Clifford algebra and spinor-valued functions: a function theory for the Dirac operator, Volume 1
This volume describes the substantial developments in Clifford analysis which have taken place during the last decade and, in particular, the role of the spin group in the study of null solutions of real and complexified Dirac and Laplace operators. The book has six main chapters. The first two (Chapters 0 and I) present classical results on real and complex Clifford algebras and show how lower-dimensional real Clifford algebras are well-suited for describing basic geometric notions in Euclidean space. Chapters II and III illustrate how Clifford analysis extends and refines the computational tools available in complex analysis in the plane or harmonic analysis in space. In Chapter IV the concept of monogenic differential forms is generalized to the case of spin-manifolds. Chapter V deals with analysis on homogeneous spaces, and shows how Clifford analysis may be connected with the Penrose transform. The volume concludes with some Appendices which present basic results relating to the algebraic and analytic structures discussed. These are made accessible for computational purposes by means of computer algebra programmes written in REDUCE and are contained on an accompanying floppy disk.
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Clifford Algebras and Spinor Spaces
Pinp q and Spinp q
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analytic basic bivector bundle C-valued called Cauchy Cauchy Theorem clearly Clifford algebra Clifford analysis Clifford number coefficients cohomology compact complex vector space compute conjugation Consequently consider coordinates Corollary corresponding decomposition defined definition denoted differential forms dimension Dirac equation Dirac operator domain eigenspaces element Euclidean exists follows Furthermore G Rp Gegenbauer polynomials given Hence homogeneous space inner product integral formula intertwining map invariant inverse irreducible isomorphic isotropic subspace Laurent Lemma Leray-Norguet residue linear M+(k manifold matrix monogenic functions Moreover multiplication non-degenerate Notice obtain oriented orthonormal basis Penrose transform polynomial of degree Proof prove pure spinors quaternionic real orthogonal space relations representation resp respectively restriction Rm+1 rotation satisfying scalar separately monogenic solutions spherical monogenics Spin Spin group Spin(m Spin+(2n spinor space subalgebra submanifold subset Theorem tubular neighbourhood universal Clifford algebra vector space Weyl equation whence