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On Reductive Lie Admissible Algebras

Published online by Cambridge University Press:  20 November 2018

Arthur A. Sagle
Arthur A. Sagle*
Affiliation:
University of Minnesota, Minneapolis, Minnesota
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A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, then

and note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 325 - 331
Copyright
Copyright © Canadian Mathematical Society 1971

References

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