Marcel Riesz's Work on Clifford Algebras

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P. Lounesto, Institute of Mathematics, Helsinki University of Technology, Finland...


MARCEL RIESZ'S WORK ON CLIFFORD ALGEBRAS P. LOUNESTO In,titute 0/ Mathematic, Hel,inki Unil1er,ity 0/ Technology SF-O!150 ESPOO, Finland Abstract. This article reviews Marcel Riesz's lecture notes on Clifford Number, and Spinor" 1958, and evaluates its effect on present research on Clifford algebras. The article begins with a critical survey of the history of Clifford algebras. Inaccuracies in citations are pointed out and mistaken priorities are rectified. Misconceptions about Clifford algebras are examined and flaws are corrected. The paper focuses on controversial issues that have been debated over the past few years, and offers definitive resolutions. Concrete counter-examples are given to misplaced claims. Particular attention is directed to the existence of a canonical linear isomorphism between the exterior algebra and the Clifford algebra. ChevalIey constructed the Clifford algebra Cl(Q) of a quadratic form Q as a subalgebra of the endomorphism algebra End(A V) of the exterior algebra A V. This construction depends on an arbitrary, not necessarily symmetric, bilinear form B such that B(x,x) = Q(x). The choice of B fixes the contraction u J 'II in A V and permits the introduction of a Clifford product xu = x J u + x "u of x E V and u E A v. This gives rise to the Clifford algebra of the symmetric bilinear form HB(x,y) + B(y, x» when the characteristic =1= 2. Marcel Riesz went backwards with his fundamental formulas x J u=

-'li.x) and x" u =

+ 'li.x)

and re-introduced the contraction and the exterior product in the Clifford algebra, and constructed, in alI characteristics =1= 2, a privileged linear isomorphism from the Clifford algebra to the exterior algebra (without antisymmetric products of vectors). [In the above 'Ii. means the grade involute of u, that is, for a homogeneous element a EA· V we have i. = (-l)·a.] Key words: Exterior algebra - contraction - Kiihler-Atiyah isomorphisms

A History of Clifford Algebras 0.1. ALGEBRAS OF HAMILTON, GRASSMANN AND CLIFFORD The first step towards a Clifford algebra was taken by Hamilton in 1843 (first published 1844), when he studied products of sums of squares and invented his quaternions for multiplicative compositions of triplets in 1R3 . The present formulation of vector algebra was extracted from the quaternion product of triplets/vectors xy = -x· y + x X y by Gibbs 1881-84 (first published 1901). Hamilton regarded quaternions as quotients of vectors and 215



wrote a rotation in the form y = ax using a unit quaternion a E 1lI, lal = 1. However, in such a rotation the axis a = a - Re( a) had to be perpendicular to the vector x (which turned in the plane orthogonal to a). The formula for rotations x --+ axa- l around an arbitrary axis a E 1R3 was presented by Cayley in 1845. [Cayley 1855 also discovered the quaternionic representation q --+ aqb of rotations in 1R4 equivalent to the decomposition Spin( 4) :::' Spin( 3) X Spin( 3).) Cayley thus came into contact with half-angles and spin representation of rotations in 1R3. However, already Olinde Rodrigues 1840 had recognized the importance of half-angles in his study on composition of rotations in 1R3. The quaternion algebra 1lI is both the even sub algebra and a proper ideal of the Clifford algebra CiO,3 on 1R3. Hamilton also considered complex quaternions d:: ® 1lI isomorphic to another Clifford algebra Ci3 = Ci 3 ,o on 1R3. Note the algebra isomorphisms CiO,3 :::' 1lI a H and Ci3 :::' M 2 (d::). Bivectors were introduced by H. Grassmann, when he created his exterior algebras in 1844. The exterior product of two vectors, the bivector a 1\ b, was interpreted geometrically in n dimensions as the parallelogram with a and b as edges, and two such exterior products were equal if their parallelograms lay in parallel planes and had the same area with the same sense of rotation (from a to b). Thus the exterior product of two vectors was anticomnmtative a 1\ b = - b 1\ a. Using a basis el, e2, ... , en for 1Rn the exterior algebra ,,1Rn had a basis 1 el, e2, ... , en el 1\ e2, el 1\ e3, ... , el " em e2 1\ e3, ... , en-l 1\ en

and was thereby of dimension 2n. W. K. Clifford studied compounds/products of two quaternion algebras, where quaternions of one algebra commuted with the quaternions of the other algebra, and applied exterior algebra to grade these tensor products of quaternion algebras and thus coined his geometric algebra in 1876 (first published in 1878). Clifford's geometric algebra was generated by n orthogonal unit vectors ell e2, ... , en which anti commuted eiej = -ejei (like Grassmann) and satisfied all ef = -1 (like Hamilton) [or then all el = 1 as in Clifford's paper 1882 - the positive definite case is missing in Budinich&Trautman 1988 p.2 r.20). The number of independent products Clifford summed esej = ei 1\ ej, i < j, of degree 2 was In(n - 1) = up the numbers of independent products of various degrees 0, 1, 2, ... , n



and thereby determined the dimension of his geometric algebra to be

He distinguished four classes of these geometric algebras characterized by the signs of (ele2 ... en)ei = ±ei(ele2 ... en) and (ele2 ... en)2 = ±1. He also introduced two algebras of lower dimension 2n - l : the subalgebra of even elements and, for odd n, a reduced algebra obtained by putting ele2·.· en = ±1 (instead ofletting ele2 ... en = el/\ e2 /\ ... /\ en to be of degree n with the requirement (ele2 .. . e n )2 = +1). 0.2.




