1 Algebraic preliminaries

We gather in this section some standard facts about Clifford algebras and their rep-

resentations. For details and proofs we refer to [LM] or [Kal].

§1.1 Clifford algebras Let Q be a quadratic form on V. The Clifford algebra

generated by V and Q, denoted by C(V, Q) is the associative unital algebra generated

by V with the relations

u - v -f v • u = —Q(u, v) - 1 V u, v 6 V

If ei, • • •, en (71 = dim V) is a basis of V in which Q is diagonal then C(V, Q) can be

alternatively characterized as the associative unital algebra generated by ei,---,e

n

modulo the relations

e{ • 6j + e3;• et- = -22(et-,ej) V ij. (1.1)

For any nonnegative integers p, / such that p -f ? 0 we denote by Rp'9 the space

W © M9 endowed with the quadratic form

Q(x®y) = \x\2-\y\2 xeW , yeRq

where | • | denotes the standard euclidian metric. Then Cp,q denotes the Clifford

algebra generated by Rp'9. When p = q = 0 we set C°'° = R.

Let e-i, • • • ep; cj, • • • tq denote the standard basis of W,g. Cp,g decomposes as a

Z2-graded algebra ("superalgebra")

C

r,q & C™®CpJq (1.2)

where C±q are the vector subspaces generated by the even/odd degree monomials in

the basis elements {et; tj).

Cp,q can be naturally equipped with a scalar product making {ej • ej] an or-

thonormal basis. Denote the corresponding norm by || • ||. (Cp'9, || • ||) is a Z2-graded

real C*-algebra with the anti-involution "*" uniquely defined by its action on the

generators:

e* = -ei , e* = t3 V ij.

It will be useful to introduce some "super" notions.

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