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Search Results (4)

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Keywords = Hochschild cohomology

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18 pages, 338 KiB  
Article
Cohomology of Graded Twisting of Hopf Algebras
by Xiaolan Yu and Jingting Yang
Mathematics 2023, 11(12), 2759; https://doi.org/10.3390/math11122759 - 18 Jun 2023
Viewed by 1340
Abstract
Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the [...] Read more.
Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms. Full article
(This article belongs to the Section A: Algebra and Logic)
22 pages, 384 KiB  
Article
Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras
by Anastasis Kratsios
Mathematics 2021, 9(3), 251; https://doi.org/10.3390/math9030251 - 27 Jan 2021
Viewed by 1706
Abstract
The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector [...] Read more.
The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C. Full article
(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)
11 pages, 232 KiB  
Article
Oriented Algebras and the Hochschild Cohomology Group
by Ali N. A. Koam
Mathematics 2018, 6(11), 237; https://doi.org/10.3390/math6110237 - 1 Nov 2018
Viewed by 1859
Abstract
Koam and Pirashivili developed the equivariant version of Hochschild cohomology by mixing the standard chain complexes computing group with associative algebra cohomologies to obtain the bicomplex C ˜ G * ( A , X ). In this paper, we form a new [...] Read more.
Koam and Pirashivili developed the equivariant version of Hochschild cohomology by mixing the standard chain complexes computing group with associative algebra cohomologies to obtain the bicomplex C ˜ G * ( A , X ). In this paper, we form a new bicomplex F ˘ G * ( A , X ) by deleting the first column and the first row and reindexing. We show that H ˘ G 1 ( A , X ) classifies the singular extensions of oriented algebras. Full article
34 pages, 402 KiB  
Article
R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation
by Victoria Lebed
Axioms 2013, 2(3), 443-476; https://doi.org/10.3390/axioms2030443 - 5 Sep 2013
Cited by 4 | Viewed by 6526
Abstract
In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case [...] Read more.
In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studied using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. This homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2013)
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