Equivariant Morita Equivalence and Twist Star Products
Speaker: Thomas Weber, University of Naples Federico II
Abstract: The notion of Drinfel’d twist gives rise to deformations of Hopf algebras and their module algebras at once. In the case of a universal enveloping algebra of a Lie algebra acting on a manifold via derivations, a twist induces a deformation quantization of the manifold. For example, the Moyal-Weyl star product emerges in this way. While it is well-known that every Poisson manifold admits a deformation quantization, it is not clear which deformations arise by this twisting procedure. One can exclude the symplectic 2-sphere and the higher symplectic Riemann surfaces to inherit such twist star products by some simple geometrical and topological arguments, but besides that not much is known. In this talk we will connect the existence of twist star products to the existence of equivariant line bundles with non-trivial Chern class. To this aim, we introduce the theory of equivariant Morita equivalence of star products and discuss some of its properties. We will show that it implies equivalence in the case of twist star products. As a consequence of this and of a result by S. Waldmann and H. Bursztyn, we will argue that non-trivial equivariant line bundles and twist star products cannot coexist on the same symplectic manifold. Therefore, one can deduce for example that there is no symplectic star product on the projective space CP^(n-1) induced by a twist based on U(gl_n(C))[[h]].
Room Reservation Information
Room Number: 320 Whitmore
Time: 2:30pm - 3:20pm