2021/07/01 by Bavula, V. V.
#13N05 #13N15 #16D30 #16P90 #16S32 #17B20 #17B63 #17B65 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2107.00321
For an arbitrary Poisson algebra \CP over an arbitrary field, an (analogue of) the Poincaré-Birkhof-Witt Theorem is proven and several presentations/constructions for its Poisson enveloping algebra \CU (\CP ) are given. As a result, explicit sets of generators and defining relations are given for \CU (\CP ) and the algebra P\CD (\CP) of Poisson differential operators on \CP. Simplicity criteria for the algebras \CU (\CP ) and P\CD (\CP ) are given. In the case when the algebra \CP is of essentially finite type, a criterion for the algebra \CU (\CP ) to be a domain is presented and a criterion for a natural epimorphism \CU (\CP )\ra P\CD (\CP ) to be an isomorphism is given. The kernel of the epimorphism is described and for large classes of Poisson algebras an explicit set of generators is given. Explicit formulae for the Gelfand-Kirillov dimension of the algebras \CU (\CP ) and P\CD (\CP) are given. In the case when the Poisson algebra \CP is a regular domain of essentially finite type an explicit simplecticity criterion for \CP is found and a criterion is presented for the algebra \CU (\CP ) to be isomorphic to the algebra \CD (\CP ) of differential operators on \CP.