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

Abstract:

Let $\mathfrak {A}$ be a finite-dimensional flexible Lie-admissible algebra over an algebraically closed field $F$ of characteristic $0$. It is shown that if ${\mathfrak {A}^ - }$ is a simple Lie algebra which is not of type ${A_n}(n \geqslant 2)$ then $\mathfrak {A}$ is a Lie algebra isomorphic to ${\mathfrak {A}^ - }$, and if ${\mathfrak {A}^ - }$ is a simple Lie algebra of type ${A_n}(n \geqslant 2)$ then $\mathfrak {A}$ is either a Lie algebra or isomorphic to an algebra with multiplication $x \ast y = \mu xy + (1 - \mu )yx - (1/(n + 1))\operatorname {Tr} (xy)I$ which is defined on the space of $(n + 1) \times (n + 1)$ traceless matrices over $F$, where $xy$ is the matrix product and $\mu \ne \frac {1} {2}$ 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 algebras $\mathfrak {A}$ such that ${\mathfrak {A}^ - }$ is reductive and the Levi-factor of ${\mathfrak {A}^ - }$ 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.