HOM-ALTERNATIVE ALGEBRAS AND HOM-JORDAN ALGEBRAS

Yıl 2010, Cilt: 8 Sayı: 8, 177 - 190, 01.12.2010

Abdenacer Makhlouf

Öz

The main feature of Hom-algebras is that the identities defining the structures are twisted by homomorphisms. The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a plus algebra of Hom-associative algebra leads to Hom-Jordan algebra.

Anahtar Kelimeler

alternative algebra, Hom-alternative algebra, Jordan algebra, Hom-Jordan algebra

Kaynakça

• H. Albuquerque and S. Majid, Quasialgebra Structure of the Octonions, J. Algebra, 220 (1999), 188-224.
• F. Ammar and A. Makhlouf, Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras, J. Algebra (to appear), arXiv:0906.1668v2 (2009).
• H. Ataguema, A. Makhlouf and S. Silvestrov, Generalization of n-ary Nambu algebras and beyond, J. Math. Phys., 50 (1), (2009).
• J.C. Baez, The octonions, Bull. of the Amer. Math. Soc., 39(2) (2001), 145–
• S. Caenepeel and I. Goyvaerts, Hom-Hopf algebras, arXiv:0907.0187v1, (2009).
• A. Dzhumadil’daev and P. Zusmanovich, The alternative operad is not Koszul, arXiv:0906.1272, (2009).
• M. Elhamdadi and A. Makhlouf, Cohomology and Formal Deformations of Left Alternative algebras, arXiv:0907.1548v1, (2009).
• Y. Fregier and A. Gohr, Lie Type Hom-algebras, arXiv:0903.3393v2, (2009).
• Y. Fregier and A. Gohr, On unitality conditions for Hom-associative algebras, arxiv:0904.4874v2, (2009).
• A. Gohr, On Hom-algebras with surjective twisting, arXiv:0906.3270v3, (2009).
• E. Goodaire, Alternative rings of small order and the hunt for Moufang circle loops, Nonassociative algebra and its applications (S˜ao Paulo, 1998), 137–146,
• Lecture Notes in Pure and Appl. Math., 211, Marcel Dekker, New York, 2000.
• J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. of Algebra, 295 (2006), 314–361.
• F. S. Kerdman, Analytic Moufang loops in the large, Algebra i Logica, 18 (1979), 523–555.
• D. Larsson and S. D. Silvestrov, Quasi- Hom-Lie algebras, Central Extensions and 2-cocycle-like identities, J. of Algebra, 288 (2005), 321–344.
• D. Larsson and S. D. Silvestrov, Quasi-Lie algebras, in ”Noncommutative Geometry and Representation Theory in Mathematical Physics”, Contemp. Math., 391, Amer. Math. Soc., Providence, RI (2005), 241–248.
• D. Larsson and S. D. Silvestrov, Quasi-deformations of sl(F) using twisted derivations, Comm. Algebra, 35 (2007), 4303 – 4318.
• A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2(2) (2008), 51–64.
• A. Makhlouf and S. D. Silvestrov, Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, Published as Chapter 17, pp 189-206, S. Silvestrov, E. Paal, V. Abramov, A. Stolin, (Eds.), Generalized Lie theory in Mathematics, Physics and Beyond, Springer-Verlag, Berlin, Heidelberg, (2008).
• A. Makhlouf and S. D. Silvestrov, Notes on Formal deformations of Hom- Associative and Hom-Lie algebras, Forum Math. (to appear). Preprints in Mathematical Sciences, Lund University, Centre for Mathematical Sciences, Centrum Scientiarum Mathematicarum, (2007:31) LUTFMA-5095-2007, 2007; arXiv:0712.3130 (2007).
• A. Makhlouf and S. D. Silvestrov, Hom-Algebras and Hom-Colgebras, J. Al- gebra Appl. (to appear) Preprints in Mathematical Sciences, Lund University, Centre for Mathematical Sciences, Centrum Scientiarum Mathematicarum, (2008:19) LUTFMA-5103-2008. (arXiv:0811.0400).
• A. I. Maltsev, Analytical loops, Matem. Sbornik., 36 (1955), 569-576 (in Rus- sian).
• K. McCrimmon, Alternative algebras, available in http://www.mathstat.uottawa.ca/ neher/Papers/alternative/ E. Paal, Note on analytic Moufang loops, Comment. Math. Univ. Carolin., (2) (2004), 349–354.
• Paal E., Moufang loops and generalized Lie-Cartan theorem, J. Gen. Lie The- ory Appl., 2 (2008), 45–49.
• N. I. Sandu, About the embedding of Moufang loops in alternative algebras II, arXiv:0804.2049, (2008).
• I.P. Shestakov, Moufang Loops and alternative algebras, Proc. Amer. Math. Soc., 132(2) (2003), 313–316.
• K. Yamaguti, On the theory of Malcev algebras Kumamoto J. Sci. Ser. A, 6 (1963), 9–45.
• D. Yau, Enveloping algebra of Hom-Lie algebras, J. Gen. Lie Theory Appl., (2) (2008), 95–108.
• D. Yau, Hom-algebras and homology, J. Lie Theory, 19(2) (2009), 409–421.
• D. Yau, Hom-bialgebras and comodule algebras, arXiv:0810.4866, (2008).
• D. Yau, Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras, J. Phys. A, 42 (2009), 165–202. D. Yau,
• The Hom-Yang-Baxter equation and Hom-Lie algebras, arXiv:0905.1887v2, (2009).
• D. Yau, The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras, arXiv:0905.1890, (2009). D. arXiv:0906.4128, (2009). groups I:quasi-triangular Hom-bialgebras,
• D. Yau, Hom-quantum groups II:cobraided Hom-bialgebras and Hom-quantum geometry, arXiv:0907.1880, (2009). D. Yau, Hom-Malsev, Hom-alternative, and Hom-Jordan algebras, arXiv:1002.3944, (2010). Abdenacer Makhlouf
• Universit´e de Haute-Alsace Laboratoire de Math´ematiques Informatique et Applications bis rue des Fr`eres Lumi`ere Mulhouse, France e-mail: [e-mail]