Central polynomials in irreducible representations of a~semisimple Lie algebra

Yu. P. Razmyslov

Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 99-124

Résumé

A central multilinear polynomial is constructed for every reductive finite-dimensional Lie algebra $\mathfrak G$ over an algebraically closed field $K$ of characteristic zero, and almost every faithful irreducible $K$-representation of $\mathfrak G$ in a vector space $V$. The central polynomial is of the form $f(z_{11},\dots,z_{1m},z_{21},\dots,z_{2m},\dots,z_{k1},\dots,z_{km})$, where $m=\dim_k\mathfrak G$ and $f$ is skew-symmetric with respect to the variables of each set $\{z_{i1},\dots,z_{im}\}$ ($ i=1,\dots,k$). The dimension of the vector space $V$ need not be finite. This result implies that, for the Lie algebra $W_n$ of all regular tangent vector fields of an $n$-dimensional affine algebraic variety, one can construct an associative multilinear polynomial $f$ such that the map $$ f\circ\mathrm{ad}: W_n\otimes\dots\otimes W_n\to\operatorname{End}_KW_n $$ is a map onto the center of the algebra $\operatorname{End}_{\mathscr E}W_n$, which is isomorphic to the algebra $\mathscr E$ of all regular functions of this variety. Bibliography: 10 titles.