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


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