A PARTIALLY DEFINED BRAIDING ON A NON-BRAIDED CATEGORY C

2019 ◽  
Vol 44 (2) ◽  
pp. 271-282
Author(s):  
M. M. Al-Shomrani
Keyword(s):  
Braided Category

scholarly journals An example of a braided category of

1998 ◽  
Vol 41 (2) ◽  
pp. 173-191
Author(s):  
Małgorzata Rowicka-Kudlicka
Keyword(s):  
Braided Category

ON THE (CO)COMMUTATIVITY CLASS OF A HOPF ALGEBRA AND CROSSED PRODUCTS IN A BRAIDED CATEGORY

2001 ◽  
Vol 29 (12) ◽  
pp. 5857-5877 ◽  
Author(s):  
J. Alonso Alvarez ◽  
J. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Hopf Algebra ◽  
Crossed Products ◽  
Braided Category

Braided Category

2004 ◽  
pp. 71-71
Author(s):  
Steven Duplij ◽  
Joshua Feinberg ◽  
Moshe Moshe ◽  
Soon-Tae Hong ◽  
Omer Faruk Dayi ◽  
...  
Keyword(s):  
Braided Category

scholarly journals The relative monoidal center and tensor products of monoidal categories

2019 ◽  
Vol 22 (08) ◽  
pp. 1950068 ◽  
Author(s):  
Robert Laugwitz
Keyword(s):  
Monoidal Category ◽  
Monoidal Categories ◽  
Braided Monoidal Category ◽  
Locally Finite ◽  
Braided Category ◽  
Enveloping Algebra ◽  
Small Quantum ◽  
Relative Version ◽  
Quantized Enveloping Algebra ◽  
Weight Modules

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over bialgebras inside a braided monoidal category, the relative center is shown to be equivalent to the category of Yetter–Drinfeld modules inside the braided category. If the braided category is given by modules over a quasitriangular Hopf algebra, then the relative center corresponds to modules over a braided version of the Drinfeld double (i.e. the double bosonization in the sense of Majid) which are locally finite for the action of the dual.

ON 4-DIMENSIONAL 2-HANDLEBODIES AND 3-MANIFOLDS

2012 ◽  
Vol 21 (12) ◽  
pp. 1250110 ◽  
Author(s):  
IVELINA BOBTCHEVA ◽  
RICCARDO PIERGALLINI
Keyword(s):  
Hopf Algebra ◽  
Complete Solution ◽  
Equivalence Theorem ◽  
Embedded Graphs ◽  
Braided Category ◽  
3 Dimensional ◽  
Ribbon Surface ◽  
Algebraic Description

We show that for any n ≥ 4 there exists an equivalence functor [Formula: see text] from the category [Formula: see text] of n-fold connected simple coverings of B3 × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category [Formula: see text] of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S3 branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S3 branched over embedded graphs. Then, we factor the functor above as [Formula: see text], where [Formula: see text] is an equivalence functor to a universal braided category [Formula: see text] freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category [Formula: see text]. From this we derive an analogous description of the category [Formula: see text] of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

scholarly journals MORE EXAMPLES OF PSEUDOSYMMETRIC BRAIDED CATEGORIES

2013 ◽  
Vol 12 (04) ◽  
pp. 1250186
Author(s):  
FLORIN PANAITE ◽  
MIHAI D. STAIC
Keyword(s):  
Hopf Algebras ◽  
Braided Category ◽  
Braided Categories

We study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.

scholarly journals The core of a weakly group-theoretical braided fusion category

2018 ◽  
Vol 29 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Sonia Natale
Keyword(s):  
Tensor Product ◽  
Product Formula ◽  
Fusion Category ◽  
Braided Category ◽  
The Core ◽  
Modular Category ◽  
Braided Fusion Category

We show that the core of a weakly group-theoretical braided fusion category [Formula: see text] is equivalent as a braided fusion category to a tensor product [Formula: see text], where [Formula: see text] is a pointed weakly anisotropic braided fusion category, and [Formula: see text] or [Formula: see text] is an Ising braided category. In particular, if [Formula: see text] is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius–Perron dimension at most 2 is necessarily group-theoretical.

Ribbon Abelian Categories as Modular Categories

1996 ◽  
Vol 05 (03) ◽  
pp. 311-403 ◽  
Author(s):  
V. Lyubashenko
Keyword(s):  
Central Extension ◽  
Vector Spaces ◽  
Finiteness Conditions ◽  
Braided Category ◽  
Ribbon Graphs ◽  
Trivalent Graphs ◽  
Abelian Categories ◽  
Modular Functor

A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given a (eventually non-semisimple) k-linear abelian ribbon braided category [Formula: see text] with some finiteness conditions we construct a functor from a central extension of N with the set of labels [Formula: see text] ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surfto k-vector spaces.

scholarly journals THE DRINFEL'D DOUBLE AND TWISTING IN STRINGY ORBIFOLD THEORY

2009 ◽  
Vol 20 (05) ◽  
pp. 623-657 ◽  
Author(s):  
RALPH M. KAUFMANN ◽  
DAVID PHAM
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Geometric Realization ◽  
Braided Category ◽  
Drinfel’D Double ◽  
Frobenius Algebras ◽  
K Theory ◽  
Orbifold Cohomology ◽  
The One ◽  
A Finite Group

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.

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