## Faculty Publications

#### Title

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

#### Document Type

Article

#### Keywords

Adjoint dimension, Adjoint operator, Flexible Lie-admissible algebra, Highest adjoint weight, Lie algebra, Reductive Lie algebra, Representation, Weight, Weyl basis

#### Journal/Book/Conference Title

Transactions of the American Mathematical Society

#### Volume

264

#### Issue

2

#### First Page

459

#### Last Page

472

#### Abstract

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.

#### Original Publication Date

1-1-1981

#### DOI of published version

10.1090/S0002-9947-1981-0603775-4

#### Recommended Citation

Okubo, Susumu and Myung, Hyo Chul, "Adjoint operators in lie algebras and the classification of simple flexible lie-admissible algebras" (1981). *Faculty Publications*. 4935.

https://scholarworks.uni.edu/facpub/4935