Araştırma Makalesi
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Deformations and Extensions of BiHom-alternative algebras

Yıl 2023, Cilt: 34 Sayı: 34, 159 - 181, 10.07.2023
https://doi.org/10.24330/ieja.1260492

Öz

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.

Kaynakça

  • 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).
Yıl 2023, Cilt: 34 Sayı: 34, 159 - 181, 10.07.2023
https://doi.org/10.24330/ieja.1260492

Öz

Kaynakça

  • 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).
Toplam 28 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Taoufik Chtıouı Bu kişi benim

Sami Mabrouk Bu kişi benim

Abdenacer Makhlouf Bu kişi benim

Erken Görünüm Tarihi 11 Mayıs 2023
Yayımlanma Tarihi 10 Temmuz 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 34 Sayı: 34

Kaynak Göster

APA Chtıouı, T., Mabrouk, S., & Makhlouf, A. (2023). Deformations and Extensions of BiHom-alternative algebras. International Electronic Journal of Algebra, 34(34), 159-181. https://doi.org/10.24330/ieja.1260492
AMA Chtıouı T, Mabrouk S, Makhlouf A. Deformations and Extensions of BiHom-alternative algebras. IEJA. Temmuz 2023;34(34):159-181. doi:10.24330/ieja.1260492
Chicago Chtıouı, Taoufik, Sami Mabrouk, ve Abdenacer Makhlouf. “Deformations and Extensions of BiHom-Alternative Algebras”. International Electronic Journal of Algebra 34, sy. 34 (Temmuz 2023): 159-81. https://doi.org/10.24330/ieja.1260492.
EndNote Chtıouı T, Mabrouk S, Makhlouf A (01 Temmuz 2023) Deformations and Extensions of BiHom-alternative algebras. International Electronic Journal of Algebra 34 34 159–181.
IEEE T. Chtıouı, S. Mabrouk, ve A. Makhlouf, “Deformations and Extensions of BiHom-alternative algebras”, IEJA, c. 34, sy. 34, ss. 159–181, 2023, doi: 10.24330/ieja.1260492.
ISNAD Chtıouı, Taoufik vd. “Deformations and Extensions of BiHom-Alternative Algebras”. International Electronic Journal of Algebra 34/34 (Temmuz 2023), 159-181. https://doi.org/10.24330/ieja.1260492.
JAMA Chtıouı T, Mabrouk S, Makhlouf A. Deformations and Extensions of BiHom-alternative algebras. IEJA. 2023;34:159–181.
MLA Chtıouı, Taoufik vd. “Deformations and Extensions of BiHom-Alternative Algebras”. International Electronic Journal of Algebra, c. 34, sy. 34, 2023, ss. 159-81, doi:10.24330/ieja.1260492.
Vancouver Chtıouı T, Mabrouk S, Makhlouf A. Deformations and Extensions of BiHom-alternative algebras. IEJA. 2023;34(34):159-81.