Noncommutative geometry of phase space (Q5916222)

A noncommutative generalization of symplectic geometry is explained. The paper consists of the following 5 sections: \S1. Distributions on class manifolds: General overview, \S2. Derivation-based noncommutative differential calculus, \S3. Lie superalgebra structures on the space of multiderivations, \S4. Schouten bracket as the measure of derivation of the coboundary operator from operators satisfying the Leibniz rule, \S5. Differential complexes and generalized functions on Poisson manifolds. \S1 reviews distributions on a class of (smooth) manifolds and proves that a distribution \(D: M\to Gr_k(TM)\) is integrable if and only if the graph of \(D(x)\) is an isotropic subspace for the bilinear form\((\cdot,\cdot): K_{D(x)}\times K_{D(x)}\to T_x(M)/D(x)\), for each \(x\in M\). Here \(K\) is the fibre bundle corresponding to the Cartan distribution (theorem 1, Definition of the Cartan distribution is given as definition 5). The author remarks that this is equivalent to the Frobenius theorem. In \S2, the graded algebra \(C_{Z(A)}(\text{Der}(A),A)= \Omega_Z(A)\) of antisymmetric \(Z(A)\)-multilinear mappings from \(\text{Der}(A)\) to \(A\) and the smallest differential graded subalgebra \(\Omega(A)\) of \(\Omega_Z(A)\) are introduced as the noncommutative generalization of the graded differential algebra of differential forms to \(A\). Here \(A\) is an algebra, \(Z(A)\) is the center of \(A\), and \(\text{Der}(A)\) is the Lie algebra of derivations of \(A\). Then noncommutative generalizations of several geometric notions such as connection and curvature, distribution and integral manifolds and Bott connection [\textit{R. Bott}, Lect. Notes Math. 279, 1--94 (1972; Zbl 0241.57010)] are given. In \S3, following \textit{A. Nijenhuis} and \textit{R. W. Richardson jun.} [Bull. Am. Math. Soc. 72, 1--29 (1966; Zbl 0136.30502)], supercommutator (Schouten bracket) on the space of multiderivations of a commutative algebra \(A\) (definition 13) and a Poisson algebra (definition 14) are defined. It is shown that the supercommutator and the second-order involutive element on an external algebra define a coboundary operator on this algebra (theorem 2) and the classical Koszul differential on the external algebra of differential forms can be represented as a supercommutator with a second-order element of some superalgebra containing the algebra of differential forms (proposition 3). As the inverse of this situation, a coboundary operator on an external algebra is shown to induce a supercommutator on this algebra. This is done introducing the Poisson algebra analogue of the \(*\)-operator (proposition 5). These results are applied to the classical case in \S5. But to find Casimir elements (functions in the center of the Poisson algebra), the Poisson bracket is extended to generalized functions (distributions) following \textit{J.-L. Brylinski} [J. Differ. Geom. 28, No. 1, 93--114 (1988; Zbl 0634.58029)]. Then it is shown for a compact symplectic manifold \(M\), \({\mathcal D}_0(M)\) the space of generalized functions that commute with all smooth functions, is one-dimensional and the function \(\langle\delta_\omega, \phi\rangle= \int_M\phi\cdot \omega^n\) is the generator of \({\mathcal D}_0(M)\) (corollar 2 of proposition 8). The paper is concluded with the introduction of Poisson ideals (definition 16). It is shown that if \(I\) is a maximal Poisson submanifold ideal of \(A\), then \(Q= A/I\) is a nondegenerate Poisson ideal (reduction of Poisson algebra; theorem 5). This paper has no introduction. No explanations on outline, motivation, object and main results are given.