Hom-type algebras

ここで Hom-type algebra と言っているのは, Hartwig, Larsson, Silvestrov [HLS06] により導入された Lie algebra の変種 Hom-Lie algebra の Hom構造を, 他の代数的構造に導入したものである。

• Hom-Lie algebra

Hom-Lie algebra の定義は簡単で, Lie algebra の定義の Jacobi identity を 自己準同型 \(\zeta :L\to L\) でずらしたものに変えただけである。

このように, 自己準同型で既存の代数的構造に捻りを加えるということは, 様々な分野に自然に現れるようで, 非常に多くのものが定義されている。 目についたものを挙げると以下のようになる。

• Hom-associative algebra ([MS10])
• Hom-Leibniz algebra
• quasi-Lie algebra
• Hom-coalgebra
• Hom-quantum group ([Yauc])
• Hom-bialgebra ([Yaua])
• Hom-Hopf algebraと Hom-quasi-bialgebra ([EM13])
• Hom-Novikov algebra ([Yaub])
• comodule Hom-algebra ([Yaua])
• comodule Hom-coalgebra ([Zha])
• Hom-Lie superalgebra ([CL])
• Hom-Lie 2-algebra ([SC])
• Hom-Lie bialgebra ([SB])
• weak monoidal Hom-Hopf algebra ([WWZ])
• biHom-associative algebra, biHom-Lie algebra, and biHom-bialgebra [Gra+]
• Hom-coring, Hom-entwining structure, entwined Hom-module [Kar]
• super Hom-Gel\('f\)and-Dorfman bialgebra and Hom-Lie conformal superalgebra [YCH]
• Hom-Lie-Rinehart algebra [MMa]
• \(3\)-Hom-Lie bialgebras [WWC]
• infinitesimal Hom-bialgebra and infinitesimal Hom-Lie bialgebra ([Yaud])
• Hom-Batalin-Vilkoviski algebra, より一般に Hom-Gerstenhaber algebra [MMb]
• Hom-Lie algebroid ([MMb]).
• Hom-Lie-Hopf algebra ([HKS]).

またこのような代数的構造のための圏論的構造として, Panaite, Schrader, Staic [PSS] が Hom-tensor category というものを導入している。 Hom-monoidal category と呼ぶべきだと思うが。 また, braided monoidal category の Hom版も定義している。

• Hom-tensor category
• Hom-braided category

Hochschild (co)homology などの (co)homology も一般化されている。

• Hochschild cohomology と Chevalley-Eilenberg cohomology の Hom-associative algebra と Hom-Lie algebra への一般化 (Ammar, Ejbehi, Makhlouf の [AEM11])
• Hochschild, cyclic, periodic cyclic (co)homology の Hom-associative algebraへの一般化 (Hassanzadeh, Shapiro, Sütlü の[HSS15])
• Gerstenhaber-Schack-type cohomology の Hom-bialgebra への一般化 (Dekkar と Makhlouf の[DM])

複数入力版もある。

• \(n\)-Hom Lie algebra [AMS]
• \(n\)-Hom Leibniz algebra [MN]

References

[AEM11]

Faouzi Ammar, Zeyneb Ejbehi, and Abdenacer Makhlouf. “Cohomology and deformations of Hom-algebras”. In: J. Lie Theory 21.4 (2011), pp. 813–836. arXiv: 1005.0456.

[AMS]

H. Ataguema, A. Makhlouf, and S. Silvestrov. Generalization of n-ary Nambu algebras and beyond. arXiv: 0812.4058.

[CL]

Bintao Cao and Li Luo. Hom-Lie superalgebra structures on finite-dimensional simple Lie superalgebras. arXiv: 1203.0136.

[DM]

Khadra Dekkar and Abdenacer Makhlouf. Gerstenhaber-Schack Cohomology for Hom-bialgebras and Deformations. arXiv: 1608.02084.

[EM13]

Mohamed Elhamdadi and Abdenacer Makhlouf. “Hom-quasi-bialgebras”. In: Hopf algebras and tensor categories. Vol. 585. Contemp. Math. Amer. Math. Soc., Providence, RI, 2013, pp. 227–245. arXiv: 1209.0988. url: https://doi.org/10.1090/conm/585/11617.

[Gra+]

Giacomo Graziani, Abdenacer Makhlouf, Claudia Menini, and Florin Panaite. BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras. arXiv: 1505.00469.

[HKS]

S. Halici, A. Karataş, and S. Sütlü. Hom-Lie-Hopf algebras. arXiv: 1910.07920.

[HLS06]

Jonas T. Hartwig, Daniel Larsson, and Sergei D. Silvestrov. “Deformations of Lie algebras using \(\sigma \)-derivations”. In: J. Algebra 295.2 (2006), pp. 314–361. arXiv: math/0408064.

[HSS15]

Mohammad Hassanzadeh, Ilya Shapiro, and Serkan Sütlü. “Cyclic homology for Hom-associative algebras”. In: J. Geom. Phys. 98 (2015), pp. 40–56. arXiv: 1504.03019. url: https://doi.org/10.1016/j.geomphys.2015.07.026.

[Kar]

Serkan Karaçuha. Hom-entwining structures and Hom-Hopf-type modules. arXiv: 1412.2002.

[MMa]

Ashis Mandal and Satyendra Kumar Mishra. Hom-Lie-Rinehart Algebras. arXiv: 1610.01477.

[MMb]

Ashis Mandal and Satyendra Kumar Mishra. On Hom-Gerstenhaber algebras and Hom-Lie algebroids. arXiv: 1707.08891.

[MN]

Abdenacer Makhlouf and Anita Naolekar. On n-Hom-Leibniz algebras and cohomology. arXiv: 1803.06840.

[MS10]

Abdenacer Makhlouf and Sergei Silvestrov. “Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras”. In: Forum Math. 22.4 (2010), pp. 715–739. arXiv: 0712.3130. url: https://doi.org/10.1515/FORUM.2010.040.

[PSS]

Florin Panaite, Paul Schrader, and Mihai D. Staic. Hom-Tensor Categories and the Hom-Yang-Baxter Equation. arXiv: 1702.08475.

[SB]

Yunhe Sheng and Chengming Bai. A new approach to hom-Lie bialgebras. arXiv: 1304.1954.

[SC]

Yunhe Sheng and Danhua Chen. Hom-Lie 2-algebras. arXiv: 1110.3405.

[WWC]

Mengping Wang, Linli Wu, and Yongsheng Cheng. Local cocycle 3-Hom-Lie Bialgebras and 3-Lie Classical Hom-Yang-Baxter Equation. arXiv: 1705.02554.

[WWZ]

Wei Wang, Shuanhong Wang, and Xiaohui Zhang. Constructing New Braided \(T\)-Categories via Weak Monoidal Hom-Hopf Algebras. arXiv: 1502.07377.

[Yaua]

Donald Yau. Hom-bialgebras and comodule Hom-algebras. arXiv: 0810.4866.

[Yaub]

Donald Yau. Hom-Novikov algebras. arXiv: 0909.0726.

[Yauc]

Donald Yau. Hom-quantum groups III: Representations and module Hom-algebras. arXiv: 0911.5402.

[Yaud]

Donald Yau. Infinitesimal Hom-bialgebras and Hom-Lie bialgebras. arXiv: 1001.5000.

[YCH]

Lamei Yuan, Sheng Chen, and Caixia He. Super Hom-Gel\('\)fand-Dorfman bialgebras and Hom-Lie conformal superalgebras. arXiv: 1507.08908.

[Zha]

Tao Zhang. Comodule Hom-coalgebras. arXiv: 1301.4152.