Adjoint operators in Lie algebras and the classification of simple flexible Lie-admissible algebras

Journal Article · Wed Apr 01 04:00:00 EST 1981 · Trans. Am. Math. Soc.; (United States)

DOI:https://doi.org/10.1090/S0002-9947-1981-0603775-4· OSTI ID:6319338

Okubo, S ^[1]; Myung, H C

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1. Univ. of Rochester, NY

Let H be a finite-dimensional flexible Lie-admissible algegra over an algebraically closed field F of characteristic 0. It is shown that if H/sup -/ is a simple Lie algegra which is not of type A/sub n/ (n greater than or equal to 2) then H is a Lie algebra isomorphic to H/sup -/, and if H/sup -/ is a simple Lie algebra of type A/sub n/ (n greater than or equal to 2) then H is either a Lie algebra or isomorphic to an algebra with multiplication x not equal to y = ..mu..xy + (1 - ..mu..)yx - (1/(n + 1))Tr(xy)I which is defined on the space of (n + 1) x (n + 1) traceless matrices over f, where xy is the matrix product and ..mu.. not equal to 1/2 is a fixed scalar in F. This result for the complex field has been previously obtained by employing an analytic method. The present classification is applied to determine all flexible Lie-admissible algebra H such that H/sup -/ is reductive and the Levi-factor of H/sup -/ is simple. The central idea is the notion of adjoint operators in Lie algebras which has been studied in physical literature in conjunction with representation theory.

• DOE Contract Number: AC02-76ER13065
• OSTI ID: 6319338
• Journal Information: Trans. Am. Math. Soc.; (United States), Journal Name: Trans. Am. Math. Soc.; (United States) Vol. 264:2; ISSN TAMTA
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
CLASSIFICATION
LIE GROUPS
MATHEMATICAL OPERATORS
MATHEMATICS
SYMMETRY GROUPS