Clifford algebras and spin representations by Wesam TALAB Monday mars

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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).

  • vector space

  • clifford algebra

  • ?? som

  • w? ?

  • dimensional complex

  • som

  • ?•w

  • construct tow maps


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Published 01 March 2006
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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 andQ:V×V−→Ca symmetric bilinear form onV, Among the repre-k sentations ofg=so(V) there exist a representations which appear inV , but not all .For build these which disappear we will use the constructions of Clifford algebras .
1 Cliffordalgebras Definition 1.1 LetVbe a complex vector space with a symmetric bilinear formQ, a Clifford algebra associated to this data is the associative algebra with unit 1 : C=C(Q) =C(V, Q) :=T(V)/I whereT(V)denotes the tensor algebra : T(V) =CV(VV). . . AndIdenotes the tow-sided ideal inT(V)generated by all elements of the formvw+wv2Q(v, w).1, for allv, wV,and 1 is the unit element of the tensor algebraT(V),in other words,C(V, Q)is an associative algebra with unity 1, which contains and is generated byVwithv.w+w.v= 2Q(v, w).1, for allv, wV, et equivalently we havev.v=Q(v, v).1for allvV. Proposition 1.2universal property If A is any associative algebra with unit 1,and a linear mappingj:V−→A is given such thatj(v).j(v) =Q(v;v).1for allvVor equivalently j(v).j(w) +j(w).j(v) = 2Q(v, w).1 . Forallv, wV, then there should be a unique homomorphism of algebras fromC(V, Q)toAextendingj. Example 1.3 LetVbe a one dimensional complex Vector space with basis(e)then(1, e) form a basis for the complex vector spaceC(V, Q)because the elemente.e..e(n once) will be inCifnis even and inVifnis odd, care.e=Q(e.e).1C L and henceC(Q) =C V, and finallydimC(Q) = 2. 1