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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.
##### MSC:
 17B50 Modular Lie (super)algebras 17A99 General nonassociative rings
##### Keywords:
algebra; Casimir element