LEADER 05916nam 22007212 450 001 9910790005003321 005 20151005020621.0 010 $a1-107-22240-0 010 $a1-280-77308-1 010 $a1-139-07669-8 010 $a9786613683854 010 $a0-511-97299-7 010 $a1-139-08124-1 010 $a1-139-07097-5 010 $a1-139-07897-6 010 $a1-139-08351-1 035 $a(CKB)2670000000159228 035 $a(EBL)692004 035 $a(OCoLC)784881794 035 $a(SSID)ssj0000633382 035 $a(PQKBManifestationID)11941503 035 $a(PQKBTitleCode)TC0000633382 035 $a(PQKBWorkID)10617557 035 $a(PQKB)10548131 035 $a(UkCbUP)CR9780511972997 035 $a(MiAaPQ)EBC692004 035 $a(Au-PeEL)EBL692004 035 $a(CaPaEBR)ebr10546443 035 $a(CaONFJC)MIL368385 035 $a(PPN)261363727 035 $a(EXLCZ)992670000000159228 100 $a20101005d2011|||| uy| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aClifford algebras $ean introduction /$fD.J.H. Garling$b[electronic resource] 210 1$aCambridge :$cCambridge University Press,$d2011. 215 $a1 online resource (vii, 200 pages) $cdigital, PDF file(s) 225 1 $aLondon Mathematical Society student texts ;$v78 300 $aTitle from publisher's bibliographic system (viewed on 05 Oct 2015). 311 $a1-107-42219-1 311 $a1-107-09638-3 320 $aIncludes bibliographical references and index. 327 $aCover; London Mathematical Society Student Texts 78: Clifford Algebras: An Introduction; Title; Copyright; Contents; Introduction; PART ONE: THE ALGEBRAIC ENVIRONMENT; 1: Groups and vector spaces; 1.1 Groups; 1.2 Vector spaces; 1.3 Duality of vector spaces; 2: Algebras, representations and modules; 2.1 Algebras; 2.2 Group representations; 2.3 The quaternions; 2.4 Representations and modules; 2.5 Module homomorphisms; 2.6 Simple modules; 2.7 Semi-simple modules; 3: Multilinear algebra; 3.1 Multilinear mappings; 3.2 Tensor products; 3.3 The trace 327 $a3.4 Alternating mappings and the exterior algebra3.5 The symmetric tensor algebra; 3.6 Tensor products of algebras; 3.7 Tensor products of super-algebras; PART TWO: QUADRATIC FORMS AND CLIFFORD ALGEBRAS; 4: Quadratic forms; 4.1 Real quadratic forms; 4.2 Orthogonality; 4.3 Diagonalization; 4.4 Adjoint mappings; 4.5 Isotropy; 4.6 Isometries and the orthogonal group; 4.8 The Cartan-Dieudonne? theorem; 4.9 The groups SO(3) and SO(4); 4.10 Complex quadratic forms; 4.11 Complex inner-product spaces; 5: Clifford algebras; 5.1 Clifford algebras; 5.2 Existence; 5.3 Three involutions 327 $a5.4 Centralizers, and the centre5.5 Simplicity; 5.6 The trace and quadratic form on A(E, q); 5.7 The group G(E; q) of invertible elements of A(E, q); 6: Classifying Clifford algebras; 6.1 Frobenius' theorem; 6.2 Clifford algebras A(E, q) with dimE = 2; 6.3 Clifford's theorem; 6.4 Classifying even Clifford algebras; 6.5 Cartan's periodicity law; 6.6 Classifying complex Clifford algebras; 7: Representing Clifford algebras; 7.1 Spinors; 7.2 The Clifford algebras Ak,k; 7.3 The algebras Bk,k+1 and Ak,k+1; 7.4 The algebras Ak+1,k and Ak+2,k; 7.5 Clifford algebras A(E, q) with dim E = 3 327 $a7.6 Clifford algebras A(E, q) with dim E = 47.7 Clifford algebras A(E, q) with dim E = 5; 7.8 The algebras A6, B7, A7 and A8; 8: Spin; 8.1 Clifford groups; 8.2 Pin and Spin groups; 8.3 Replacing q by ?q; 8.4 The spin group for odd dimensions; 8.5 Spin groups, for d = 2; 8.6 Spin groups, for d = 3; 8.7 Spin groups, for d = 4; 8.8 The group Spin5; 8.9 Examples of spin groups for d >= 6; 8.10 Table of results; PART THREE: SOME APPLICATIONS; 9: Some applications to physics; 9.1 Particles with spin 1/2; 9.2 The Dirac operator; 9.3 Maxwell's equations; 9.4 The Dirac equation 327 $a10: Clifford analyticity10.1 Clifford analyticity; 10.2 Cauchy's integral formula; 10.3 Poisson kernels and the Dirichlet problem; 10.4 The Hilbert transform; 10.5 Augmented Dirac operators; 10.6 Subharmonicity properties; 10.7 The Riesz transform; 10.8 The Dirac operator on a Riemannian manifold; 11: Representations of Spind and SO(d); 11.1 Compact Lie groups and their representations; 11.2 Representations of SU(2); 11.3 Representations of Spind and SO(d) for d<=4; 12: Some suggestions for further reading; The algebraic environment; Quadratic spaces; Clifford algebras 327 $aClifford algebras and harmonic analysis 330 $aClifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions. They are famously used in proofs of the Atiyah-Singer index theorem, to provide double covers (spin groups) of the classical groups and to generalize the Hilbert transform. They also have their place in physics, setting the scene for Maxwell's equations in electromagnetic theory, for the spin of elementary particles and for the Dirac equation. This straightforward introduction to Clifford algebras makes the necessary algebraic background - including multilinear algebra, quadratic spaces and finite-dimensional real algebras - easily accessible to research students and final-year undergraduates. The author also introduces many applications in mathematics and physics, equipping the reader with Clifford algebras as a working tool in a variety of contexts. 410 0$aLondon Mathematical Society student texts ;$v78. 606 $aClifford algebras 615 0$aClifford algebras. 676 $a512.57 700 $aGarling$b D. J. H.$056885 801 0$bUkCbUP 801 1$bUkCbUP 906 $aBOOK 912 $a9910790005003321 996 $aClifford algebras$93850917 997 $aUNINA