducible bimodules over these factors is constructed. A tensor category with [Formula: see text] fusion rule is obtained from these bimodules.

scholarly journals Realizing the Braided Temperley–Lieb–Jones C*-Tensor Categories as Hilbert C*-Modules

2020 ◽  
Vol 380 (1) ◽  
pp. 103-130
Author(s):  
Andreas Næs Aaserud ◽  
David E. Evans
Keyword(s):  
Orthonormal Basis ◽  
Full Subcategory ◽  
Tensor Category ◽  
Compact Operators ◽  
Finitely Generated ◽  
Tensor Categories ◽  
C Algebra

Abstract We associate to each Temperley–Lieb–Jones C*-tensor category $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ T L J ( δ ) with parameter $$\delta $$ δ in the discrete range $$\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}$$ { 2 cos ( π / ( k + 2 ) ) : k = 1 , 2 , … } ∪ { 2 } a certain C*-algebra $${\mathcal {B}}$$ B of compact operators. We use the unitary braiding on $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ T L J ( δ ) to equip the category $$\mathrm {Mod}_{{\mathcal {B}}}$$ Mod B of (right) Hilbert $${\mathcal {B}}$$ B -modules with the structure of a braided C*-tensor category. We show that $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ T L J ( δ ) is equivalent, as a braided C*-tensor category, to the full subcategory $$\mathrm {Mod}_{{\mathcal {B}}}^f$$ Mod B f of $$\mathrm {Mod}_{{\mathcal {B}}}$$ Mod B whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.

scholarly journals Pointed finite tensor categories over abelian groups

2017 ◽  
Vol 28 (11) ◽  
pp. 1750087
Author(s):  
Iván Angiono ◽  
César Galindo
Keyword(s):  
Abelian Group ◽  
Cohomology Class ◽  
Hopf Algebras ◽  
Abelian Groups ◽  
Tensor Category ◽  
Tensor Categories ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of the category of corepresentations of finite-dimensional pointed Hopf algebras with abelian group of group-like elements only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of the category of corepresentations of a finite-dimensional pointed Hopf algebras with abelian group of group-like elements.

scholarly journals RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES

2008 ◽  
Vol 10 (supp01) ◽  
pp. 871-911 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Matrix Element ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Tensor Category ◽  
General Version ◽  
Natural Structure ◽  
Tensor Categories ◽  
Modular Tensor Category ◽  
Verlinde Formula ◽  
Completely Reducible

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)= 0 for n < 0, V(0)= ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V-modules is rigid, balanced and nondegenerate. In particular, the category of V-modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V-module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis.

scholarly journals Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1038
Author(s):  
Steven Duplij
Keyword(s):  
Tensor Product ◽  
Tensor Category ◽  
Baxter Equation ◽  
Algebraic Structures ◽  
Monoidal Categories ◽  
Graded Algebras ◽  
Tensor Categories ◽  
Lie Brackets

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

scholarly journals K-theory of AF-algebras from braided C*-tensor categories

2020 ◽  
Vol 32 (08) ◽  
pp. 2030005 ◽  
Author(s):  
Andreas Næs Aaserud ◽  
David Emrys Evans
Keyword(s):  
Tensor Category ◽  
Group Analysis ◽  
Loop Group ◽  
Polynomial Rings ◽  
Tensor Categories ◽  
Conformal Field ◽  
Special Cases ◽  
Rank One ◽  
Af Algebras

Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans and Gould (1994) explicitly described the [Formula: see text]-theory of certain unital AF-algebras [Formula: see text] as (quotients of) polynomial rings. In this paper, we show that in each case the multiplication in the polynomial ring (quotient) is induced by a ∗-homomorphism [Formula: see text] arising from a unitary braiding on a C*-tensor category and essentially defined by Erlijman and Wenzl (2007). We also present some new explicit calculations based on the work of Gepner, Fuchs and others. Specifically, we perform computations for the rank two compact Lie groups SU(3), Sp(4) and G2 that are analogous to the Evans–Gould computation for the rank one compact Lie group SU(2). The Verlinde rings are the fusion rings of Wess–Zumino–Witten models in conformal field theory or, equivalently, of certain related C*-tensor categories. Freed, Hopkins and Teleman (early 2000s) realized these rings via twisted equivariant [Formula: see text]-theory. Inspired by this, our long-term goal is to realize these rings in a simpler [Formula: see text]-theoretical manner, avoiding the technicalities of loop group analysis. As a step in this direction, we note that the Verlinde rings can be recovered as above in certain special cases.

scholarly journals Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

2019 ◽  
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Hopf Algebras ◽  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Closed Field ◽  
Characteristic P ◽  
Tensor Categories ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Chevalley Property

Abstract We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p&gt;2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon \in k\mathcal{G}$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(\mathcal{G},\epsilon )$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\textrm{Vec}$, so $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p&gt;0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p&gt;0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory.

scholarly journals On Non-trivially Associated Tensor Categories

2018 ◽  
Vol 11 (4) ◽  
pp. 1027-1045
Author(s):  
Bashayer Al-harbi ◽  
Wafa M. Fakieh ◽  
Mohammed Mosa Al-shomrani
Keyword(s):  
Tensor Category ◽  
Left Coset ◽  
Tensor Categories ◽  
Mathematical Formulas

The purpose of this article is to provide mathematical formulas for some operationson the objects of a non-trivially associated tensor category constructed from a factorization of a group into a subgroup and a set of left coset representatives. A detailed example is provided.

scholarly journals Finite symmetric tensor categories with the Chevalley property in characteristic 2

2020 ◽  
pp. 2140010
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Function Algebra ◽  
Equivalence Classes ◽  
Closed Field ◽  
Arbitrary Field ◽  
Tensor Categories ◽  
Characteristic 2 ◽  
Chevalley Property

We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. Namely, we prove that every such category [Formula: see text] admits a symmetric fiber functor to the symmetric tensor category [Formula: see text] of representations of the triangular Hopf algebra [Formula: see text]. Equivalently, we prove that there exists a unique finite group scheme [Formula: see text] in [Formula: see text] such that [Formula: see text] is symmetric tensor equivalent to [Formula: see text]. Finally, we compute the group [Formula: see text] of equivalence classes of twists for the group algebra [Formula: see text] of a finite abelian [Formula: see text]-group [Formula: see text] over an arbitrary field [Formula: see text] of characteristic [Formula: see text], and the Sweedler cohomology groups [Formula: see text], [Formula: see text], of the function algebra [Formula: see text] of [Formula: see text].

scholarly journals A classification of SU(d)-type C*-tensor categories

2014 ◽  
Vol 25 (09) ◽  
pp. 1450081 ◽  
Author(s):  
Bas P. A. Jordans
Keyword(s):  
Tensor Category ◽  
Hecke Algebra ◽  
Sufficient Condition ◽  
Tensor Categories ◽  
Fusion Ring ◽  
Type C ◽  
Markov Trace

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.

Rigid C*-Tensor Categories and their Realizations as Hilbert C*-Bimodules

2018 ◽  
Vol 62 (2) ◽  
pp. 367-393 ◽  
Author(s):  
Wei Yuan
Keyword(s):  
Tensor Category ◽  
Tensor Categories

AbstractWe realize every small rigid C*-tensor category with simple unit object as Hilbert C*-bimodules.

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