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Extra resources for Orthogonal and Symplectic Clifford Algebras: Spinor Structures
Are linear p-forms, and (Ji 1\ (Ji = (Ji 1\ (Ji = AW i8l (Ji) = 4W i8l (Ji - (Ji i8l (Ji) O. Generally, if Xl, X2, . . ,Xp is a strictly increasing sequence of indices, (JXl 1\ (JX2 1\ . . 1\ (JXI' = A( (JXl i8l (JX2 i8l . . i8l (JXI') so that (7) By the reasoning used to compute the dimension of I\P(E'), the C~ exterior products obtained in (7) from all sequences (XI, X2, . . , Xp) form a basis of N(E') if the (JI, (J2, . . ,(In form a basis of E' . In particular, if we set p = n, if cpl, cp2, .
The ei 1 ® ei 2 ® . . i p) . '2 ••. 'p. (tp1 , tp 2 , . . , tpP) tpI. , _ - I 2 tp il tp i 2 if the tpfp are the components of the forms change we will have • • • p tp i p , tpl , .. , tpP. Under a coordinate 23 TENSOR, EXTERIOR AND SYMMETRIC ALGEBRAS. In general, for the sequence of spaces the space of (p + q)-linear forms on E; x . . x E; x E1'+l x . X E1'+ q is denoted by El 18' . . 18' E1' 18' E;+l 18' . . 18' E;+q, or 18'1' E ®q E* if all E, = E . ip 18' (Jip+! i p+q). A mixed tensor in E 18' E* or E* 18' E can be linearly identified with an endomorphism of E : End E = E 18' E* = E* 18' E .
N p . In this first part , we will only recall some results, omitting all proofs, only to specify our definitions , our notations and our methods. We will assume that the reader is already somewhat familiar with tensor algebras. 1 A p-linear mapping of E I x E 2 X . . e. a mapping which is linear in any of its p arguments, when the p - 1 other ones are fixed , is called a p-linear form . epp : (Xl, X2 , . . , xp) E E I X E 2 X .. E p --+ epP(XI' X2 , . , xp) E K. The set of linear p-forms is a vector space on K and will be called the space of covariant tensors of order p.