Hopf algebras with triality

Benkart, Georgia; Madariaga, Sara; Pérez-Izquierdo, José

doi:10.1090/S0002-9947-2012-05656-X

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Hopf algebras with triality

Authors: Georgia Benkart, Sara Madariaga and José M. Pérez-Izquierdo
Journal: Trans. Amer. Math. Soc. 365 (2013), 1001-1023
MSC (2010): Primary 16T05, 20N05, 17D99
Posted: August 21, 2012
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Abstract | References | Similar Articles | Additional Information

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.

[1] Richard Hubert Bruck, A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 20. Reihe: Gruppentheorie, Springer Verlag, Berlin, 1958. MR 0093552 (20 #76)
[2] Stephen Doro, Simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 377–392. MR 0492031 (58 #11195)
[3] 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 (2006f:20079)
[4] George Glauberman, On loops of odd order. II, J. Algebra 8 (1968), 393–414. MR 0222198 (36 #5250)
[5] Alexander Grishkov, Lie algebras with triality, J. Algebra 266 (2003), no. 2, 698–722. MR 1995132 (2004h:17019), http://dx.doi.org/10.1016/S0021-8693(03)00162-5
[6] 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 (2006d:20122), http://dx.doi.org/10.1017/S0305004105008388
[7] Alexander N. Grishkov and Andrei V. Zavarnitsine, Groups with triality, J. Algebra Appl. 5 (2006), no. 4, 441–463. MR 2239539 (2007g:20062), http://dx.doi.org/10.1142/S021949880600182X
[8] Alexander N. Grishkov and Andrei V. Zavarnitsine, Sylow’s theorem for Moufang loops, J. Algebra 321 (2009), no. 7, 1813–1825. MR 2494749 (2010h:20159), http://dx.doi.org/10.1016/j.jalgebra.2008.08.035
[9] J. I. Hall, On Mikheev’s construction of enveloping groups, Comment. Math. Univ. Carolin. 51 (2010), no. 2, 245–252. MR 2682477 (2011h:20135)
[10] Jonathan I. Hall, Moufang loops and groups with triality are essentially the same thing, submitted.
[11] 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, http://dx.doi.org/10.1007/BF02259849
[12] 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 (94g:20098), http://dx.doi.org/10.1070/RM1993v048n02ABEH001029
[13] José M. Pérez-Izquierdo, Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math. 208 (2007), no. 2, 834–876. MR 2304338 (2008f:17006), http://dx.doi.org/10.1016/j.aim.2006.04.001
[14] José M. Pérez-Izquierdo and Ivan P. Shestakov, An envelope for Malcev algebras, J. Algebra 272 (2004), no. 1, 379–393. MR 2029038 (2004j:17040), http://dx.doi.org/10.1016/S0021-8693(03)00389-2
[15] Hala O. Pflugfelder, Quasigroups and loops: introduction, Sigma Series in Pure Mathematics, vol. 7, Heldermann Verlag, Berlin, 1990. MR 1125767 (93g:20132)
[16] Richard D. Schafer, An introduction to nonassociative algebras, Dover Publications Inc., New York, 1995. Corrected reprint of the 1966 original. MR 1375235 (96j:17001)
[17] 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 (2000d:00001)
[18] 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, 1982. Translated from the Russian by Harry F. Smith. MR 668355 (83i:17001)

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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05656-X
PII: S 0002-9947(2012)05656-X
Keywords: Triality, Hopf algebras, Moufang-Hopf algebras, Moufang loops, Malcev algebras, Lie algebras, groups, nonassociative algebra
Received by editor(s): August 4, 2010
Received by editor(s) in revised form: November 7, 2010, and June 21, 2011
Posted: 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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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