Ring theory from symplectic geometry
The desire for a noncommutative differential geometry in the spirit of Poisson and symplectic geometry provides an incentive to isolate the ring theory which appears explicitly and implicitly in the traditional geometry. The paper under review is a sort of primer on symplectic algebra for ring theoretists which goes far enough to clarify some ideas that appeared before in the work of other authors and to answer some open questions. Although the study has been inspired by geometry, it treats properties of noncommutative algebras which stand also on their own. First the authors point out a fundamental bifurcation in the theory of Poisson algebras: If a prime Poisson algebra is not commutative, then the Poisson bracket is equal to the commutator bracket up to appropriate scalar. Then the paper deals with the general definition of symplectic algebra. The flexibility allowed by the authors distills the algebraic essence of many typically symplectic properties. In particular, the general version of symplectic algebra is broad enough to encompass algebras with differential operators on commutative affine domains. The authors generalize the theorem of Dixmier that the derivations of the Weyl algebra are inner and give a noncommutative analogue of a theorem of Calabi. They also show that a simple algebra with a symplectic structure supported by a finite differential expression, satisfies a polynomial identity. Then the paper concentrates on more classical commutative Poisson algebras and show that, under some extra smoothness assumptions, the Poisson bracket is compatible with the symplectic structure if the module of the algebra derivations is generated by Poisson-inner derivations. Finally, the authors consider symplectic algebras with supporting two-form which is closed with respect to some differential. They answer a question of Dubois-Violette and obtain some connections with the Lie structure of symplectic algebras.
Poisson algebras with polynomial identities
scientific article; zbMATH DE number 1121681
scientific article