AUTOMORPHISMS AND TWISTED FORMS OF DIFFERENTIAL LIE CONFORMAL SUPERALGEBRAS

Abstract

Given a conformal superalgebra A over an algebraically closed field k of characteristic zero, a twisted loop conformal superalgebra L based on A has a differential conformal superalgebra structure over the differential Laurent polynomial ring D. In this context, L is a D_m/D–form of A \otimes D with respect to an étale extension of differential rings D_m/D, and hence is a \hat{D}/D–form of A. Such a perspective reduces the problem of classifying the twisted loop conformal superalgebras based on A to the computation of the non-abelian cohomology set of its automorphism group functor. The primary goal of this dissertation is to classify the twisted loop conformal superalgebras based on A when A is one of the N=1,2,3 and (small or large) N=4 conformal superalgebras. To achieve this, we first explicitly determined the automorphism group of the \hat{D}–conformal superalgebra A\otimes\hat{D} in each case. We then computed the corresponding non-abelian continuous cohomology set, and obtained the classification of our objects up to isomorphism over D. Finally, by applying the so-called “centroid trick”, we deduced from isomorphisms over D to isomorphisms over k, thus accomplishing the classification over k. Additionally, in order to understand the representability of the automorphism group functors of the N=1,2,3 and small N = 4 conformal superalgebras, we discuss the (R,d)–points of these automorphism group functors for an arbitrary differential ring (R,d). In particular, if R is an integral domain with certain additional assumptions in the small N = 4 case), these automorphism groups have been completely determined.