Hopf algebras with triality
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- by Georgia Benkart, Sara Madariaga and José M. Pérez-Izquierdo PDF
- Trans. Amer. Math. Soc. 365 (2013), 1001-1023 Request permission
Abstract:
In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and Zavarnitsine to groups with triality.References
- Richard Hubert Bruck, A survey of binary systems, Reihe: Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0093552, DOI 10.1007/978-3-662-35338-7
- Stephen Doro, Simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 377–392. MR 492031, DOI 10.1017/S0305004100054669
- S. M. Gagola III and J. I. Hall, Lagrange’s theorem for Moufang loops, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 45–64. MR 2160355
- George Glauberman, On loops of odd order. II, J. Algebra 8 (1968), 393–414. MR 222198, DOI 10.1016/0021-8693(68)90050-1
- Alexander Grishkov, Lie algebras with triality, J. Algebra 266 (2003), no. 2, 698–722. MR 1995132, DOI 10.1016/S0021-8693(03)00162-5
- Alexander N. Grishkov and Andrei V. Zavarnitsine, Lagrange’s theorem for Moufang loops, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 1, 41–57. MR 2155504, DOI 10.1017/S0305004105008388
- Alexander N. Grishkov and Andrei V. Zavarnitsine, Groups with triality, J. Algebra Appl. 5 (2006), no. 4, 441–463. MR 2239539, DOI 10.1142/S021949880600182X
- Alexander N. Grishkov and Andrei V. Zavarnitsine, Sylow’s theorem for Moufang loops, J. Algebra 321 (2009), no. 7, 1813–1825. MR 2494749, DOI 10.1016/j.jalgebra.2008.08.035
- J. I. Hall, On Mikheev’s construction of enveloping groups, Comment. Math. Univ. Carolin. 51 (2010), no. 2, 245–252. MR 2682477
- Jonathan I. Hall, Moufang loops and groups with triality are essentially the same thing, submitted.
- P. O. Mikheev, On the embedding of Mal′tsev algebras into Lie algebras, Algebra i Logika 31 (1992), no. 2, 167–173, 221 (Russian, with Russian summary); English transl., Algebra and Logic 31 (1992), no. 2, 106–110 (1993). MR 1289030, DOI 10.1007/BF02259849
- P. O. Mikheev, Groups that envelop Moufang loops, Uspekhi Mat. Nauk 48 (1993), no. 2(290), 191–192 (Russian, with Russian summary); English transl., Russian Math. Surveys 48 (1993), no. 2, 195–196. MR 1239875, DOI 10.1070/RM1993v048n02ABEH001029
- José M. Pérez-Izquierdo, Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math. 208 (2007), no. 2, 834–876. MR 2304338, DOI 10.1016/j.aim.2006.04.001
- José M. Pérez-Izquierdo and Ivan P. Shestakov, An envelope for Malcev algebras, J. Algebra 272 (2004), no. 1, 379–393. MR 2029038, DOI 10.1016/S0021-8693(03)00389-2
- Hala O. Pflugfelder, Quasigroups and loops: introduction, Sigma Series in Pure Mathematics, vol. 7, Heldermann Verlag, Berlin, 1990. MR 1125767
- Richard D. Schafer, An introduction to nonassociative algebras, Dover Publications, Inc., New York, 1995. Corrected reprint of the 1966 original. MR 1375235
- Jonathan D. H. Smith and Anna B. Romanowska, Post-modern algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1673047, DOI 10.1002/9781118032589
- K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F. Smith. MR 668355
Additional Information
- Georgia Benkart
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 34650
- Email: benkart@math.wisc.edu
- Sara Madariaga
- Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006, Logroño, España
- Email: sara.madariaga@unirioja.es
- José M. Pérez-Izquierdo
- Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006, Logroño, España
- Email: jm.perez@unirioja.es
- Received by editor(s): August 4, 2010
- Received by editor(s) in revised form: November 7, 2010, and June 21, 2011
- Published electronically: August 21, 2012
- Additional Notes: The second and third authors would like to thank Spanish Ministerio de Educación y Ciencia and FEDER MTM 2007-67884-C04-03 and the University of La Rioja. The second author was also supported by the Spanish MICINN grant AP2007-01986 and ATUR 09/22.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 1001-1023
- MSC (2010): Primary 16T05, 20N05, 17D99
- DOI: https://doi.org/10.1090/S0002-9947-2012-05656-X
- MathSciNet review: 2995381