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Involutory families of functions on dual spaces of Lie algebras of type $$G+_{\phi}V$$. (English. Russian original) Zbl 0664.17006
Russ. Math. Surv. 42, No. 6, 227-228 (1987); translation from Usp. Mat. Nauk 42, No. 6, 183-184 (1987).
Let G be a semisimple (complex or real) Lie algebra, and $$\pi$$ be its linear representation in a space V. Then $$\pi$$ defines the Lie algebra $$L=G+V$$ which is a semidirect sum of G and V. If f and g are analytical functions on $$L^*$$ then the formula $$\{f,g\}(x)=(x,[df,dg])$$ defines the Poisson bracket. One says that f and g are in involution if $$\{$$ f,g$$\}$$ $$\equiv 0$$. The aim of the paper is the proof of the following statement: There is a full involutive set of functions on $$L^*$$, which consists of polynomials.
Reviewer: A.Klimyk

##### MSC:
 17B05 Structure theory for Lie algebras and superalgebras 58D15 Manifolds of mappings
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