Clifford's geometric algebra was reinvented in 1880, just two years after its first publication, by Rudolf Lipschitz, who later acknowledged Clifford's prior discovery in his book, see R. Lipschitz: Untersuchungen tiber die Summen von Quadraten, 1886 [Budinich&Trautman 1988 p. 2 r. 18 refer to 1886 but seem to be unaware of 1880]. In his study on sums of squares Lipschitz considered a representation of rotations by complex numbers and quaternions and generalized this to higher dimensional rotations in 1Rn. Lipschitz thus gave the first geometrical application of Clifford algebras in 1880 [Budinich&Trautman 1988 p. 54 r.10 again refer only to 1886]. He expressed a rotation y = (1 +A)(1 -A)-IX (written here in modern notation with an antisymmetric matrix A) in the form y - Ay = x + Ax or as y +y. B = x+ B·x where BEN 1Rn is the bivector determined by Ax = B·x (= -x· B). He rewrote this expression with an even Clifford number a E Clci,n in the form ya = ax, thus representing the rotation as y = axa- l . (Lipschitz wrote yal = az where z = xe l \ y = yell, al = elael l .) In modern terms, he used the exterior exponential a = 1 + B + B /\ B + . .. of the bivector B so that the normalized element a/lal was in the spin group Spin(n). [Delanghe&Sommen&Soucek 1992 p.127 follow here Helmstetter 1986 and attribute these results incorrectly to Sato&Miwa&Jimbo 1978.) Vahlen 1897 found an explicit expression for the multiplication rule of two basis elements in Clo,n e "'le"'2 I 2


e"'''ef31ef32 n I 2


B.. - (_1)2:i>j"'i{3je"'1+f31 e "'2+f32 e;;'. I 2

• ••

e",,,+{3,, n

where the exponents are 0 or 1 (added here modulo 2, though for Vahlen 1+ 1 = 2 so that the summation was over i > j) - a formula reinvented frequentlyafterwards [Brauer&WeyI1935, Deheuvels 1981 p. 294, or disguised with index sets as in Brackx&Delanghe&Sommen 1982 p. 2, Carmichael 1984 p. 228, Chevalley 1946 p. 62, Artin 1957 p. 186, or then hidden among permutations like in Delanghe&Sommen&Soucek 1992 pp. 58-59 who gratuitously



attribute this result to Kiihler 1960/62]. In 1902 Vahlen initiated the study of Mobius transformations of vectors in 1Rn (or paravectors in 1R +1Rn) by 2 X 2-matrices with entries in Clo,n. This important study was re-initiated by Ahlfors in the 1980's. E. Cartan 1908 p. 464 identified the Clifford algebras Clp,q as matrix algebras with entries in 1R, = = = = and so (u /\ v) J w = u J (v J w). • In case of a non-degenerate Q we have verified the following properties of the contraction




x J y = for x, y E V



( c)

( U A v) J w = u J (v J w)

for u, v, w E AV

see Hehnstetter 1982. These properties uniquely detennine the contraction also for an arbitrary, not necessarily non-degenerate Q. The identity (c) introduces a scalar multiplication on AV making it a left module over AV. The identity (b) means that x E V operates like a derivation. Evidently, x J a E AIe-I V for a E AIe V. Introduce the (Clifford) product of x E V and u E AV by XU= xAu+x J u and extend this product by linearity and associativity to all of AV making it isomorphic to Cl( Q). For instance, the product of a simple bivector x AyE A2 V and an arbitrary element u E AV is given by (x A y)u = x A Y A u + x A (y J u) - Y A (x J u) + x J (y J u) where we have first expanded (x 1\ y)u = (xy - x J y)u then used x(yu) = x A Y A u + x 1\ (y J u) + x J (y AU) + x J (y J u) and the derivation rule x J (y A u) = (x J y) Au - Y A (x J u). Exercises. Show that 1. XA(yJu)-YA(xJu)=xJ(YAu)-yJ(xAu) for x,yEV 2. x A Y A (z J u) - x A z A (y J u) + y A z A (x J u) = x J (y A z A u) - y J (x A z A u) + z J (x A Y AU) 3. x A (y J (z J u)) - y A (x J (z J u)) + z A (x J (y J u)) = (x A y) J (z Au) - (x A z) J (y A u) + (y A z) J (x AU) = x J (y J (z 1\ u)) - x J (z J (y 1\ u)) + y J (z J (x A u)) 4. (x A Y A z)u = x A Y 1\ (z J u) - x A z A (y J u) + Y 1\ z 1\ (x J u) +x A (y J (z J u)) - y A (x J (z J u)) + z A (x J (y J u)) +x A y A z Au + x J (y J (z J u)) 5. a J b E Ai - i V for a E Ai V, b E Ai V In the last exercises we have a non-degenerate Q. Define u L v by = (we say that v contracts u and also u is contracted by the contractor v). 6. u J v = Va L Ua - Va LUI + VI L Ua + VI LUI = V L u - 2va LUI (Va = even(v), UI = odd(u))

