Supergeometry, noncommutative geometry and invariant theory

This thesis covers various questions in supergeometry, noncommutative geometry and invariant theory. A brief summary is given below. The fundamental theorems of invariant theory characterize the ring of invariants in C[Mr×p] under a canonical action of special linear group SLr(C). In this case, the generators correspond to the r × r minors and satisfy Pl¨ucker relations. We present here a super version of this question. We prove the first fundamental theorem for the case of special linear supergroup SL(r|s). By employing an approach based on the Jacobi’s identity, we establish certain relations, what we refer to as super Pl¨ucker relations. For the case of SL(1|1), we prove that these super Pl¨ucker relations completely characterize the ring of invariants. In a recent work by Buachalla and Somberg, Lusztig’s positive root vectors, with respect to a distinguished choice of reduced decomposition of the longest element of the Weyl group, were shown to give a quantum tangent space for every A-series full quantum flag manifold Oq(Fn). Moreover, the associated differential calculus Ω(0,•)q(Fn) was shown to have classical dimension. Here we examine in detail the rank two case, namely the full quantum flag manifold of Oq(SU3). In particular, we examine the ∗-differential calculus associated to Ω(0,•)q(F3) and its non-commutative complex geometry. A quantization of the complex Minkowski space due to R. Fioresi and others, is well-known. We extend this approach to the case of N = 2 Minkowski superspace. We give the superalgebra of N = 2 antichiral quantum superfields realized as a subalgebra of the quantum supergroup Cq[SL(4|2)]. The multiplication law in the quantum supergroup induces a coaction on the set of antichiral superfields. We also realize the quantum deformation of the Minkowski superspace as a quantum principal bundle.