Classification of simple flexible Lie-admissible algebras

Journal Article · Fri Jun 01 04:00:00 EDT 1979 · Hadronic J.; (United States)

OSTI ID:6615089

Okubo, S ^[1]; Myung, H C

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1. Univ. of Rochester, NY

Let A be a finite-dimensional flexible Lie-admissible algebra over the complex field such that A/sup -/ is a simple Lie algebra. It is shown that either A is itself a Lie algebra isomorphic to A/sup -/ or A/sup -/ is a Lie algebra of type A/sub n/ (n greater than or equal to 2). In the latter case, A is isomorphic to the algebra defined on the space of (n + 1) x (n + 1) traceless matrices with multiplication given by x * y = ..mu..xy + (1 - ..mu..)yx - (1/(n + 100 Tr (xy) E where ..mu.. is a fixed scalar, xy denotes the matrix operators in Lie algebras which has been studied in theoretical physics. We also discuss a broader class of Lie algebras over arbitrary field of characteristic not equal to 2, called quasi-classical, which includes semisimple as well as reductive Lie algebras. For this class of Lie algebras, we can introduce a multiplication which makes the adjoint operator space into an associative algebra. When L is a Lie algebra with nondegenerate killing form, it is shown that the adjoint operator algebra of L in the adjoint representation becomes a commutative associative algebra with unit element and its dimension is 1 or 2 if L is simple over the complex field. This is related to the known result that a Lie algebra of type A/sub n/ (n greater than or equal to 2) alone has a nonzero completely symmetric adjoint operator in the adjoint representation while all other algebras have none. Finally, Lie-admissible algebras associated with bilinear form are investigated.

• DOE Contract Number: AC02-76ER13065
• OSTI ID: 6615089
• Journal Information: Hadronic J.; (United States), Journal Name: Hadronic J.; (United States) Vol. 2:3; ISSN HAJOD
• Country of Publication: United States
• Language: English

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