2000/09/18 by Henrique Bursztyn, Stefan Waldmann, Bursztyn, Henrique +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #math-ph #math.MP #math.QA #math.SG
paper · pdf · doi:10.48550/arxiv.math/0009170
14 pages
arxiv created 2000/09/18 · openalex publication_date 2000/09/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by deformation quantization, we consider in this paper ^*-algebras \mathcal A over rings \ring C = \ringR(i), where \ring R is an ordered ring and i2 = -1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) \mathcal A-valued inner product. For A=C^∞(M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star-product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C^∞(M) and Γ^∞(\End(E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C^*-algebras. We also discuss the semi-classical geometry arising from these deformations.