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Journal articles on the topic 'Braided crossed module'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Noohi, Behrang. "Group cohomology with coefficients in a crossed module." Journal of the Institute of Mathematics of Jussieu 10, no. 2 (June 17, 2010): 359–404. http://dx.doi.org/10.1017/s1474748010000186.

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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.

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2

Davydov, Alexei, and Dmitri Nikshych. "The Picard crossed module of a braided tensor category." Algebra & Number Theory 7, no. 6 (September 19, 2013): 1365–403. http://dx.doi.org/10.2140/ant.2013.7.1365.

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3

Shi, Gui-Qi, Xiao-Li Fang, and Blas Torrecillas. "Generalized Yetter–Drinfeld (quasi)modules and Yetter–Drinfeld–Long bi(quasi)modules for Hopf quasigroups." Journal of Algebra and Its Applications 18, no. 02 (February 2019): 1950034. http://dx.doi.org/10.1142/s0219498819500348.

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As generalizations of Yetter–Drinfeld module over a Hopf quasigroup, we introduce the notions of Yetter–Drinfeld–Long bimodule and generalize the Yetter–Drinfeld module over a Hopf quasigroup in this paper, and show that the category of Yetter–Drinfeld–Long bimodules [Formula: see text] over Hopf quasigroups is braided, which generalizes the results in Alonso Álvarez et al. [Projections and Yetter–Drinfeld modules over Hopf (co)quasigroups, J. Algebra 443 (2015) 153–199]. We also prove that the category of [Formula: see text] having all the categories of generalized Yetter–Drinfeld modules [Formula: see text], [Formula: see text] as components is a crossed [Formula: see text]-category.

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4

Ma, Tianshui, Linlin Liu, and Haiying Li. "A class of braided monoidal categories via quasitriangular Hopf π-crossed coproduct algebras." Journal of Algebra and Its Applications 14, no. 02 (October 19, 2014): 1550010. http://dx.doi.org/10.1142/s0219498815500103.

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Let π be a group and (H = {Hα}α∈π, μ, η) a Hopf π-algebra. First, we introduce the concept of quasitriangular Hopf π-algebra, and then prove that the left H-π-module category [Formula: see text], where (H, R) is a quasitriangular Hopf π-algebra, is a braided monoidal category. Second, we give the construction of Hopf π-crossed coproduct algebra [Formula: see text]. At last, the necessary and sufficient conditions for [Formula: see text] to be a quasitriangular Hopf π-algebra are derived, and in this case, [Formula: see text] is a braided monoidal category.

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5

Yan, Dongdong, and Shuanhong Wang. "Drinfel’d construction for Hom–Hopf T-coalgebras." International Journal of Mathematics 31, no. 08 (June 23, 2020): 2050058. http://dx.doi.org/10.1142/s0129167x20500585.

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Let [Formula: see text] be a Hom–Hopf T-coalgebra over a group [Formula: see text] (i.e. a crossed Hom–Hopf [Formula: see text]-coalgebra). First, we introduce and study the left–right [Formula: see text]-Yetter–Drinfel’d category [Formula: see text] over [Formula: see text], with [Formula: see text], and construct a class of new braided T-categories. Then, we prove that a Yetter–Drinfel’d module category [Formula: see text] is a full subcategory of the center [Formula: see text] of the category of representations of [Formula: see text]. Next, we define the quasi-triangular structure of [Formula: see text] and show that the representation crossed category [Formula: see text] is quasi-braided. Finally, the Drinfel’d construction [Formula: see text] of [Formula: see text] is constructed, and an equivalent relation between [Formula: see text] and the representation of [Formula: see text] is given.

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6

Ma, Tianshui, and Huihui Zheng. "An extended form of Majid’s double biproduct." Journal of Algebra and Its Applications 16, no. 04 (April 2017): 1750061. http://dx.doi.org/10.1142/s021949881750061x.

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Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

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7

Fernández-Fariña, A., and M. Ladra. "Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras." Journal of Algebra and Its Applications 19, no. 09 (September 27, 2019): 2050176. http://dx.doi.org/10.1142/s0219498820501765.

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In this paper, the categories of braided categorical associative algebras and braided crossed modules of associative algebras are studied. These structures are also correlated with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.

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8

Fernández-Fariña, Alejandro, and Manuel Ladra. "Braiding for categorical algebras and crossed modules of algebras II: Leibniz algebras." Filomat 34, no. 5 (2020): 1443–69. http://dx.doi.org/10.2298/fil2005443f.

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In this paper, we study the category of braided categorical Leibniz algebras and braided crossed modules of Leibniz algebras, and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras using the Loday-Pirashvili category.

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9

Arvasi, Z., M. Koçak, and E. Ulualan. "BRAIDED CROSSED MODULES AND REDUCED SIMPLICIAL GROUPS." Taiwanese Journal of Mathematics 9, no. 3 (September 2005): 477–88. http://dx.doi.org/10.11650/twjm/1500407855.

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10

Quang, N. T., C. T. K. Phung, and P. T. Cuc. "Braided equivariant crossed modules and cohomology of Γ-modules." Indian Journal of Pure and Applied Mathematics 45, no. 6 (December 2014): 953–75. http://dx.doi.org/10.1007/s13226-014-0098-z.

