Deformations and Extensions of BiHom-alternative algebras

Year 2023, , 159 - 181, 10.07.2023

Taoufik Chtıouı Sami Mabrouk Abdenacer Makhlouf

https://doi.org/10.24330/ieja.1260492

Abstract

The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and
Hom-alternative algebras, their structure is defined with two
commuting multiplicative linear maps. We study cohomology and
one-parameter formal deformation theory of left BiHom-alternative
algebras. Moreover, we study central and $T_\theta$-extensions of
BiHom-alternative algebras and their relationship with
cohomology.
Finally, we investigate generalized derivations and give some relevant results.

Keywords

BiHom-alternative algebra, representation, cohomology, formal deformation, extension

References

• H. Ataguema and A. Makhlouf, Deformations of ternary algebras, J. Gen. Lie Theory Appl., 1(1) (2007), 41-55.
• H. Ataguema and A. Makhlouf, Notes on cohomologies of ternary algebras of associative type, J. Gen. Lie Theory Appl., 3(3) (2009), 157-174.
• M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian., 66(2) (1997), 151-201.
• S. Caenepeel and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra, 39(6) (2011), 2216-2240.
• J. S. Carter, A. S. Crans, M. Elhamdadi and M. Saito, Cohomology of categorical self-distributivity, J. Homotopy Relat. Struct., 3(1) (2008), 13-63.
• J. S. Carter, A. S. Crans, M. Elhamdadi and M. Saito, Cohomology of the adjoint of Hopf algebras, J. Gen. Lie Theory Appl., 2(1) (2008), 19-34.
• T. Chtioui, S. Mabrouk and A. Makhlouf, BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras, Rocky Mountain J. Math., 50(1) (2020), 69-90.
• T. Chtioui, S. Mabrouk and A. Makhlouf, BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras, Bull. Math. Soc. Sci. Math. Roumanie, 63(111)(1) (2020), 3-21.
• A. Das, Cohomology of BiHom-associative algebras, J. Algebra Appl., 21(1) (2022), 2250008 (22 pp).
• M. Elhamdadi and A. Makhlouf, Cohomology and formal deformations of alternative algebras, J. Gen. Lie Theory Appl., 5 (2011), G110105 (10 pp).
• M. Elhamdadi and A. Makhlouf, Deformations of Hom-Alternative and Hom-Malcev algebras, Algebras Groups Geom., 28(2) (2011), 117-145.
• A. Fialowski, Deformation of Lie algebras, Math. USSR Sb., 55(2) (1986), 467-473.
• M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2), 79 (1964), 59-103.
• M. Gerstenhaber, On the deformation of rings and algebras. II., Ann. of Math., 84 (1966), 1-19.
• M. Gerstenhaber, On the deformation of rings and algebras. III., Ann. of Math. (2), 88 (1968), 1-34.
• M. Gerstenhaber, On the deformation of rings and algebras. IV., Ann. of Math. (2), 99 (1974), 257-276.
• M. Goze and A. Makhlouf, On the rigid complex associative algebras, Comm. Algebra, 18(12) (1990), 4031-4046.
• M. Goze and A. Makhlouf, Classification of rigid associative algebras in lowdimensions, in Lois d'algebres et varietes algebriques (Colmar, 1991), Travauxen Cours, Hermann, Paris, 50 (1996), 5-22.
• M. Goze and E. Remm, Valued deformations of algebras, J. Algebra Appl., 3(4) (2004), 345-365.
• G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras, SIGMA Symmetry Integrability Geom. Methods Appl., 11 (2015), 086 (34 pp).
• J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra, 295(2) (2006), 314-361.
• L. Liu, A. Makhlouf, C. Menini and F. Panaite, BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras, Georgian Math. J., 28(4) (2021), 581-594.
• A. Makhlouf, Degeneration, rigidity and irreducible components of Hopf algebras, Algebra Colloq., 12(2) (2005), 241-254.
• A. Makhlouf, A comparison of deformations and geometric study of varieties of associative algebras, Int. J. Math. Math. Sci., (2007), 18915 (24 pp).
• A. Makhlouf, Hom-alternative algebras and Hom-Jordan algebras, Int. Electron. J. Algebra, 8 (2010), 177-190.
• A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2(2) (2008), 51-64.
• M. Markl and J. D. Stasheff, Deformation theory via deviations, J. Algebra, 170(1) (1994), 122-155.
• D. Yau, Hom-Novikov algebras, J. Phys. A, 44(8) (2011), 085202 (20 pp).