Some remarks on the Akivis algebras and the Pre-Lie algebras

Yuqun Chen; Yu Li

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 3, page 707-720
  • ISSN: 0011-4642

Abstract

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In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in X * * forms a linear basis of the free Pre-Lie algebra PLie ( X ) generated by the set X . For completeness, we give the details of the proof of Shirshov’s Composition-Diamond lemma for non-associative algebras.

How to cite

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Chen, Yuqun, and Li, Yu. "Some remarks on the Akivis algebras and the Pre-Lie algebras." Czechoslovak Mathematical Journal 61.3 (2011): 707-720. <http://eudml.org/doc/196373>.

@article{Chen2011,
abstract = {In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in $X^\{**\}$ forms a linear basis of the free Pre-Lie algebra $\{\rm PLie\}(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov’s Composition-Diamond lemma for non-associative algebras.},
author = {Chen, Yuqun, Li, Yu},
journal = {Czechoslovak Mathematical Journal},
keywords = {non-associative algebra; Akivis algebra; universal enveloping algebra; Pre-Lie algebra; Gröbner-Shirshov basis; nonassociative algebra; Akivis algebra; universal enveloping algebra; pre-Lie algebra; Gröbner-Shirshov basis},
language = {eng},
number = {3},
pages = {707-720},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some remarks on the Akivis algebras and the Pre-Lie algebras},
url = {http://eudml.org/doc/196373},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Chen, Yuqun
AU - Li, Yu
TI - Some remarks on the Akivis algebras and the Pre-Lie algebras
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 707
EP - 720
AB - In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra ${\rm PLie}(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov’s Composition-Diamond lemma for non-associative algebras.
LA - eng
KW - non-associative algebra; Akivis algebra; universal enveloping algebra; Pre-Lie algebra; Gröbner-Shirshov basis; nonassociative algebra; Akivis algebra; universal enveloping algebra; pre-Lie algebra; Gröbner-Shirshov basis
UR - http://eudml.org/doc/196373
ER -

References

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