Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 16, 2020

Regular Hom-algebras admitting a multiplicative basis

  • Antonio J. Calderón Martín EMAIL logo

Abstract

Let (,μ,α) be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽. A basis ={ei}iI of is called multiplicative if for any i,jI, we have that μ(ei,ej)𝔽ek and α(ei)𝔽ep for some k,pI. We show that if admits a multiplicative basis, then it decomposes as the direct sum =rr of well-described ideals admitting each one a multiplicative basis. Also, the minimality of is characterized in terms of the multiplicative basis and it is shown that, in case , in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

MSC 2010: 17A60; 17A30

Funding statement: Supported by the PCI of the UCA “Teoría de Lie y Teoría de Espacios de Banach”, by the PAI with project numbers FQM298, FQM7156 and by the project of the Spanish Ministerio de Educación y Ciencia MTM2013-41208-P.

References

[1] L. Alvarez-Gaumé, C. Gomez and G. Sierra, Quantum group interpretation of some conformal field theories, Phys. Lett. B 220 (1989), no. 1–2, 142–152. 10.1016/0370-2693(89)90027-0Search in Google Scholar

[2] J. Arnlind, A. Makhlouf and S. Silvestrov, Construction of n-Lie algebras and n-ary Hom–Nambu–Lie algebras, J. Math. Phys. 52 (2011), no. 12, Article ID 123502. 10.1063/1.3653197Search in Google Scholar

[3] S. Benayadi and A. Makhlouf, Hom–Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Geom. Phys. 76 (2014), 38–60. 10.1016/j.geomphys.2013.10.010Search in Google Scholar

[4] L. Cai and Y. Sheng, Purely Hom–Lie bialgebras, Sci. China Math. 61 (2018), no. 9, 1553–1566. 10.1007/s11425-016-9102-ySearch in Google Scholar

[5] A. J. Calderón Martín, On split Lie algebras with symmetric root systems, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 3, 351–356. 10.1007/s12044-008-0027-3Search in Google Scholar

[6] A. J. Calderón Martín and F. J. Navarro Izquierdo, Arbitrary algebras with a multiplicative basis, Linear Algebra Appl. 498 (2016), 106–116. 10.1016/j.laa.2015.01.021Search in Google Scholar

[7] T. L. Curtright and C. K. Zachos, Deforming maps for quantum algebras, Phys. Lett. B 243 (1990), no. 3, 237–244. 10.1016/0370-2693(90)90845-WSearch in Google Scholar

[8] M. Elhamdadi and A. Makhlouf, Deformations of Hom-alternative and Hom–Malcev algebras, Algebras Groups Geom. 28 (2011), no. 2, 117–145. Search in Google Scholar

[9] B. Guan and L. Chen, Restricted Hom–Lie algebras, Hacet. J. Math. Stat. 44 (2015), no. 4, 823–837. 10.15672/HJMS.2015449439Search in Google Scholar

[10] J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006), no. 2, 314–361. 10.1016/j.jalgebra.2005.07.036Search in Google Scholar

[11] C. Kassel, Cyclic homology of differential operators, the Virasoro algebra and a q-analogue, Comm. Math. Phys. 146 (1992), no. 2, 343–356. 10.1007/BF02102632Search in Google Scholar

[12] A. Kitouni, A. Makhlouf and S. Silvestrov, On (n+1)-Hom–Lie algebras induced by n-Hom–Lie algebras, Georgian Math. J. 23 (2016), no. 1, 75–95. 10.1515/gmj-2015-0063Search in Google Scholar

[13] D. Larsson and S. D. Silvestrov, Quasi-Lie algebras, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math. 391, American Mathematical Society, Providence (2005), 241–248. 10.1090/conm/391/07333Search in Google Scholar

[14] Y. Liu, L. Chen and Y. Ma, Representations and module-extensions of 3-hom-Lie algebras, J. Geom. Phys. 98 (2015), 376–383. 10.1016/j.geomphys.2015.08.013Search in Google Scholar

[15] Y. Ma, L. Chen and J. Lin, One-parameter formal deformations of Hom–Lie–Yamaguti algebras, J. Math. Phys. 56 (2015), no. 1, Article ID 011701. 10.1063/1.4905733Search in Google Scholar

[16] A. Makhlouf, Hom-alternative algebras and Hom–Jordan algebras, Int. Electron. J. Algebra 8 (2010), 177–190. Search in Google Scholar

[17] A. Makhlouf, Paradigm of nonassociative Hom-algebras and Hom-superalgebras, Proceedings of Jordan Structures in Algebra and Analysis Meeting, Editorial Círculo Rojo, Almería (2010), 143–177. Search in Google Scholar

[18] A. Makhlouf and F. Panaite, Yetter–Drinfeld modules for Hom-bialgebras, J. Math. Phys. 55 (2014), no. 1, Article ID 013501. 10.1063/1.4858875Search in Google Scholar

[19] A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for hom-associative algebras, Glasg. Math. J. 58 (2016), no. 3, 513–538. 10.1017/S0017089515000294Search in Google Scholar

[20] A. Makhlouf and S. Silvestrov, Notes on 1-parameter formal deformations of Hom-associative and Hom–Lie algebras, Forum Math. 22 (2010), no. 4, 715–739. 10.1515/forum.2010.040Search in Google Scholar

[21] A. A. Sagle, On simple extended Lie algebras over fields of characteristic zero, Pacific J. Math. 15 (1965), 621–648. 10.2140/pjm.1965.15.621Search in Google Scholar

[22] B. Sun and L. Chen, Rota–Baxter multiplicative 3-ary Hom-Nambu-Lie algebras, J. Geom. Phys. 98 (2015), 400–413. 10.1016/j.geomphys.2015.08.011Search in Google Scholar

[23] C. Wang, Q. Zhang and Z. Wei, Hom-Leibniz superalgebras and Hom–Leibniz Poisson superalgebras, Hacet. J. Math. Stat. 44 (2015), no. 5, 1163–1179. 10.15672/HJMS.2015449664Search in Google Scholar

[24] J. Zhao, L. Chen and L. Ma, Representations and T*-extensions of hom-Jordan–Lie algebras, Comm. Algebra 44 (2016), no. 7, 2786–2812. 10.1080/00927872.2015.1065843Search in Google Scholar

[25] J. Zhao, L. Yuan and L. Chen, Deformations and generalized derivations of Hom–Lie conformal algebras, Sci. China Math. 61 (2018), no. 5, 797–812. 10.1007/s11425-016-9063-0Search in Google Scholar

[26] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov, Rings that are Nearly Associative, Pure Appl. Math. 104, Academic Press, New York, 1982. Search in Google Scholar

Received: 2019-01-08
Accepted: 2019-12-12
Published Online: 2020-07-16
Published in Print: 2021-08-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 24.4.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2020-2068/html
Scroll to top button