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11

Liu, Huili, Tao Yang, and Lingli Zhu. "Yetter–Drinfeld Modules for Group-Cograded Hopf Quasigroups." Mathematics 10, no. 9 (April 21, 2022): 1388. http://dx.doi.org/10.3390/math10091388.

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Let H be a crossed group-cograded Hopf quasigroup. We first introduce the notion of p-Yetter–Drinfeld quasimodule over H. If the antipode of H is bijective, we show that the category YDQ(H) of Yetter–Drinfeld quasimodules over H is a crossed category, and the subcategory YD(H) of Yetter–Drinfeld modules is a braided crossed category.

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12

Zhou, Xuan, and Tao Yang. "New BraidedT-Categories over Weak Crossed Hopf Group Coalgebras." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/626394.

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LetHbe a weak crossed Hopf group coalgebra over groupπ; we first introduce a kind of newα-Yetter-Drinfel’d module categories𝒲𝒴𝒟α(H)forα∈πand use it to construct a braidedT-category𝒲𝒴𝒟(H). As an application, we give the concept of a Long dimodule categoryH𝒲ℒHfor a weak crossed Hopf group coalgebraHwith quasitriangular and coquasitriangular structures and obtain thatH𝒲ℒHis a braidedT-category by translating it into a weak Yetter-Drinfel'd module subcategory𝒲𝒴𝒟(H⊗H).

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13

Bespalov, Yuri, and Bernhard Drabant. "Hopf (bi-)modules and crossed modules in braided monoidal categories." Journal of Pure and Applied Algebra 123, no. 1-3 (January 1998): 105–29. http://dx.doi.org/10.1016/s0022-4049(96)00105-3.

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14

ODABAŞ, Alper, and Erdal ULUALAN. "Braided regular crossed modules bifibered over regular groupoids." TURKISH JOURNAL OF MATHEMATICS 41 (2017): 1385–403. http://dx.doi.org/10.3906/mat-1604-63.

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15

Bespalov, Yu N. "Crossed modules, quantum braided groups, and ribbon structures." Theoretical and Mathematical Physics 103, no. 3 (June 1995): 621–37. http://dx.doi.org/10.1007/bf02065863.

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16

FUKUSHI, Takeo. "Perfect braided crossed modules and their mod-q analogues." Hokkaido Mathematical Journal 27, no. 1 (February 1998): 135–46. http://dx.doi.org/10.14492/hokmj/1351001255.

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17

Arvasi, Z., and E. Ulualan. "3-types of simplicial groups and braided regular crossed modules." Homology, Homotopy and Applications 9, no. 1 (2007): 139–61. http://dx.doi.org/10.4310/hha.2007.v9.n1.a5.

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18

Zhu, Haixing. "The crossed structure of Hopf bimodules." Journal of Algebra and Its Applications 17, no. 09 (August 23, 2018): 1850172. http://dx.doi.org/10.1142/s0219498818501724.

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Let [Formula: see text] be a Hopf algebra with bijective antipode. We first define some generalized Hopf bimodules. Next, we show that these Hopf bimodules form a new tensor category with a crossed structure, which is equivalent to the category of some generalized Yetter–Drinfeld modules introduced by Panaite and Staic. Finally, based on this equivalence, we verify that the category of Hopf bimodules admits the structure of a braided [Formula: see text]-category in the sense of Turaev.

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19

Laugwitz, Robert. "Comodule algebras and 2-cocycles over the (Braided) Drinfeld double." Communications in Contemporary Mathematics 21, no. 04 (May 31, 2019): 1850045. http://dx.doi.org/10.1142/s0219199718500451.

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We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

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20

HASEGAWA, MASAHITO. "A quantum double construction in Rel." Mathematical Structures in Computer Science 22, no. 4 (May 18, 2012): 618–50. http://dx.doi.org/10.1017/s0960129511000703.

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We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

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21

Chen, Quanguo, and Dingguo Wang. "Constructing New Crossed Group Categories Over Weak Hopf Group Algebras." Mathematica Slovaca 65, no. 3 (January 1, 2015). http://dx.doi.org/10.1515/ms-2015-0035.

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AbstractLet π be a group. The main purpose of this paper is to provide further examples of crossed π-categories in the sense of Turaev. For this, we first introduce the notion of weak Hopf π-algebra as the dual notion of weak Hopf π-coalgebra and investigate the properties of weak Hopf π-algebra keeping close to weak Hopf algebra in sense of Böhm et al. It is shown that the category of the copresentations of weak Hopf π-algebra is braided crossed π-category. Finally, we shall consider the notion of weak Doi-Hopf group module in the weak Hopf π-algebra setting, and discuss the separability of a class of functors for the category of weak Doi-Hopf π-modules to the category of comodule over a suitable coalgebras. Also, the applications of our theories are presented.

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22

Huebschmann, J. "Braids and crossed modules." Journal of Group Theory 15, no. 1 (January 1, 2012). http://dx.doi.org/10.1515/jgt.2011.095.

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23

LADRA, Manuel, and Alejandro FERNÁNDEZ-FARİÑA. "UNIVERSAL CENTRAL EXTENSIONS OF BRAIDED LIE CROSSED MODULES." Hacettepe Journal of Mathematics and Statistics, December 31, 2022, 1–16. http://dx.doi.org/10.15672/hujms.901199.

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