Enveloping algebras of Malcev algebras

Murray R. Bremner; Irvin R. Hentzel; Luiz A. Peresi; Marina V. Tvalavadze; Hamid Usefi

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 2, page 157-174
  • ISSN: 0010-2628

Abstract

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We first discuss the construction by Pérez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops.

How to cite

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Bremner, Murray R., et al. "Enveloping algebras of Malcev algebras." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 157-174. <http://eudml.org/doc/37749>.

@article{Bremner2010,
abstract = {We first discuss the construction by Pérez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops.},
author = {Bremner, Murray R., Hentzel, Irvin R., Peresi, Luiz A., Tvalavadze, Marina V., Usefi, Hamid},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Malcev algebras; universal enveloping algebras; universal alternative envelopes; differential operators; Bol algebras; analytic loops; Mal'tsev algebra; universal enveloping algebra; universal alternative envelope; differential operator; Bol algebra; analytic loop},
language = {eng},
number = {2},
pages = {157-174},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Enveloping algebras of Malcev algebras},
url = {http://eudml.org/doc/37749},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Bremner, Murray R.
AU - Hentzel, Irvin R.
AU - Peresi, Luiz A.
AU - Tvalavadze, Marina V.
AU - Usefi, Hamid
TI - Enveloping algebras of Malcev algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 157
EP - 174
AB - We first discuss the construction by Pérez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops.
LA - eng
KW - Malcev algebras; universal enveloping algebras; universal alternative envelopes; differential operators; Bol algebras; analytic loops; Mal'tsev algebra; universal enveloping algebra; universal alternative envelope; differential operator; Bol algebra; analytic loop
UR - http://eudml.org/doc/37749
ER -

References

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