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Covariant and contravariant spinors
- Source :
- Clifford Algebras and Lie Theory ISBN: 9783642362156
- Publication Year :
- 2013
- Publisher :
- Springer Berlin Heidelberg, 2013.
-
Abstract
- Let V be a real or complex vector space with a non-degenerate symmetric bilinear form B, and let q: ∧(V)→Cl(V) be the quantization map. The space q(∧2(V)) is a Lie subalgebra under commutation in the Clifford algebra. This subspace is canonically isomorphic to the orthogonal Lie algebra of V, and the restriction of the exponential map for the Clifford algebra is identified with the exponential map for the spin group. One of the problems addressed in this chapter is to give explicit formulas for these Clifford exponentials, in terms of exterior algebra exponentials of the corresponding elements in ∧2(V). These questions will be studied using the spin representation for the vector space V ∗⊕V, with bilinear form given by the pairing. One of the outcomes of this discussion is the construction of a remarkable ∧(V)-valued function on the orthogonal Lie algebra, which will play a role in our discussion of the Duflo theorem in Chapter 7.
Details
- ISBN :
- 978-3-642-36215-6
- ISBNs :
- 9783642362156
- Database :
- OpenAIRE
- Journal :
- Clifford Algebras and Lie Theory ISBN: 9783642362156
- Accession number :
- edsair.doi...........7b00f45d936e378644139b7696d6aa03
- Full Text :
- https://doi.org/10.1007/978-3-642-36216-3_4