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Niveau: Supérieur, Doctorat, Bac+8

Clifford algebras and spin representations by Wesam TALAB , Monday 6 mars 2006 introduction In all this document V denotes a finite dimensional complex vector space and Q : V ? V ?? C a symmetric bilinear form on V, Among the repre- sentations of g = so(V ) there exist a representations which appear in ?kV , but not all . For build these which disappear we will use the constructions of Clifford algebras . 1 Clifford algebras Definition 1.1 Let V be a complex vector space with a symmetric bilinear form Q, a Clifford algebra associated to this data is the associative algebra with unit 1 : C = C(Q) = C(V,Q) := T (V )/I where T (V ) denotes the tensor algebra : T (V ) = C ? V ? (V ? V )? . . . And I denotes the tow-sided ideal in T (V ) generated by all elements of the form v?w+w?v?2Q(v, w).1 , for all v, w ? V ,and 1 is the unit element of the tensor algebra T (V ),in other words, C(V,Q) is an associative algebra with unity 1, which contains and is generated by V with v.w + w.v = 2Q(v, w).

Clifford algebras and spin representations by Wesam TALAB , Monday 6 mars 2006 introduction In all this document V denotes a finite dimensional complex vector space and Q : V ? V ?? C a symmetric bilinear form on V, Among the repre- sentations of g = so(V ) there exist a representations which appear in ?kV , but not all . For build these which disappear we will use the constructions of Clifford algebras . 1 Clifford algebras Definition 1.1 Let V be a complex vector space with a symmetric bilinear form Q, a Clifford algebra associated to this data is the associative algebra with unit 1 : C = C(Q) = C(V,Q) := T (V )/I where T (V ) denotes the tensor algebra : T (V ) = C ? V ? (V ? V )? . . . And I denotes the tow-sided ideal in T (V ) generated by all elements of the form v?w+w?v?2Q(v, w).1 , for all v, w ? V ,and 1 is the unit element of the tensor algebra T (V ),in other words, C(V,Q) is an associative algebra with unity 1, which contains and is generated by V with v.w + w.v = 2Q(v, w).

- vector space
- clifford algebra
- ?? som
- w? ?
- dimensional complex
- som
- ?•w
- construct tow maps

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Published by | mijec |

Published | 01 March 2006 |

Reads | 27 |

Language | English |

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