224 7.


Lv =


J tto

+ Vo J

ttl - VI

(VI = odd(v), Uo = even(u))


8. a J b = (_l),U-i)b L a for a E



9. a E ,,10 V, x E V, x J a


10. b E /\,. V is simple

(a J b) A b


+ 111 J




J u - 2V1 J

b E I\ i V

x = a J b for some b E



,,"'+1 V

for all a E "Ic-l V.

Remark. The dot-notation a· b may be used for the contraction, when it is clear from the context which one of the factors is contracted and which is the contractor. This dot product a' b can be used, when at least ODe of the factors is homogeneous. If both the factors are homogeneous, then we agree that the one with lower (or not higher) degree is the contractor .:


a·b = a J b for a el\v,

be/\V, i:5:i



a·b = aLb for a e /\v, b E !\V, and when precisely one fact or i s known to be homogeneous we agree that it is the contractor

a·u = aJu for a e /\v, u E I\V ;

u-a=uLa for a E I\V, U EI\V_



Note that the contraction obeys the rules 1 J u tI, tt E 1\ V, and x J 1 0, x E V, whereas the dot product obeys the rule 1 - tI = U - 1 = u . Note also that the dot product does not act like a scalar multiplication on the left 1\ V-module 1\ V, that is, (a li b)· u i:- a· (b· ttl. As an exercise the reader may verify that for a = el, b = el + ez and u = e l + el II e2 in 1\ IR.2 all the expressions

a J (b Ju), a J (b L u), a L(b J u), a L(b Lu) and (aA b) Ju, (a A b) Lu are unequal with the exception of (a li b ) J u= a J (b J u) = ? - The lack of linearity over /\ V renders less useful any extension of u· v for arbitrary u, v E /\ V. Such an extension was actually introduced under the name of 'hmer product' by Hestenes&Sobczyk 1984 p.6 who also demonstrated its futility by displAying only formulas with at least one homogeneous factor. The non-/\ V -linear 'iruler product ' is not consistent with the contraction (in the sense that the 'inner product' is not a special case of the contraction), because tt· v might differ simultaneously both from u J v and u Lv. Boudet 1992 p. 345 mentioned a formalization of the



'inner product' but his rules are not sufficient to permit the evaluation of (X 1\ y) . u when U E /\ V, U rf. V (though they do permit a construction of the 'inner product' with an additional rule (x 1\ y) . U = X· (y . u) where u E /\k V, k ;::: 2). • The above remark shows how the asymmetric contration solves a problem of Hestenes&Sobczyk 1984, who postulate the 'inner product' to be 0 [p.6, r. 12 formula (1.21b)) if one of the factors is a scalar, and run into difficulties on p. 20 rows 8-18 formula (2.9). However, as the following example shows the problem is deeper than that since the 'inner product' is not equal to the contraction even though scalars would be excluded. Example. Let el, e2 be an orthonormal basis for R 2. Compute

in the sense of Hestenes&Sobczyk. The same elements have the contractions (el - el 1\ e2) J (e2 (el - el 1\ e2) L (e2

+ el 1\ e2) = 1 + e2

+ el 1\ e2) =

1 - el.

This shows that neither of the contractions coincide with the 'inner product' of Hestenes&Sobczyk. •

Remark. We may re-obtain the dot product in terms of the Clifford product as follows (char i- 2) a· b

= (ab)li-il



for a E /\ V, bE /\ V

where (U)k is the k-vector part of U E /\ V ::: Cl( Q). For two homogeneous elements a E /\i V, b E /\i V the Clifford product decomposes into k-vector parts ab = (ab)i+i + (ab)i+i-2 + ... + (ab)li-il where al\b = (ab)i+i and a·b = (ab)li-il' In particular, ab =I al\b+aJb, when i =11 even for i < j, as can be seen by the counter-example a

2 3 3 ab = e134 E /\R 4

= e12 E /\R\ b = e234 E /\R\

[this blunder lurks in Salamon, 1989, p. 170). 1.2.


