Adjoint Operators In Lie Algebras And The Classification Of Simple Flexible Lie-Admissible Algebras
Adjoint dimension, Adjoint operator, Flexible Lie-admissible algebra, Highest adjoint weight, Lie algebra, Reductive Lie algebra, Representation, Weight, Weyl basis
Transactions of the American Mathematical Society
Let A be a finite-dimensional flexible Lie-admissible algebra over an algebraically closed field F of characteristic 0. It is shown that if A-is a simple Lie algebra which is not of type An(n > 2) then A-is a Lie algebra isomorphic to A-and if 21 ˜ is a simple Lie algebra of type An(n > 2) then A is either a Lie algebra or isomorphic to an algebra with multiplication x*y = 1/2 (1 —)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 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 algebras A such that A-is reductive and the Levi-factor of A-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. © 1981 American Mathematical Society.
Department of Mathematics
Original Publication Date
DOI of published version
Okubo, Susumu and Myung, Hyo Chul, "Adjoint Operators In Lie Algebras And The Classification Of Simple Flexible Lie-Admissible Algebras" (1981). Faculty Publications. 4935.