Free generic Poisson fields and algebras. (English) Zbl 1434.17027
Summary: The free generic Poisson algebras (\(GP\)-algebras) over a field \(k\) of characteristic 0 are studied. We prove that certain properties of free Poisson algebras are true for free \(GP\)-algebras as well. In particular, the universal multiplicative enveloping algebra \(U=U(GP(x_1,\dots,x_n))\) of a free \(GP\)-field \(GP(x_1,\dots,x_n)\) is a free ideal ring. Besides, the Poisson and polynomial dependence of two elements are equivalent in \(GP(x_1,\dots,x_n)\). As a corollary, all automorphisms of the free \(GP\)-algebra \(GP\{x,y\}\) are tame and we have the isomorphisms of groups of automorphisms \(\operatorname{Aut} GP\{x,y\}\cong\operatorname{Aut} P\{x,y\}\cong\operatorname{Aut} k[x,y]\).
MSC:
| 17B63 | Poisson algebras |
| 16S40 | Smash products of general Hopf actions |
| 17A50 | Free nonassociative algebras |
| 17A36 | Automorphisms, derivations, other operators (nonassociative rings and algebras) |
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