Cl(Q) C End(/\ V)

Chevalley, 1954, pp. 38-42, introduced a linear operator such that A V. The composite linear map () = ¢ 0 1/J was the right inverse of the natural map A V -> Cl(Q) and

/\ V



t End(/\ V) t /\ V

was the identity mapping on A V. The faithful representation 1/J sent Cl( Q) onto an isomorphic subalgebra of End(A V). Chevalley's identification works fine also with a contraction defined by an arbitrary - not necessarily symmetric - bilinear form B such that B(x, x) = Q(x) and

= B (x, y)

for x, y E V

( a)

x J y




(u /\ v) J w

=u J

(v J w) for u, v, w E A V

see Helmstetter 1982. As before x J a E Ak- l V for a E Ak V and x J (Xl /\ x2 /\ ... /\ Xk) k

= Z)-1)i- 1 B(x,Xi)Xl/\ X2/\ ... /\Xi-l/\Xi+1/\ ... /\ xk

i=l and the faithful representation 1/J sends the Clifford algebra Cl( Q) onto an isomorphic subalgebra of End(A V), which, however, as a subspace depends

onB. Remark. Chevalley introduced his identification Cl(Q) C End(A V) in order to be able to include the exceptional case of characteristic 2. In characteristic f= 2 the theory of quadratic forms is the same as the theory of symmetric bilinear forms and Chevalley's identification gives the Clifford algebra of the symmetric bilinear form = t(B(x,y) + B(y,x)) satisfying xy + yx = 2. I For arbitrary Q but char K f= 2 there is the natural choice of the unique symmetric bilinear form B such that B(x,x) = Q(x) giving rise to the canonical/privileged linear isomorphism Cl( Q) -7 A V. The case char K = 2 is quite different. In general there are no symmetric bilinear forms such that B( x, x) = Q (x) and in the case that there is such a symmetric bilinear form, then it is not unique, since any alternating bilinear form (is also symmetric and) could be added to the symmetric bilinear form without changing



Q. [Recall that antisymmetric means B(x,y) = -B(y,x) and alternating B(x, x) = OJ alternating is always antisymmetric, though in characteristic 2 antisymmetric is not necessarily alternating.] Thereby the contraction is not unique, and there is an ambiguity in I{)x. In characteristic 2 the theory of quadratic forms is not the same as the theory of symmetric bilinear forms. Example. Let dimV = 2, B(x,y) = aZlYl + bz l Y2 + CZ2Yl + dZ 2Y2 and Q(x) = B(x,x). The contraction xJy = B(x,y) gives the Clifford product xv = x A v + x J v of x E V, v E "V. We will determine the matrix of v - uv, U = Uo + Ul el + U2e2 + U12el A e2 with respect to the basis 1, ell e2, el A e2 for The matrix of el is obtained by the following computations


= 0100) = aOOO)

(first column

= el J el = a (second column ele2 = el A e2 + el J e2 = el A e2 + b (third column = bOOl) el (el A e2) = el J (el A e2) = (el J ed A e2 - (el J e2) A el = ae2 el el


and the matrix of el A e2 by (el A e2) 1 = el A e2

= (el A e2) L el = el A (e2 L ed - e2 A (el L el) = eel (el A e2)e2 = (ele2 - el J e2)e2 = - (el J e2)e2 = del - be2 (el A e2)(el A e2) = (ele2 - el J e2)(el A e2) = el(e2 A (el A e2) + e2 J (el A e2» - (el J e2)(el A e2) = el(ee2 - ded - b(el A e2) = -ad + be + (-b + c){el A e2)' (el A e2 )el

So we have the following matrix representations


b 0 0 0 0 0 1

e 0



a 0

0 el A ea

e -a


0 or in general U

C Ul U2 U12


0 d -b 0

aUl + eU2 Uo + eU12 - au 12 -U2


0 0 -1

d 0 0 0

-aYbO) -b+ e bUl + dU2 dU12 Uo - bU12 Ul

-(ad - bolu" -(bUl + dual Uo

aUl + cU2 + (-b + e)ul2





Evidently, the conunutationrelations ele2+e2el = b+e and ei are satisfied, and we have the following multiplication table



-bel + ae2 -del + ee2 -ad + be + (-b + e)el 1\ e2


-el 1\ e2 + e eel - ae2

In characteristic

= a,

2 we find 1 1 '2 (el e2 - e2 e d = el 1\ e2 + '2 (b - e)

and more generally for x = xlel + X2e2, y = Ylel + Y2e2 1

'2 (xy -

yx) = (XlY2 - x2yd el 1\ e2 +


'2 (b -

e)(xlY2 - x2yd

= x 1\ Y + A( x, y)

with an alternating scalar valued form A(x,y) = t(B(x,y) - B(y,x)). For non-zero A(x,y) the quotient x 1\ y/A(x,y) is independent of X,y E V. Note that the matrix of Hxy - yx) is traceless. The synunetric bilinear form associated with Q(x) is X· Y =


