By Marcel Riesz (auth.), E. Folke Bolinder, Pertti Lounesto (eds.)

Marcellliesz's lectures introduced on October 1957 -January 1958 on the Uni versity of Maryland, collage Park, were formerly released purely infor mally as a manuscript entitled CLIFFORD NUMBERS AND SPINORS (Chap ters I - IV). because the identify says, the lecture notes encompass 4 Chapters I, II, III and IV. even if, within the preface of the lecture notes lliesz refers to Chapters V and VI which he couldn't end. bankruptcy VI is pointed out on pages 1, three, sixteen, 38 and 156, which makes it believable that lliesz was once good conscious of what he used to be going to incorporate within the ultimate lacking chapters. the current e-book makes lliesz's vintage lecture notes in most cases on hand to a much broader viewers and attempts a bit to fill in a single of the final lacking chapters. This booklet additionally attempts to guage lliesz's impression at the current study on Clifford algebras and attracts distinctive awareness to lliesz's contributions during this box - usually misunderstood.

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The tensor algebra T{E) over E is the infinite dimensional algebra whose basis is formed by all products where 0 , h < 00 and (i l , i 2 , ••• , i h ) is any set of indices 1,2, ••• , n with possible repetitions. 1) with real or complex coeffiCients, respectively. 1) is formed by juxtaposition and is extended by distributivity to any pair of elements of T{E). This multiplication is associative - 28 - but not commutative. 1) which contain exactly h factors. T(E) is the direct sum of these Th(E) (O~ h< 00 ) Then and Th(E) ® Tk (E) C Th + keEl, Le.

7) and in - 53- (-1) "xl-Xl; ... 8) shows • that it occurs with opposite signs. 5). 5) we conclude the following fact wiich is of utmost importance for the sequel. The operators [Pe' < i ••• i P' ] (p=O,l, ••• ,nj ex 1 e""p 0< 2 <:: ••• "" 0< P ~ n) corresponding to a basis e l , e 2 , ••• , en of E are linearly independent. 5), imply the linear dependence of the p-vectors [eO( i ••• i eO( J of 1 P the tensor algebra T. However, the independence of these pvectors was proved in an entirely trivial way ir.

1 2 < ... ,8 j ) == 1 r l' We observe that these expressions vanish when two J. f3 are equal. 3) and vice versa, which proves that S p is independent of the choice of the basis. lletric tensor of order 11 or a P I' -~. is called a skewThe simplest elements, namely the COllDl1utators of p vectors, are called P -vectors. 5) [ali a 2 ;···; apJ = -1;p. [ aI' a 2 ,···, a pJ and call the left hand side the simPle p -vector Spanned by the vectors aI' a 2 , ••• ,a p' A simple p-vector is zero if and only if the vectors which span it are linearly dependent.