By using the rigidity of quaternionic bimodules, we present a family of finite groups acting on the nth 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 LeviCivita 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, 1l. We show that this groupoid is equivalent to the groupoid of self equivalences and natural transformations of the category of right IHImodules. 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 nonneutral 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.
New Approaches to Quaternionic Algebra and Geometry
Finlay N., Thompson
1999
Abstract
By using the rigidity of quaternionic bimodules, we present a family of finite groups acting on the nth 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 LeviCivita 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, 1l. We show that this groupoid is equivalent to the groupoid of self equivalences and natural transformations of the category of right IHImodules. 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 nonneutral 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.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.14242/122643
URN:NBN:IT:SISSA122643