'2 (B(x,y) + B(y,x)) =

aXlYl +


'2 (b + e)(xlY2 + x2yd + dX2Y2

and we have xy + yx = 2x . y for X,y EVe Cf(Q). Orthogonal vectors x..l y anticonunute xy = -yx and (xy)2 = _x 2y2 even though xy = x 1\ Y + A(x,y). (In this special case A(x,y) = B(x,y) :f 0 while x·y = 0 implies B(y,x) = -B(x,y).) • It is convenient to regard A V as the subalgebra of End(A V) with the canonical choice of the synunetric B = o. We may also regard Cf(Q) as a sub algebra of End(A V) obtained with some B such that B(x, x) = Q(x) and choose the synunetric B in char :f 2. The following example shows that for Q = 0 and B = 0, Chevalley's process results in the original multiplication of the exterior algebra A V but that for Q = 0 and alternating B :f 0, the process gives an isomorphic but different exterior multiplication on A V.

Example. Take a special case of the previous example, the Clifford algebra with Q 0 and B(x,y) b(XlY2 - 2:2Yl). Send the matrix of the exterior product (with the synunetric bilinear form = 0) to a matrix of the isomorphic Clifford product (determined by the alternating bilinear form = B)








In this case J3(x)J3(y) = J3(x A Y + B(x, y)).

In particular, J3(el)J3(e2) = J3(elAe2+b), J3(e2)J3(ed = -J3(elAe2+b) and ,8( el Ae2 + b),8( e;) = 0, ,8( ei),8( el Ae2 + b) = O. We will meet this situation again in the section on the 'Uniqueness and the definition by generators and relations' except that here the exterior algebra and the Clifford algebra (determined by the alternating B) are regarded as different subspaces but isomorphic subalgebras of End(A V). • The above example shows that those who refute the existence of k-vectors in a Clifford algebra Cf(Q) over K, char K f- 2, should also exclude fixed subspaces A k V c A


In general, consider two copies of Cf(Q) in End(A V) so that Q(x) = B1(x, x) = B2(X, x) determines J31(X),81(Y) = ,81(X A y + B1(x, y)) and ,82(X),82(Y) = ,82(X A y + B 2 (x,y)). A transition between the two copies is given by an alternating bilinear form B(x,y) = B1(x,y) - B 2 (x,y) and ,8(x),8(y) = ,8(xy + B(x,y)). In char K f- 2 this means that the symmetric bilinear form such that = Q(x) gives rise to the natural choice xy = x A Y + among the Clifford products xy + B( x, y) with an alternating B. In other words, the Clifford product xy has a distinguised decomposition into a sum xAy+ ofascalar = !(xy+yx) andabivector xAy= !(xy - yx) among the possible decompositions with antisymmetric part XAY = !(xy-yx) equaling a new kind ofbivector xAy = xAy+ B(x, y). [Similarly, a completely antisymmetric product of three vectors equals a new kind of 3-vector x A y A z = x 1\ Y 1\ z + xB(y,z) + yB(z,x) + zB(x,y).) Example. Let K = {O,1}, dim V = 2 and Q(xlel + X2e2) = XIX2. There are only two bilinear forms Bi such that Bi(X, x) = Q(x), namely B1(x,y) = XIY2 and B 2 (x,y) = X2Yt, and neither is symmetric. The difference A = B 1 -B 2 , A(x,y) = XIY2-X2Yl (= XIY2+X2yd is alternating (and thereby symmetric). There are two representations of Cf (Q) in End (A V), namely for B 1 :












Uo -


for B 2 :



Ul U2 U12


+ Ul2 0 -U2


-"1 )




0 0

o0 )

Uo Ul


U2 Uo



These representations have the following multiplication tables



Bl el e2 el A e2

B2 el e2 el A e2

el A e2 el e2 0 -el 1 + el A e2 0 0 -el A e2 0 -e2 -el A e2 e2 el A e2 el 0 0 el A e2 0 1- el A e2 e2 0 el el A e2

with respect to the basis 1, el, e2, el Ae2 for " V. In this case there are only two linear isomorphisms "V -+ Cl(Q) which are identity mappings when restricted to K + V and which preserve parity (send even elements to even elements and odd to odd). It is easy to verify that the above multiplication tables of Cl( Q) are actually the only representations in In this case there are no canonical linear isomorphisms -+ Cl(Q), in other words, neither of the above multiplication tables can be preferred over the other. In particular, N V cannot be canonically embedded in Cl(Q), and there are no bivectors in characteristic 2! •

"V "V.

