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# Lie Groups MAT 261B

UCD

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This 2 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 261B at University of California - Davis taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/187433/mat-261b-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15

1 Representations of Clifford Algebras Recall a Clifford algebra is a unital associative Lie algebra with generators ei and commutation relations eh ej hij where h is a nondegenerate matrix Note that over C there is exactly one Clifford algebra of each dimension 11 up to isomorphism This will be denoted ClnC When one speci es a chosen matrix for h one often writes Clnh Also recall a familiar representation of Cl2n C over the Grassmann algebra An De ne ai 3 and bj ej multiplication from the left One sees that aibj 6g and aiaj 12111211 0 Claim This is the only irreducible representation of Cl2nC and every other nitedimensional rep resentation is a direct sum of copies of this one Proof In this context one refers to ai as annihilation operators and bj as creation operators Consider the following De nition a vacuum vector is a nonzero 39i39 such that for all i ai39i39 0 It is straightforward to show that such a vector always exists and that it is unique up to scaling if and only if the corresponding representation is irreducible Since the vector 1 E An is a unique vacuum vector this representation is irreducible Let E be a nitedimensional representation of ClnC Let S be the set of all vacuum vectors in E S is clearly a linear space Choose a basis L1 Then each fl generates a subrepresentation El span 12111212 J 1 Hence each E is isomorphic to the representation described above and E LEZ This completes the proof The following outlines an extension of this fact to Clifford algebras with odd dimension Let h be a diagonal 2m l by 2m l relation matrix for a Clifford algebra Let h be the 2m by 2m upperleft block of h Then Cl2mh is a subalgebra of Cl2m lh Hence every representation of Cl2mh can be restricted to a representation of Cl2m and every representation of Cl2m h can be extended to a representation of Cl2m l h on the same space Note since h is diagonal the matrix representatives Pk of ek satisfy the following relations 1quot2 hi and ia j a FiFj 4 11 Suppose F i l comprise a representation of Cl2mh Construct F2m1 aF1F2 lam One can show that Fk mfl comprises an irreducible representation of Cl2m lh for exactly two values of a This provides th distinct irreducible representations of Cl 2721 l it One can also show that there are no other irreducible representations and that every other representation is completely reducible This construction can also yield projective representations of the orthogonal groups Let n 2m and hbe a symmetric n by n matrix ofcommutation relations for Pk Under a linear change of variables Ti b l h is replaced by it Note h h if b E On h by de nition of the orthogonal group and if so Pk generate the same algebra Le 12 provides an automorphism If the representation given by Pk is irreducible then that given by fk is also and these representations are equivalent Hence there exists an intertwiner de ned up to a constant factor it obeys fk UarkUgl Where a b li One can show that for compositions of such intertvviners7 Uaa const Ua Ua by de nition this means that we constructed a projective representation of On7 hi

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