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Hom-type algebras

ここで Hom-type algebra ているのは , Hartwig, Larsson, Silvestrov [ HLS06 ] により 導入 された Lie algebra Hom-Lie algebra Hom 導入 したものである

  • Hom-Lie algebra

Hom-Lie algebra , Lie algebra Jacobi identity ζ : L L でずらしたものに えただけである

このように , りを えるということは , れるようで , くのものが されている についたものを ると 以下 のようになる

またこのような のための として , Panaite, Schrader, Staic [ PSS ] Hom-tensor category というものを 導入 している Hom-monoidal category ぶべきだと うが , braided monoidal category Hom して いる

  • Hom-tensor category
  • Hom-braided category

Hochschild (co)homology などの (co)homology されている

  • Hochschild cohomology Chevalley-Eilenberg cohomology Hom-associative algebra Hom-Lie algebra への (Ammar, Ejbehi, Makhlouf [ AEM11 ] )
  • Hochschild, cyclic, periodic cyclic (co)homology Hom-associative algebra への (Hassanzadeh, Shapiro, Sütlü [ HSS15 ] )
  • Gerstenhaber-Schack-type cohomology Hom-bialgebra への (Dekkar Makhlouf [ DM17 ] )

もある

  • n -Hom Lie algebra [ AMS09 ]
  • n -Hom Leibniz algebra [ MN ]

References

[AEM11]     Faouzi Ammar, Zeyneb Ejbehi, and Abdenacer Makhlouf. Cohomology and deformations of Hom-algebras. J. Lie Theory , 21(4):813–836, 2011, arXiv:1005.0456 .

[AMS09]     H. Ataguema, A. Makhlouf, and S. Silvestrov. Generalization of n -ary Nambu algebras and beyond. J. Math. Phys. , 50(8):083501, 15, 2009, arXiv:0812.4058 .

[CL13]     Bintao Cao and Li Luo. Hom-Lie superalgebra structures on finite-dimensional simple Lie superalgebras. J. Lie Theory , 23(4):1115–1128, 2013, arXiv:1203.0136 .

[DM17]     Khadra Dekkar and Abdenacer Makhlouf. Gerstenhaber-Schack cohomology for Hom-bialgebras and deformations. Comm. Algebra , 45(10):4400–4428, 2017, arXiv:1608.02084 .

[EM13]     Mohamed Elhamdadi and Abdenacer Makhlouf. Hom-quasi-bialgebras. In Hopf algebras and tensor categories , volume 585 of Contemp. Math. , pages 227–245. Amer. Math. Soc., Providence, RI, 2013, arXiv:1209.0988 .

[GMMP15]     Giacomo Graziani, Abdenacer Makhlouf, Claudia Menini, and Florin Panaite. BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras. SIGMA Symmetry Integrability Geom. Methods Appl. , 11:Paper 086, 34, 2015, arXiv:1505.00469 .

[HLS06]     Jonas T. Hartwig, Daniel Larsson, and Sergei D. Silvestrov. Deformations of Lie algebras using σ -derivations. J. Algebra , 295(2):314–361, 2006, arXiv:math/0408064 .

[HSS15]     Mohammad Hassanzadeh, Ilya Shapiro, and Serkan Sütlü. Cyclic homology for Hom-associative algebras. J. Geom. Phys. , 98:40–56, 2015, arXiv:1504.03019 .

[Kar]     Serkan Karacuha. Hom-entwining structures and Hom-Hopf-type modules, arXiv:1412.2002 .

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

[MM18]     Ashis Mandal and Satyendra Kumar Mishra. Hom-Lie-Rinehart algebras. Comm. Algebra , 46(9):3722–3744, 2018, arXiv:1610.01477 .

[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. Forum Math. , 22(4):715–739, 2010, arXiv:0712.3130 .

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

[SB14]     Yunhe Sheng and Chengming Bai. A new approach to hom-Lie bialgebras. J. Algebra , 399:232–250, 2014, arXiv:1304.1954 .

[SC13]     Yunhe Sheng and Danhua Chen. Hom-Lie 2-algebras. J. Algebra , 376:174–195, 2013, 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 .

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

[Yau10]     Donald Yau. Hom-bialgebras and comodule Hom-algebras. Int. Electron. J. Algebra , 8:45–64, 2010, arXiv:0810.4866 .

[YCH17]     La Mei Yuan, Sheng Chen, and Cai Xia He. Hom-Gel’fand-Dorfman super-bialgebras and Hom-Lie conformal superalgebras. Acta Math. Sin. (Engl. Ser.) , 33(1):96–116, 2017, arXiv:1507.08908 .

[Zha]     Tao Zhang. Comodule Hom-coalgebras, arXiv:1301.4152 .