The above example shows that Lawson&Michelsohn pp. 10-11 are wrong when they allow characteristic 2 and claim that there is a canonical linear isomorphism" V -+ Cl(Q). Chevalley used unnecessarily complicated and abstract methods, since he wanted to include the case char K = 2 and employed endomorphisms of to get a faithful representation of Cl(Q). The need for a simplification of Chevalley's presentation is obvious. For instance, B. L. van der Waerden: On Clifford algebras, 1966, said that 'the ideas underlying Chevalley's proof (p. 40) are not easy to discern' and gave another proof, equivalent but easier to follow. (Also Crumeyrolle 1990 p. xi claims that 'Chevalley's book proved too abstract for most physicists' and in a Bull. AMS review Lam 1989 p. 122 admits that 'Chevalley's book on spinors is ... not the easiest book to read. ') It might be helpful to get acquainted with a simpler and more direct method of relating " V and Cl( Q) due to Marcel lliesz: Clifford Numbers and Spinors, 1958, pp. 61-67. lliesz introduced a second product in Cl( Q) making it isomorphic with" V without resorting to the usual completely antisymmetric Clifford product of vectors and constructed a privileged linear isomorphism Cl( Q) -+ " V.





Start from the Clifford algebra Cl( Q) over K, char K -::J. 2. The isometry x -+ -x of V is extended to an automorphism of Cl( Q) called the grade involution u -+ U. Define the exterior product of x E V and u E Cl( Q)



by (xu + ux),

x Au =

u Ax =

(ux + xu)

and extend this exterior product by linearity to all of Cl( Q) which then becomes isomorphic to AV. The exterior products of two vectors x A y = (xy - yx) are simple bivectors and they span A2 V. The exterior product of a vector and a bivector x A B = (xB +Bx) is a 3-vector in A3 V. The subspace of k-vectors is constructed recursively by



We may deduce associativity of the exterior product as follows. First, the definition implies for x, y, z E V 1 x A (y A z) = 4" (xyz - xzy + yzx - zyx)

1 (x A y) A z = 4" (xyz - yxz

+ zxy -


Then xA(yAz)=(xAy)Az since xyz - zyx

= xyz -

zyx + (zy

+ yz)x -

x(yz + zy)

= yzx -


This last result implies 1 x A Y A z = "6 (xyz + yzx + zxy - zyx - xzy - yxz)

when char K -=f 2, 3 (note the resemblance with anti symmetric tensors). [Similarly, we may conclude that xyz + zyx = x(yz + zy) - (xz + zx)y + z(xy + yx) is a vector in V.] lliesz's construction shows that bivectors exist in all characteristics -=f 2. Introduce the contraction of u E Cl( Q) by x E V so that xJ u


(xu - ux)

and show that this contraction is a derivation of Cl( Q) while x J (uv)


(xuv - Uvx)

= !2 (xuv -


(xuv - uvx)

uxv + uxv - uvx)

= (x J u)v + u(x J v).



Thus one and the same contraction is indeed a derivation for both the exterior product and the Clifford product. (Kahler 1962 p. 435 (4.4) and p. 456 (10.3) was aware of the equations x J (u 1\ v) = (x J u) 1\ v + it 1\ (x J v) and x J (uv) = (x J u)v +it( x J v).) Provided with the scalar multiplication ( u 1\ v) J w = u J (v J w) the exterior algebra A V and the Clifford algebra Cl(Q) are linearly isomorphic as left A V-modules. Exercises. Show that (char K



1. x and x 1\ y anti commute for vectors x, y E V 2. x and x J B anticommute for a bivector B E A 2 V 3. (x 1\ y)2 = (x J y)2 - x 2y2 (Lagrange's identity) 4. (x 1\ Y 1\ z) J u = (x 1\ y) J (z J u) = x J (y J (z J u»

5. (xyz - zyx)2 E K, x 1\ Y 1\ z

= Hxyz -


= (_l)iib 1\ a for a E Ai V and bE Ai V u 1\ v = Vo 1\ Uo + Vo 1\ UI + VI 1\ Uo - VI 1\ UI = v 1\ u -

6. a 1\ b 7.

u,v E A V,


= even(u),


= even(v),


= odd(u),



1\ UI where

= odd(v)

8. Bu = B 1\ u + HBu - uB) + B J u for BEN V [Hint: (x 1\ y) 1\ u = (x 1\ y) J u = x 1\ (y 1\ u) + x J (y J u)] 9. Q(u) = (uu)o, =

(u J v)o (=

the scalar part of

In the last exercise we have a non-degenerate Q: 10. Q on V extends to a neutral or anisotropic Q on Ct.

u J v)

Riesz's construction shows that Crumeyrolle 1990 pp. 42-45, p. 292 is mistaken when he excludes characteristic 2 and claims that there is no canonical linear isomorphism A V -+ Cl(Q). Crumeyrolle seems to take it both ways when he agrees that in char f:. 2 there is a natural choice for Cl(Q) c End(A V) given by the unique symmetric B such that B(x,x) = Q(x). Crumeyrolle and Lawson&Michelsohn have opposite opinions about the existence of a canonical linear isomorphism A V -+ Cl(Q) but both are wrong: Lawson&Michelsohn because they allow char 2 and Crumeyrolle because he excludes char 2. Curiously, Crumeyrolle excludes char 2 even though he uses Chevalley's method developed specifically for char 2. Neither Crumeyrolle nor Lawson&Michelsohn have understood that Chevalley presented his regular-type representations of Cl(Q) in End(A V) in order to be able to handle also the exceptional case of char 2. [This motive of Chevalley has also escaped Harvey 1990 p. 180 and Budinich&Trautman 1988 p. 58.] (Roughly speaking, Lawson&Michelsohn claim that two numbers x, y in a field K always have a mean !(x + y) within that field K, but this is false in the field of two elements K = {O, I} and Crumeyrolle claims that no



such mean exist in K, but he is wrong while he excludes char 2.)



