New Approaches to Quaternionic Algebra and Geometry

By using the rigidity of quaternionic bimodules, we present a family of finite groups acting on the n-th tensor powers of the algebra of quaternions. These finite groups are used to present the Lorentz group and some of its representations on IHI®n. We discuss some new approaches to quaternionic differential geometry, in particular for Lorentzian geometry. The Levi-Civita connection for a quaternionic Frobenius metric is presented using quaternionic functions and some group algebra. By including the algebra of quaternions with its group of automorphisms we present a quaternionic groupoid, 1-l. We show that this groupoid is equivalent to the groupoid of self equivalences and natural transformations of the category of right IHI-modules. The Euclidean conformal group in four dimensions appears as a tensor product on this groupoid. After a description of the general theory of stacks and gerbes, we introduce the notion of a quaternionic gerbe. Some basic examples are given and, following Brylinski [3], we present a non-neutral gerbe associated to a given principle S0(3)-bundle. In the same way that the transition functions for a principle bundle can be organized into a cocycle, we demonstrate the construction of a "cocycle" for quaternionic gerbes. We give two presentations of this cocycle, first explicitly in terms of (IHI* -+ Aut(IHI) )-valued functions, and then as a family of quaternionic bitorsors and their maps. Cocycle and coboundary conditions are presented and we show that equivalence classes of cocycles classify quaternionic gerbes. A conformal four manifold carries a canonical quaternionic gerbe related to the tangent bundle. We present this "tangent gerbe" explicitly in the form of a cocycle.