Bimodule deformations, Picard groups and contravariant connections

We study deformations of invertible bimodules and the behavior of Picard groups under deformation quantization. While K_0-groups are known to be stable under formal deformations of algebras, Picard groups may change drastically. We identify the semiclassical limit of bimodule deformations as contravariant connections and study the associated deformation quantization problem. Our main focus is on formal deformation quantization of Poisson manifolds by star products.Comment: 32 pages. Minor corrections in Sections 5 and 6, typos fixed. Revised version to appear in K-theor

Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization

In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the *-representation theory of such *-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C^*-algebras. Throughout this paper the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C^*-algebras, we show that the GNS construction of *-representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two *-algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate *-representations of the two *-algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over *-algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules.Comment: LaTeX2e, 51pages, minor typos corrected and Note/references adde

Lie groupoids and the Frolicher-Nijenhuis bracket

The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different characterizations of multiplicative vector-valued forms.Comment: 16 pages. Appeared in special volume of the Bull. Braz. Math. Society in 2013 (IMPA 60 years

The characteristic classes of Morita equivalent star products on symplectic manifolds

In this paper we give a complete characterization of Morita equivalent star products on symplectic manifolds in terms of their characteristic classes: two star products $\star$ and $\star'$ on $(M,\omega)$ are Morita equivalent if and only if there exists a symplectomorphism $\psi:M\longrightarrow M$ such that the relative class $t(\star,\psi^*(\star'))$ is 2 \pi \im-integral. For star products on cotangent bundles, we show that this integrality condition is related to Dirac's quantization condition for magnetic charges.Comment: 22 pages. A few corrections made to Sections 3.2 and 3.

Dirac structures, moment maps and quasi-Poisson manifolds

We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and hamiltonian actions, to the realm of Dirac geometry. As an example, we show how hamiltonian quasi-Poisson manifolds fit into this framework by constructing an ``inversion'' procedure relating quasi-Poisson bivectors to twisted Dirac structures.Comment: 36 pages. Typos and signs fixed. To appear in Progress in Mathematics, Festschrift in honor of Alan Weinstein, Birkause