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Elduque-Myung Type Mutations of Hom-algebras

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Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures

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Abstract

Elduque-Myung type mutations of general, possibly non-associative, algebras and of several classes of Hom-algebras are explored. Several extensions of the results from scalar and non-scalar mutations of associative algebras to mutations of non-associative algebras and of several kinds of hom-algebras of hom-associative type are obtained, including general conditions, in terms of Hom-associators, commutators and general mutation elements (mutations parameters), for mutated Hom-algebras or non-associative algebras to be Hom-associative, Hom-flexible, 3-Hom-Power associative, as well as results on reductions of double parameter mutations to single parameter mutations in algebras and Hom-algebras. Several formulas for mutations of associative algebras are extended to arbitrary possibly non-associative algebras and Hom-algebras.

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References

  1. Elduque, A., Myung, H. C.: Mutations of Alternative Algebras, Springer, Dordrecht, 1994

    Book  MATH  Google Scholar 

  2. Frégier, Y., Gohr, A.: On Hom type algebras, J. Gen. Lie Theory Appl. 4, Article ID G101001, 16 pp. (2010) (arXiv:0903.3393, 2009)

    Google Scholar 

  3. Frégier, Y., Gohr, A.: On unitality conditions for Hom-associative algebras, arXiv:0904.4874 (2009)

    Google Scholar 

  4. Fregier, Y., Gohr, A., Silvestrov, S.: Unital algebras of Hom-associative type and surjective or injective twistings, J. Gen. Lie Theory Appl. 3(4) (2009), 285–295

    Article  MathSciNet  MATH  Google Scholar 

  5. Gohr, A., On Hom-algebras with surjective twisting, J. Algebra, 324, 1483–1491 (2010) (arXiv:0906.3270, 2009)

    Google Scholar 

  6. Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using \(\sigma \)-derivations, J. Algebra, 295(2), 314–361 (2006) (Preprints in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, Lund University, 52 pp. (2003))

    Google Scholar 

  7. Larsson, D., Silvestrov, S. D.: Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra 288,321–344 (2005) (Preprints in Mathematical Sciences 2004:3, LUTFMA-5038-2004, Centre for Mathematical Sciences, Lund University (2004))

    Google Scholar 

  8. Larsson, D., Silvestrov, S. D.: Quasi-Lie algebras, In: Fuchs, J., Mickelsson, J., Rozemnblioum, G., Stolin, A., Westerberg, A.: Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math. 391, Amer. Math. Soc., Providence, RI, 241–248 (2005) (Preprints in Mathematical Sciences 2004:30, LUTFMA-5049-2004, Centre for Mathematical Sciences, Lund University (2004))

    Google Scholar 

  9. Larsson, D., Silvestrov, S. D.: Graded quasi-Lie algebras, Czechoslovak J. Phys. 55(11), 1473–1478 (2005)

    Article  MathSciNet  Google Scholar 

  10. Larsson, D., Silvestrov, S. D.: Quasi-deformations of \(sl_2(\mathbb {F})\) using twisted derivations, Comm. Algebra, 35, 4303–4318 (2007) (Preprint in Mathematical Sciences 2004:26, LUTFMA-5047-2004, Centre for Mathematical Sciences, Lund University (2004). arXiv: math/0506172 [math.RA] (2005))

    Google Scholar 

  11. Sigurdsson, G., Silvestrov, S.: Graded quasi-Lie algebras of Witt type, Czechoslovak J. Phys. 56, 1287–1291 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sigurdsson, G., Silvestrov, S.: Lie color and Hom-Lie algebras of Witt type and their central extensions, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer-Verlag, Berlin, Heidelberg, Ch. 21, 247–255 (2009)

    Google Scholar 

  13. Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras, Int. Elect. Journ. Alg. 8, 177–190 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Makhlouf, A., Silvestrov, S. D.: Hom-algebra structures, J. Gen. Lie Theory Appl. 2(2), 51–64 (2008) (Preprints in Mathematical Sciences 2006:10, LUTFMA-5074-2006, Centre for Mathematical Sciences, Lund University (2006))

    Google Scholar 

  15. Makhlouf, A., Silvestrov, S.: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22(4), 715–739 (2010) (Preprints in Mathematical Sciences 2007:31, LUTFMA-5095-2007, Centre for Mathematical Sciences, Lund University. arXiv: 0712.3130v1 [math.RA] (2007))

    Google Scholar 

  16. Myung, H. C.: Malcev-admissible algebras, Progress in Math. 64 Birkhäuser, 1986

    Google Scholar 

  17. Myung, H. C.: Lie-admissible algebras, Hadronic J. 1, 169–193 (1978)

    MathSciNet  MATH  Google Scholar 

  18. Myung, H. C.: Lie algebras and flexible Lie-admissible algebras, Hadronic Press INC, Hadronic Press Monographs in Mathematics, 1, Massachusetts, 1982

    Google Scholar 

  19. Myung, H. C., Okubo, S., Santilli, R. M.: Applications of Lie-admissible algebras in physics, Vol I, II, Hadronic Press INC, Nonantum, Mass. 1978

    MATH  Google Scholar 

  20. Santilli, R. M.: Imbedding of Lie algebras in nonassociative structures, Nuovo Cimento A, Ser. X, 51, 570–576 (1967)

    MATH  Google Scholar 

  21. Okubo, S.: Introduction to octonion and other non-associative algebras in physics, Cambridge Univ. Press, 1995

    Book  MATH  Google Scholar 

  22. Silvestrov, S.: Paradigm of quasi-Lie and quasi-Hom-Lie algebras and quasi-deformations, In: “New techniques in Hopf algebras and graded ring theory”, K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 165–177 (2007)

    Google Scholar 

  23. Yau, D.: Enveloping algebra of Hom-Lie algebras, J. Gen. Lie Theory Appl., 2, 95–108 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yau, D.: Hom-Malcev, Hom-alternative, and Hom-Jordan algebras. Int. Electron. J. Algebra, 11, 177–217 (2012)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Germán García Butenegro & Sergei Silvestrov

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  1. Germán García Butenegro
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  2. Sergei Silvestrov
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Correspondence to Sergei Silvestrov .

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Editors and Affiliations

  1. International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou, Benin

    Mahouton Norbert Hounkonnou

  2. CAM-FMEN, University of Niš, Niš, Serbia

    Melanija Mitrović

  3. Government College University, Lahore, Pakistan

    Mujahid Abbas

  4. Abbottabad Campus, COMSATS University Islamabad, Abbottabad, Pakistan

    Madad Khan

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Butenegro, G.G., Silvestrov, S. (2023). Elduque-Myung Type Mutations of Hom-algebras. In: Hounkonnou, M.N., Mitrović, M., Abbas, M., Khan, M. (eds) Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-031-39334-1_7

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