Graded Differential Geometry of Graded Matrix Algebras

We study the graded derivation-based noncommutative differential geometry of the Z2-graded algebra M(n|m) of complex (n+m)× (n+m)-matrices with the “usual block matrix grading” (for n 6= m). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that M(n|m) is a “noncommutative graded manifold” in a stricter sense: There is a natural body map and the cohomologies of M(n|m) and its body coincide (as in the case of ordinary graded manifolds). 1991 MSC: 17B56, 17B70, 46L87, 58A50, 58B30, 58C50, 81T60 PACS: 02.40.-k, 11.30.Pb