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Concept
Clifford bundle
Summary
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold M which is called the Clifford bundle of M.
Let V be a (real or complex) vector space together with a symmetric bilinear form . The Clifford algebra Cl(V) is a natural (unital associative) algebra generated by V subject only to the relation
for all v in V. One can construct Cl(V) as a quotient of the tensor algebra of V by the ideal generated by the above relation.
Like other tensor operations, this construction can be carried out fiberwise on a smooth vector bundle. Let E be a smooth vector bundle over a smooth manifold M, and let g be a smooth symmetric bilinear form on E. The Clifford bundle of E is the fiber bundle whose fibers are the Clifford algebras generated by the fibers of E:
The topology of Cl(E) is determined by that of E via an associated bundle construction.
One is most often interested in the case where g is positive-definite or at least nondegenerate; that is, when (E, g) is a Riemannian or pseudo-Riemannian vector bundle. For concreteness, suppose that (E, g) is a Riemannian vector bundle. The Clifford bundle of E can be constructed as follows. Let ClnR be the Clifford algebra generated by Rn with the Euclidean metric. The standard action of the orthogonal group O(n) on Rn induces a graded automorphism of ClnR. The homomorphism
is determined by
where vi are all vectors in Rn. The Clifford bundle of E is then given by
where F(E) is the orthonormal frame bundle of E. It is clear from this construction that the structure group of Cl(E) is O(n). Since O(n) acts by graded automorphisms on ClnR it follows that Cl(E) is a bundle of Z2-graded algebras over M. The Clifford bundle Cl(E) can then be decomposed into even and odd subbundles:
If the vector bundle E is orientable then one can reduce the structure group of Cl(E) from O(n) to SO(n) in the natural manner.
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