The following definition is favoured by physicists:

Definition. An associative algebra over K with unit 1 is the Clifford algebra Cl of a non-degenerate Q on V if it contains V and K = K . 1 as distinct subspaces so that

= Q(x)

for any x E V




V generates Cl as an algebra over K


Cl is not generated by any proper subspace of V.

The universal property is guaranteed by including the third property (3). Using an orthonormal basis ell e2, ... , en of R n , generating Clp,q, the condition (1) can be expressed as

while the condition (3) becomes Porteous' 1969 requirement ele2'" en f:. ±1 which is needed only in the signatures p - q = 1 mod 4. The relations (La) also generate a lower-dimensional non-universal algebra in any signature p - q = 1 mod 4 in which all the basis elements ei commute with el e2 ... en and (el e2 ... e n )2 = 1. However, no similar counter-examples exist in even dimensions. Therefore, whereas it is correct to introduce the Clifford algebra of the Minkowski space-time without the condition (3), in arbitrary dimensions it is controversial to omit condition (3). The above definition gives a unique algebra only for non-degenerate (nonsingular) quadratic forms Q. In particular, it is not good for a degenerate Q as is shown by the following two counter-examples in case Q = O. 1. Define for x, y E V the product xy K Efl V an algebra of dimension n + 1.

= O.

This makes the direct sum

2. Introduce a product in AR 3 by eiej = ei A ej for all i,j = 1,2,3 and el e2e3 = O. Thus the subspace R R 3 A2 R 3 of A R 3 is a 7-dimensional algebra generated by Rand R3.

+ +

This shows that it is not possible to replace the condition (3) by the requirement that only parallel vectors commute. We could include arbitrary quatratic forms Q by requiring instead of condition (3) that the product of any set of linearly free vectors in V should not belong to K. However, even this would leave some 'ambiguity' in the definition by generators and relations. The above definition results in a unique algebra only 'up to isomorphism'. Here are two more examples to clarify the meaning of this statement:



3. The multiplication table of the exterior algebra basis 1, el, e2, el /\ e2 is /\

o o o

Introduce a second product on

A el e2 el /\ e2

Arn. 2 with respect to the

Arn. 2 with multiplication table



e2 el /\ e2 + b

-el /\ e2 - b -bel



el /\ e2 -bel -be2 _b 2 - 2bel /\ e2

where b > o. Denote the second product by u Av. Rearrange the multiplication table of the second product into the form

A el /\ e2 + b

o o

0 0


which shows that we have generated a new exterior algebra Arn. 2 on rn. 2 , different from Arn. 2 but isomorphic with Arn. 2 • In other words, we have introduced a linear mapping a: Arn. 2 -+ Arn. 2 for which a(ed = ei, i = 1,2 and a( el /\ e2) = el Ae2 = el /\ e2 + b so that it is the identity on rn. 2 and gives an isomorphism between the two products a( u /\ v) = a( u) Aa( v). 4. An orthonormal basis el, e2 for rn. 2 satisfying eiej+ejei = 2liij generates the Clifford algebra Cl 2 = Cl 2 ,o with basis 1, el, e2, e12 = el e2 (= el/\ e2). We have the following multiplication table for Cl 2 el e2 e12

el e2 e12 1 e12 e2 -e12 1 -el -e2 el -1

Introduce a second product on Cl 2 with multiplication table el e2 e12

el e2 el2 1 e2 - bel e12 + b -e12 - b 1 -el - be2 -e2 - bel el - be2 -1 - b2 - 2be12

The anticommutation relations eiej + ejei = 2liij are satisfied also by the new product, and one may directly verify the associativity. As the real number b varies we have a family of different but isomorphic Clifford algebras on rn. 2 •





x 2 = Q(X)

The Clifford algebra Cl is the universal associative algebra over K generated by V with the relations x 2 = Q(x), x E V. Let Q be the quadratic form on a vector space V over a field K, and let A be an associative algebra over K with unit lA. A linear mapping V -+ A, x -+ If'x such that (If'X)2

= Q(x) ·IA

for all x E V

is called a Clifford map. The sub algebra of A generated by K = K . lA and V (or more precisely by the images of K and V in A) is called an algebra of quadratic form Q. The Clifford algebra Cl is an algebra of the quadratic form Q with a Clifford map V -+ Cl, x -+ IX such that for any Clifford map cp: V -+ A there exists a unique algebra homomorphism 1/1 : Cl -+ A making the following diagram commutative

VLCl cp'\.

