scholarly journals The Picard crossed module of a braided tensor category

2013 ◽  
Vol 7 (6) ◽  
pp. 1365-1403 ◽  
Author(s):  
Alexei Davydov ◽  
Dmitri Nikshych
Keyword(s):  
Tensor Category ◽  
Crossed Module

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

Non-Abelian Cohomology of Groups

1997 ◽  
Vol 4 (4) ◽  
pp. 313-331
Author(s):  
H. Inassaridze
Keyword(s):  
Exact Sequence ◽  
Crossed Module ◽  
Dimension 2 ◽  
Crossed Bimodule ◽  
Cohomology Of Groups ◽  
Cohomology Sequence

Abstract Following Guin's approach to non-abelian cohomology [Guin, Pure Appl. Algebra 50: 109–137, 1988] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2.

Internal Crossed Modules

2003 ◽  
Vol 10 (1) ◽  
pp. 99-114 ◽  
Author(s):  
G. Janelidze
Keyword(s):  
Crossed Module ◽  
Crossed Modules

Abstract We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.

scholarly journals The Structure of the Grothendieck Rings of Wreath Product Deligne Categories and their Generalisations

2019 ◽  
Author(s):  
Christopher Ryba
Keyword(s):  
Hopf Algebra ◽  
Wreath Product ◽  
Tensor Category ◽  
Product Category ◽  
Closed Field ◽  
Grothendieck Ring ◽  
Product Categories ◽  
Witt Vectors ◽  
Grothendieck Rings ◽  
Lambda Ring

Abstract Given a tensor category $\mathcal{C}$ over an algebraically closed field of characteristic zero, we may form the wreath product category $\mathcal{W}_n(\mathcal{C})$. It was shown in [10] that the Grothendieck rings of these wreath product categories stabilise in some sense as $n \to \infty $. The resulting “limit” ring, $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$, is isomorphic to the Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$ as defined by [9] (although it is also related to $FI_G$-modules). This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ that is free as a $\mathbb{Z}$-module, we construct a ring $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ that specialises to $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$ when $R = \mathcal{G}(\mathcal{C})$. We give a description of $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ using generators very similar to the basic hooks of [5]. We also show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is a $\lambda $-ring wherever $R$ is and that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is (unconditionally) a Hopf algebra. Finally, we show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in $(W\otimes _{\mathbb{Z}} R)^\times $, where $W$ is the ring of Big Witt Vectors.

scholarly journals Group cohomology with coefficients in a crossed module

2010 ◽  
Vol 10 (2) ◽  
pp. 359-404 ◽  
Author(s):  
Behrang Noohi
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Geometric Approach ◽  
Group Cohomology ◽  
Theoretic Approach ◽  
Crossed Module ◽  
Explicit Approach ◽  
Crossed Modules ◽  
Cohomology Sequence

AbstractWe compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

ON A SUBFACTOR ANALOGUE OF THE SECOND COHOMOLOGY

2002 ◽  
Vol 14 (07n08) ◽  
pp. 733-757 ◽  
Author(s):  
MASAKI IZUMI ◽  
HIDEKI KOSAKI
Keyword(s):  
Tensor Category ◽  
Equivalence Classes

The set of equivalence classes of Longo's Q-systems is shown to serve as a right subfactor analogue of the second cohomology. This "cohomology" is computed for several classes of subfactors, which tells us if a subfactor is uniquely determined up to inner conjugacy by the associated canonical endomorphism. As a byproduct of our analysis, two different groups of order 64 are shown to possess the same representation category as an abstract tensor category. Thus, this category has two permutation symmetries with non-isomorphic groups via the Doplicher–Roberts duality.

SEMI-COMPLETE CROSSED MODULES OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250096
Author(s):  
A. AYTEKIN ◽  
J. M. CASAS ◽  
E. Ö. USLU
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Crossed Module ◽  
Crossed Modules ◽  
Sufficient And Necessary Conditions

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

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 Homotopical aspects of Lie algebras

1993 ◽  
Vol 54 (3) ◽  
pp. 393-419 ◽  
Author(s):  
Graham J. Ellis
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Crossed Module ◽  
Low Dimensional

AbstractThe Hurewicz theorem, Mayer-Vietoris sequence, and Whitehead's certain exact sequence are proved for simplicial Lie algebras. These results are applied, using crossed module techniques, to obtain information on the low dimensional homology of a Lie algebra, and information on aspherical presentations of Lie algebras.

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