Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras...

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Bibliographic Details
Published in:Communications in algebra 2017-07, Vol.45 (7), p.2809-2820
Main Author: Umirbaev, Ualbai
Format: Article
Language:English
Subjects:
Algebra
Commutators
Derivation
Derivations
free algebras
Jacobian matrices
left-symmetric algebras
Lie groups
Mathematical analysis
Primary 17D25, 17A42, 14R15
Secondary 17A36, 17A50
Symmetry
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Summary:A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2016.1233206