The existence of invariant twisted products (deformations of the associative algebra of C=-functions) on the cotangent bundles of classical groups and Stiefel manifolds is proved by explicit constructions. All these products are positive.
On a symplectic manifold W, we can define Classical Mechanics by means of two algebraic structures defined on the vector space-N = C ~ (W; R)
(1) a sructure of associative algebra defined by the usual product of functions.
(2) a structure of Lie algebra defined by the Poisson bracket ~ . We have shown ([1], [2]) that it is reasonable to study if we can obtain by suitable deformations of these two structures a model isomorphic to the Quantum Mechanics. An important theorem of J. Vey [5] proves the existence of suitable deformations of the Poisson Lie algebra. The general problem of the existence of suitable associative (non commutative) deformations of the associative algebra N (or ,v-products for N) appears as much more difficult. Some procedures for defining such deformations has been given [1] in the context of the coadjoint representations of Lie algebras.
We prove here the existence of good invariant *u-products for the symplectic manifolds defined by the cotangent bundles of classical groups and Stiefel manifolds. A study concerning the Grassmann manifolds is given.
For the construction, we use for one part procedures of quotient and product, and for another part invariant 'pseudo-metrics'.
1. NOTION OF ,v-PRODUCT
The definitions and notations are those of [ 1 ]. In particular Qr is a bidifferential operator on N of maximum order r (r > 1) in each argument, null on the constants, such that the principal symbol of Qr coincides with the principal symbol of U (for an arbitrary symplectic connection F). We take Q0 (u, v) = uv, Q1 (u, v) = P (u, v). We suppose that Qr is symmetric in (u, v) ifr is even, skewsymmetric if r is odd (condition of symmetry).