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Central polynomials in irreducible representations of a semisimple Lie algebra
Yu. P. Razmyslov
Abstract:
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.
UDC:
519.4
MSC: 17B20,
17B10 Received: 17.09.1982
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