Central polynomials for adjoint representations of simple Lie algebras exist

A. A. Kagarmanov ; Yu. P. Razmyslov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 4, pp. 1015-1025.

Résumé

Yu. P. Razmyslov has proved that for any finite dimensional reductive Lie algebra $\mathcal G$ over a field $K$ of zero characteristic ($\dim_{K}\mathcal G=m$) and for its arbitrary associative enveloping algebra $U$ with non-empty center $Z(U)$ there exists a central polynomial which is multilinear and skew-symmetric in $k$ sets of $m$ variables for a certain positive integer $k$. This result is now proved for adjoint representations of classical simple Lie algebras of type $A_s,B_s,C_s,D_s$ and matrix Lie algebra $M_n$ over fields of positive characteristic.