32 Chapter 2 Remark 2.7.3. By changing B into -B, we get similar identities for D g . In particular, 1 D g,2 = C g + B ( g , g ) . (2.7.18) 4 2.8 The Clifford-Heisenberg algebra of g g Now we use temporarily the conventions of section 1.5 in the case where V = g. In particular we will consider the Clifford algebra c (g g ) and the Heisenberg algebra b (g g ), and we disregard for the moment the bilinear form B. Similar considerations will be applied to p, k. Of course, g g = (p p ) (k k ) , so that c (g g ) = c (p p ) c (k k ) , g g (2.8.1) b (g g ) = b (p p ) b (k k ) . (2.8.2) p p We define the operators d , d c (g g )b (g g ), d , d c (p p ) k k b (p p ) and d , d c (k k ) b (k k ) as in (1.5.1). Of course, d = d + d , g g g p k d g = d p + d . k (2.8.3) Also, the operators d , d are G-invariant. If we consider p, k as Euclidean vector spaces, using (1.6.7), we may view p p k k the operators d , d as acting on · (p ) L 2 (p), and d , d as acting on · (k ) L 2 (k). This is what we will do in the sequel, because it is the most suitable representation in a geometric context. Still, it is very easy to go back to the original representation. If Y g, we split Y in the form Y = Y p + Y k , p k (2.8.4) with Y p, Y k. From now on, the Clifford operators will be associated with (g, B). First, we consider the Euclidean vector space p, and the associated Clifford algebras c (p) , c (p). Recall that the operators D p , E p associated to p were defined in (1.6.1), (1.6.9). We will not distinguish these objects from their action on · (p ) S · (p ), which was described in (1.6.7)­(1.6.9). This will also be done in the sequel for other operators. Let e 1 , . . . , e m be an orthonormal basis of p. By (1.6.9), we get m D p = i=1 c (e i ) e i , E p = c (Y p ) . (2.8.5) By (1.6.2), d + d p p 1 = (D p + E p ) . 2 (2.8.6)