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Casimir elements of \(\mathbb Z\)-forms of modular Lie algebras. (English. Russian original) Zbl 1037.17018
Russ. Math. 46, No. 3, 28-31 (2002); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2002, No. 3, 32-35 (2002).
Let \(G\) be an arbitrary algebra over a field \( k\) and \(L_x\) the multiplication operator of \(x\in G\). Extend \(L_x\) to be a derivation \(D_{x\text{ }}\)of the symmetric algebra \(S(G)\) generated by \(G\). Let \(D\) be the associative algebra generated by \(\{D_x,x\in G\}\) and \(D_L\) the Lie algebra associated to \(D\). If \(P\) and \(P^{*}\) are mutually dual submodules of the \(D_{L}\)-module \(S(G)\) and \(\{u_i\}\), \(\{u_i^{*}\} \) their dual bases, then \(z=z(P,P^{*})\) is called a generalized Casimir element. The author proves that if \(z\) is \(D_{L}\)-invariant in \(S(G),\) \(\deg z>1\) (and \(\deg z\) is not divisible by \(p\) if \(\text{Char\,} k=p>0\)) and \(\dim _kG<\infty \), then \(z=z(P,P^{*})\) with \(P\subset G.\) This is an analogue to a theorem of L. P. Bedratyuk [Symmetric invariants of modular Lie algebras, Ph. Thesis, Moscow (1995)] for the central elements of the universal enveloping algebra of a modular Lie algebra. An application to the so called “\(\mathbb{Z}\)-form” of a simple Lie algebra over \(F_p\) is also given.
17B50 Modular Lie (super)algebras
17A99 General nonassociative rings