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

Susumu Okubo, Hyo Chul Myung
1981 Transactions of the American Mathematical Society  
Let 31 be a finite-dimensional flexible Lie-admissible algebra over an algebraically closed field F of characteristic 0. It is shown that if 21 ~ is a simple Lie algebra which is not of type An (n > 2) then 21 is a Lie algebra isomorphic to 21", and if 9t is a simple Lie algebra of type An (n > 2) then 21 is either a Lie algebra or isomorphic to an algebra with multiplication x » y = pxy + ( 1 -¡i)yx -(l/(n + l))Tr(xy)/ which is defined on the space of (n + 1) X (n + 1) traceless matrices over
more » ... less matrices over F, where xy is the matrix product and p ¥={ 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 21 such that 21 ~ is reductive and the Levi-factor of 21 ~ 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.
doi:10.1090/s0002-9947-1981-0603775-4 fatcat:56chctj77bcgheutlkcgaohlem