Browse > Article
http://dx.doi.org/10.4134/JKMS.j180850

α-TYPE HOCHSCHILD COHOMOLOGY OF HOM-ASSOCIATIVE ALGEBRAS AND BIALGEBRAS  

Hurle, Benedikt (Departement de Mathematiques Universite de Haute Alsace)
Makhlouf, Abdenacer (Departement de Mathematiques Universite de Haute Alsace)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.6, 2019 , pp. 1655-1687 More about this Journal
Abstract
In this paper we define a new type of cohomology for multiplicative Hom-associative algebras, which generalizes Hom-type Hochschild cohomology and fits with deformations of Hom-associative algebras including the deformation of the structure map ${\alpha}$. Moreover, we provide various observations and similarly a new type cohomology of Hom-bialgebras extending the Gerstenhaber-Schack cohomology for Hom-bialgebras and fitting with formal deformations including deformations of the structure map.
Keywords
cohomology; Hom-associative algebra; deformation; Hombialgebra; $L_{\infty}$-structure;
Citations & Related Records
연도 인용수 순위
  • Reference
1 F. Ammar, Z. Ejbehi, and A. Makhlouf, Cohomology and deformations of Hom-algebras, J. Lie Theory 21 (2011), no. 4, 813-836.
2 K. Dekkar and A. Makhlouf, Gerstenhaber-Schack cohomology for Hom-bialgebras and deformations, Comm. Algebra 45 (2017), no. 10, 4400-4428. https://doi.org/10.1080/00927872.2016.1265124   DOI
3 Y. Fregier, M. Markl, and D. Yau, The $L_{\infty}$-deformation complex of diagrams of algebras, New York J. Math. 15 (2009), 353-392.
4 Y. Fregier and M. Zambon, Simultaneous deformations of algebras and morphisms via derived brackets, J. Pure Appl. Algebra 219 (2015), no. 12, 5344-5362. https://doi.org/10.1016/j.jpaa.2015.05.018   DOI
5 M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103. https://doi.org/10.2307/1970484   DOI
6 M. Gerstenhaber and S. D. Schack, On the deformation of algebra morphisms and diagrams, Trans. Amer. Math. Soc. 279 (1983), no. 1, 1-50. https://doi.org/10.2307/1999369   DOI
7 M. Gerstenhaber and S. D. Schack, On the cohomology of an algebra morphism, J. Algebra 95 (1985), no. 1, 245-262. https://doi.org/10.1016/0021-8693(85)90104-8   DOI
8 J. T. Hartwig, D. Larsson, and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra 295 (2006), no. 2, 314-361. https://doi.org/10.1016/j.jalgebra.2005.07.036   DOI
9 C. Laurent-Gengoux, A. Makhlouf, and J. Teles, Universal algebra of a Hom-Lie algebra and group-like elements, J. Pure Appl. Algebra 222 (2018), no. 5, 1139-1163. https://doi.org/10.1016/j.jpaa.2017.06.012   DOI
10 M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157-216. https://doi.org/10.1023/B:MATH.0000027508.00421.bf   DOI
11 J.-L. Loday and B. Vallette, Algebraic Operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-30362-3
12 M. Markl, Intrinsic brackets and the $L_{\infty}$-deformation theory of bialgebras, J. Homotopy Relat. Struct. 5 (2010), no. 1, 177-212.
13 G. Sharygin, Deformation quantization and the action of Poisson vector fields, Lobachevskii J. Math. 38 (2017), no. 6, 1093-1107. https://doi.org/10.1134/s1995080217060129   DOI
14 A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2(2008), no. 2, 51-64. https://doi.org/10.4172/1736-4337.1000115   DOI
15 A. Makhlouf and S. D. Silvestrov, Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22 (2010), no. 4, 715-739. https://doi.org/10.1515/FORUM.2010.040   DOI