A PARTIALLY DEFINED BRAIDING ON A NON-BRAIDED CATEGORY C

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.

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ON 4-DIMENSIONAL 2-HANDLEBODIES AND 3-MANIFOLDS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512501106 ◽

2012 ◽

Vol 21 (12) ◽

pp. 1250110 ◽

Cited By ~ 1

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.

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MORE EXAMPLES OF PSEUDOSYMMETRIC BRAIDED CATEGORIES

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501861 ◽

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.

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The core of a weakly group-theoretical braided fusion category

International Journal of Mathematics ◽

10.1142/s0129167x1850012x ◽

2018 ◽

Vol 29 (02) ◽

pp. 1850012 ◽

Cited By ~ 3

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.

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Ribbon Abelian Categories as Modular Categories

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216596000229 ◽

1996 ◽

Vol 05 (03) ◽

pp. 311-403 ◽

Cited By ~ 10

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.

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THE DRINFEL'D DOUBLE AND TWISTING IN STRINGY ORBIFOLD THEORY

International Journal of Mathematics ◽

10.1142/s0129167x09005431 ◽

2009 ◽

Vol 20 (05) ◽

pp. 623-657 ◽

Cited By ~ 7

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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