HOM-MALTSEV, HOM-ALTERNATIVE, AND HOM-JORDAN ALGEBRAS

Year 2012, Volume: 11 Issue: 11 , 177 - 217 , 01.06.2012

Donald Yau

https://izlik.org/JA53XA79ZC

Abstract

Hom-Maltsev(-admissible) algebras are defined, and it is shown
that Hom-alternative algebras are Hom-Maltsev admissible. With a new defi-
nition of a Hom-Jordan algebra, it is shown that Hom-alternative algebras are
Hom-Jordan-admissible. Hom-type generalizations of some well-known identities
in alternative algebras, including the Moufang identities, are obtained.

Keywords

Hom-Maltsev algebras , Hom-Maltsev admissible algebras , Homalternative algebras , Hom-Moufang identities , Hom-Jordan algebras , HomJordan-admissible algebras

References

• A.A. Albert, A structure theory for Jordan algebras, Ann. Math., 48 (1947), –567.
• A.A. Albert, Power-associative rings, Trans. Amer. Math. Soc., 64 (1948), –593.
• F. Ammar and A. Makhlouf, Hom-Lie superalgebras and Hom-Lie admissible superalgebras, J. Algebra, 324 (2010), 1513–1528.
• H. Ataguema, A. Makhlouf, and S. Silvestrov, Generalization of n-ary Nambu algebras and beyond, J. Math. Phys., 50(8) (2009), 083501.
• J.C. Baez, The octonions, Bull. Amer. Math. Soc., 39 (2002), 145–205.
• R.H. Bruck and E. Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc., 2 (1951), 878–890.
• ´E. Cartan, Les groupes r´eels simples Şnis et continus, Ann. ´Ecole Norm., 31 (1914), 263–355.
• Y. Fr´egier, A. Gohr, and S. Silvestrov, Unital algebras of Hom-associative type and surjective or injective twistings, J. Gen. Lie Theory Appl., 3 (2009), –295.
• A. Gohr, On hom-algebras with surjective twisting, J. Algebra, 324 (2010), –1491.
• F. G¨ursey and C.-H. Tze, On The Role of Division, Jordan and Related Algebras in Particle Physics, World ScientiŞc, Singapore, 1996.
• J.T. Hartwig, D. Larsson, and S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra, 295 (2006), 314–361.
• N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc., Providence, RI, 1968.
• P. Jordan, J. von Neumann, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math., 35 (1934), 29–64.
• F.S. Kerdman, Analytic Moufang loops in the large, Alg. Logic, 18 (1980), –347.
• E.N. Kuz’min, The connection between Mal’cev algebras and analytic Moufang loops, Alg. Logic, 10 (1971), 1–14.
• A. Makhlouf, Hom-alternative algebras and Hom-Jordan algebras, Int. Elec- tron. J. Algebra, 8 (2010), 177–190. A. Makhlouf, Paradigm of nonassociative Hom-algebras and Hom- superalgebras, Proceedings of Jordan Structures in Algebra and Analysis
• Meeting, 143-177, Editorial C´ırculo Rojo, Almer´ıa, 2010.
• A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008), 51–64.
• A. Makhlouf and S. Silvestrov, Hom-algebras and Hom-coalgebras, J. Algebra Appl., 9 (2010), 1–37.
• A.I. Mal’tsev, Analytic loops, Mat. Sb., 36 (1955), 569–576.
• R. Moufang, Zur struktur von alternativk¨orpern, Math. Ann., 110 (1935), –430.
• H.C. Myung, Malcev-admissible Algebras, Progress in Math. 64, Birkh¨auser, Boston, MA, 1986.
• P.T. Nagy, Moufang loops and Malcev algebras, Sem. Sophus Lie, 3 (1993), –68.
• S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics, Cambridge Univ. Press, Cambridge, UK, 1995.
• J.M. P´erez-Izquierdo and I.P. Shestakov, An envelope for Malcev algebras, J. Algebra, 272 (2004), 379–393.
• L.V. Sabinin, Smooth Quasigroups and Loops, Kluwer Academic, The Netherlands, 1999.
• A.A. Sagle, Malcev algebras, Trans. Amer. Math. Soc., 101 (1961), 426–458.
• R.D. Schafer, An Introduction to Nonassociative Algebras, Dover, New York, T.A. Springer and F.D. Veldkamp, Octonions, Jordan Algebras, and Excep- tional Groups, Springer, Berlin, 2000.
• J. Tits and R.M. Weiss, Moufang Polygons, Springer-Verlag, Berlin, 2002.
• D. Yau, Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl., (2008), 95–108.
• D. Yau, Hom-algebras and homology, J. Lie Theory, 19 (2009), 409-421.
• D. Yau, Hom-bialgebras and comodule Hom-algebras, Int. Electron. J. Alge- bra, 8 (2010), 45–64.
• D. Yau, Hom-Novikov algebras, J. Phys. A, 44 (2011) 085202.
• D. Yau, The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi- triangular bialgebras, J. Phys. A, 42 (2009), 165202 (12pp).
• D. Yau, The Hom-Yang-Baxter equation and Hom-Lie algebras, J. Math. Phys., 52 (2011), 053502.
• D. Yau, The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras, arXiv:0905.1890. Yau, arXiv:1001.5000. Hom-bialgebras and Hom-Lie bialgebras,
• D. Yau, On n-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geometry Phys., accepted, arXiv:1004.2080.
• D. Yau, Hom-quantum groups I: quasi-triangular Hom-bialgebras, J. Phys. A, accepted, arXiv:0906.4128.
• D. Yau, Hom-quantum groups II: cobraided Hom-bialgebras and Hom- quantum geometry, arXiv:0907.1880.
• D. Yau, Hom-quantum groups III: representations and module Hom-algebras, arXiv:0911.5402. Donald Yau
• Department of Mathematics The Ohio State University at Newark University Drive Newark, OH 43055, USA e-mail: [e-mail]