scholarly journals A relative tensor product of subfactors over a modular tensor category

2017 ◽  
Vol 107 (11) ◽  
pp. 1963-1970
Author(s):  
Yasuyuki Kawahigashi
Keyword(s):  
Tensor Product ◽  
Tensor Category ◽  
Modular Tensor Category

MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1807-1824 ◽  
Author(s):  
VLADIMIR G. TURAEV
Keyword(s):  
Quantum Groups ◽  
Tensor Category ◽  
Jones Polynomial ◽  
Knot Invariants ◽  
Modular Tensor Category ◽  
Modular Category

The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to the notion of modular tensor category in the sense of G. Moore and N. Seiberg. For simplicity we restrict ourselves in this paper to the case of closed 3-manifolds.

scholarly journals The Tensor Category of Linear Maps and Leibniz Algebras

1998 ◽  
Vol 5 (3) ◽  
pp. 263-276
Author(s):  
J. L. Loday ◽  
T. Pirashvili
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Type Theorem ◽  
Tensor Category ◽  
Leibniz Algebra ◽  
Vector Spaces ◽  
Associative Algebras ◽  
Enveloping Algebra ◽  
Leibniz Algebras ◽  
Linear Maps

Abstract We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor–Moore type theorem for Leibniz algebras.

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 CFT Correlators for Cardy Bulk Fields via String-Net Models

2021 ◽  
Author(s):  
Christoph Schweigert ◽  
◽  
Yang Yang ◽  
◽  
◽  
...  
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Mapping Class Group ◽  
Class Group ◽  
Tensor Category ◽  
Geometric Method ◽  
Mapping Class ◽  
Frobenius Algebra ◽  
Conformal Field ◽  
Modular Tensor Category

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

scholarly journals DECOMPOSING THE TUBE CATEGORY

2019 ◽  
Vol 62 (2) ◽  
pp. 441-458
Author(s):  
LEONARD HARDIMAN ◽  
ALASTAIR KING
Keyword(s):  
Tensor Category ◽  
Matrix Algebras ◽  
Modular Tensor Category ◽  
Primitive Idempotents

AbstractThe tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring through irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.

scholarly journals Q-Systems and Extensions of Completely Unitary Vertex Operator Algebras

2021 ◽  
Author(s):  
Bin Gui
Keyword(s):  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Central Charge ◽  
Tensor Category ◽  
Modular Tensor Category ◽  
Conformal Extension ◽  
Type C ◽  
Q System

Abstract Complete unitarity is a natural condition on a CFT-type regular vertex operator algebra (VOA), which ensures that its modular tensor category (MTC) is unitary. In this paper we show that any CFT-type unitary (conformal) extension $U$ of a completely unitary VOA $V$ is completely unitary. Our method is to relate $U$ with a Q-system $A_U$ in the $C^*$-tensor category $\textrm{Rep}^{\textrm{u}}(V)$ of unitary $V$-modules. We also update the main result of [ 30] to the unitary cases by showing that the tensor category $\textrm{Rep}^{\textrm{u}}(U)$ of unitary $U$-modules is equivalent to the tensor category $\textrm{Rep}^{\textrm{u}}(A_U)$ of unitary $A_U$-modules as unitary MTCs. As an application, we obtain infinitely many new (regular and) completely unitary VOAs including all CFT-type $c&lt;1$ unitary VOAs. We also show that the latter are in one-to-one correspondence with the (irreducible) conformal nets of the same central charge $c$, the classification of which is given by [ 29].

Reference Signal Based Tensor Product Expansion for EOG-Related Artifact Separation in EEG

2017 ◽  
Vol E100.A (11) ◽  
pp. 2230-2237
Author(s):  
Akitoshi ITAI ◽  
Arao FUNASE ◽  
Andrzej CICHOCKI ◽  
Hiroshi YASUKAWA
Keyword(s):  
Tensor Product ◽  
Reference Signal

On degeneracy problem of NFSRs via semi-tensor product

2020 ◽  
Author(s):  
Xinyu Zhao ◽  
Biao Wang ◽  
Shuqian Zhu ◽  
Jun-e Feng
Keyword(s):  
Tensor Product ◽  
Degeneracy Problem

On the Buchstaber Subring in MSp∗

1998 ◽  
Vol 5 (5) ◽  
pp. 401-414
Author(s):  
M. Bakuradze
Keyword(s):  
Tensor Product ◽  
Characteristic Classes ◽  
Generalized Quaternion ◽  
Symplectic Cobordisms

Abstract A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.

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