11/1 A

This definition says that all Clifford maps may be obtained from I : V which is thereby universal.



The definition by the univeral property is meaningful for an algebraist who knows categories and morphisms up to the theory of universal objects. A category contains objects and morphisms between the objects. Invertible morphisms are called isomorphisms. In a category there is an initial (resp. final) universal object U, iffor any object A, there is a unique morphism a: U -+ A (resp. A -+ U). The universal objects are unique up to isomorphism. In many categories there exists trivially the final universal object, which often reduces to o. Clifford algebra is the initial universal object in the category of algebras of quadratic forms.

Example. Consider the category of algebras of the quadratic form on lRp,q In this category the initial universal object is the Clifford algebra Clp,q of dimension 2n and the final universal object is o. Between these two objects there are no other objects, when p - q :f. 1 mod 4. However, there are 4 objects in this category, when p - q 1 mod 4; between Clp,q and 0 there are two algebras of both dimension 2n - 1 ; in one we have the relation el e2 ... en = 1 and in the ohter el e2 ... en = -1; these two algebras are not isomorphic in the category of algebras of quadratic forms (the identity mapping on lRp,q does not extend to an isomorphism from one algebra to the other); however, they are isomorphic as associative algebras (in the category




of all real algebras).

The above definition of Clifford algebras is most suitable for an algebraist who wants to study Clifford algebras over commutative rings (and who does not insist on injectivity of mappings K A and V A). However, this approach does not guarantee the existence, which is given by constructing the Clifford algebra as the quotient algebra of the tensor algebra (which in turn is regarded by algebraists as the mother of all algebras). 1.6.


®V/I(Q) Chevalley 1954 p. 37 constructs the Clifford algebra Ci(Q) as the quotient algebra ®V/I(Q) of the tensor algebra ®V with respect to the two-sided ideal I( Q) generated by the elements x ® x - Q(x) where x E V. See also N. Bourbaki 1959 p.139 and T. Y. Lam 1973 p.l03. The tensor algebra approach gives a proof of existence by construction - suitable for an algebraist who is interested in rapid access to the main properties of Clifford algebras.

* * * * In characteristic zero we may avoid quotient structures by making the exterior algebra AV concrete as the subspace of antisymmetric tensors in ®V. For example, if x,y E V, then x A y = ® y - y ® x) E A2 V. More generally, a simple k-vector Xl A X2 A ... A Xk is identified with Alt(Xl ® X2 ® ... ® Xk)


;! E

sign(7r) x,..(l) ® X 1f(2) ® ... ® x,..(k),


where the linear operator Alt : ®V AV, called alternation, is a projection operator Alt(®V) = AV satisfying u A v = Alt(u ® v). [Of course, the subspace Alt( ®V) is not closed under the tensor product, though for - B A) E A2 V.l instance for A, BEN V we have A ® B - B ® A = Similarly, we may obtain a linear isomorphism AV Ci(Q) by identifying simple k-vectors with antisymmetric Clifford products

thus splitting the Clifford algebra Cl(Q) into fixed subspaces of k-vectors Any orthogonal basis ell e2, ... , en of V gives a correspondence

Ak V c Cl( Q).

of bases for AV and Ci(Q). Some authors use unnecessarily complicated tools in this context, such as a filtration of Cl( Q), needed only in characteristic 2, even though the above method does not allow division by 2.



2. Chapter 2: 'Rotations and Reflections' 3. Chapter 3: 'Canonical Representation of Isometries' Chapter 2 on rotations and reflections and the Section 4.18 in Chapter 4 on symmetric, absorbing, expelling and idempotent sets and norms have been edited by John Horvath. Chapter 3 does not deal with Clifford algebras. 4. Chapter 4: 'Representation ofIsometries by Infinitesimal Transformations and Clifford Bivectors' Among the four chapters of Riesz's monograph this last chapter has had the greatest influence on physical applications of Clifford algebras. Chapter 4 is a thorough investigation on squaring, exponentiation, decomposition, contraction and differentiation of bivectors in Clifford algebras. In particular, Riesz condensed Maxwell's equations into a single equation with Clifford bivectors (following here Juvet&Schidlof 1932 and A. Mercier 1935). However, to prevent this review from becoming too long I will stop here and leave the reader the excitement of exploring the texts of Riesz. Final Remark Marcel Riesz's monograph is an endeavour to make Clifford algebras elementary and easily accessible to a wide public. This was also observed by J. L. Tits in Mathematical Reviews 31 (1966) #6177. Tits focused on rigour, though he observed that pedagogical aspects are the main concern of Riesz. This renders Tits' review useless and unfair. Acknowledgements I would like to thank Ronald Shaw (Hull) for correcting my misconceptions about quadratic forms in charactetistic 2 and Jacques Helmstetter (Grenoble) for teaching me the theory of universal objects and for emphasizing to me the significance of the contraction. Bibliography

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