Our recent article: Composite bundles in Clifford algebras. Gravitation theory. Part 1

Our recent article: G. Sardanashvily, A. Yarygin, “Composite bundles in Clifford algebras. Gravitation theory. Part 1” in arXiv: 1512.07581

Abstract. Based on a fact that complex Clifford algebras of even dimension are isomorphic to the matrix ones, we consider bundles in Clifford algebras whose structure group is a general linear group acting on a Clifford algebra by left multiplications, but not a group of its automorphisms. It is essential that such a Clifford algebra bundle contains spinor subbundles, and that it can be associated to a tangent bundle over a smooth manifold. This is just the case of gravitation theory. However, different these bundles need not be isomorphic. To characterize all of them, we follow the technique of composite bundles. In gravitation theory, this technique enables us to describe different types of spinor fields in the presence of general linear connections and under general covariant transformations.

Contents

1 Introduction 

2 Clifford algebras 

• 2.1 Real Clifford algebras
• 2.2 Complex Clifford algebras 

3 Automorphisms of Clifford algebras 

• 3.1 Automorphisms of real Clifford algebras
• 3.2 Pin and Spin groups
• 3.3 Automorphisms of complex Clifford algebras 

4 Spinor spaces of complex Clifford algebras 

5 Reduced structures 

• 5.1 Reduced structures in gauge theory
• 5.2 Lorentz reduced structures in gravitation theory 

6 Spinor structures 

• 6.1 Fibre bundles in Clifford algebras
• 6.2 Composite bundles in Clifford algebras

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