Finite Dimensional Representations of Clifford ...

CRN-94-03

Finite Dimensional Representations of Clifford Algebras of Polynomials

N. Fleury and M. Rausch de Traubenberg

Physique Th´eorique Centre de Recherches Nucl´eaires et Universit´e Louis Pasteur B.P. 20 F-67037 Strasbourg Cedex

Adv. Apl. Clif. Alg. 4 (1994) 123

Abstract Finite dimensional representations of Clifford algebras of polynomials are built using generalized Clifford algebras, and their matricial representations are explicitly constructed.

The aim of this note is to make a comparison between Clifford algebras of polynomials [1-9] and generalized Clifford algebras [10-15]. The latter ones can be considered as the basic tools for a systematic construction of a finite dimensional (but not faithful) representation of the former [15]. In the sequel such a representation Lm (f, n, p) is named the linearizing algebra of f , m is the dimension of the representation and f is a polynomial of degree n with p indeterminates defined on C(a I more

general field can also be considered). We just consider homogeneous polynomials, because inhomogeneous can  polynomials  be mapped into homogeneous ones using the projective x x 1 p coordinates xn0 f . ,···, x0 x0 There is a one-to-one correspondence between n-th order symmetric tensors gi1 ···in and polynomials because one can write f (x1 , · · · , xp ) =

p X

{i}=1

xi1 · · · xin gi1 ···in .

(1)

The Roby’s [1] Clifford algebra of f is the consequence of the relation f (x1 , · · · , xp ) · 1 = (x1 g1 + · · · xp gp )n ,

(2)

where 1 is the identity and gi (i = 1 · · · p) are p indeterminates submitted to the constraints I=

(

) 1 X giσ(1) · · · giσ(n) − gi1 ···in = 0 , n!

(3)

σ∈Σn

with Σn the group of permutations of n elements and I a two-sided ideal. This relation comes from the identification of the r.h.s. of (1) and (2). The Clifford algebra C(f, n, p) of f is then defined by C(f, n, p) = C[x I 1 , · · · , xp ]/I .

(4)

This is clearly an extention of complex numbers or, more generally, of Clifford algebras. The Clifford algebras of quadratic polynomials could have been also constructed on IR, but in general for n − ic polynomials, C(f, n, p) has to be constructed on C(the I closure

of IR), because primitive roots of unity are needed. It has been pointed out [1,3,7] that, unlike the Clifford case, one cannot, in general, obtain a finite faithful representation of C(f, n, p). This is due to the fact that the conditions (3) are not sufficient to get a finite dimensional C(f, n, p), and the Clifford algebra of f does not lead to a finite linearization

of f .

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However, it has been shown that one can get, in any case, a finite dimensional representation C(f, n, p) which leads to a finite linearization of f [15]. In other words one can build a surjective homomorphism fm between C(f, n, p) and Lm (f, n, p), the kernel of which

is nothing else but the ideal generated by some supplementary constraints compatible with (3) Lm (f, n, p) = C(f, n, p)/Kerfm

The major difference between C(f, n, p) and Lm (f, n, p) is that the Clifford algebra of f is

defined by the n-linear constraints (3), whereas the universal linearizing algebra by some

quadratic constraints (L), which will be defined below, in such a way that (L) lead to (3) although the converse is not true. For quadratic polynomials this difference does not hold 2 and C(f, 2, p) = L(f, 2, p) = C(p) , the usual Clifford algebra.

Now, let us recall how we can define the Lm (f, n, p)-algebra. The main property which

leads to a finite dimensional algebra is the following: for a systematic construction of Lm (f, n, p), we need to consider and linearize only two basic polynomials, each of them generating its L-algebra through quadratic constraints. These two polynomials are respectively the sum S(x) = xn1 + · · · + xnp , and the product P (x) = x1 · · · xn [15].

Indeed, any polynomial can be understood as a sum of monomials f (x) =

k P

Mi (x),

i=1

where each monomial Mi appears as a special case of P . After the linearization of all the monomials by means of the P −procedure, f appears just as a S-type polynomial and can be linearized as such; some attention has to be paid at this point, because to use S-polynomial, one needs a commuting set of matrices [15]. Similar results on factorization and summation of polynomials have been obtained in [3] but without explicit linearization. The reduction of the general case to the special cases of sums and products is made clearer by the following observation : Let f be a form in p variables of degree n and suppose f is a sum of k monomials. Let f ′ be a form in kp variables obtained by renaming the variables of f in such a way that the variables in different monomials are all distinct. For example if f (x, y) = ax3 +bx2 y+cxy 2 +dy 3 then f ′ (x1 , · · · , x6 ) = ax31 + bx22 x3 + cx4 x25 + dx36 . There is a homomorphism from C(f, n, p) to C(f ′ , n, kp) given by sending a variable to the sum of the new variables that took its place. k P So in our example x would go to x1 + x2 + x4 and y to x3 + x5 + x6 . So f = Mi is sent i=1

to f ′ =

k P

i=1

Mi′ . There is a homomorphism from C(f ′ , n, kp) to C(S, n, p) ⊗ C(M1′ , n, p1 ) ⊗

· · · ⊗ C(Mk′ , n, pk ) and then to C(S, n, p) ⊗ C(P1′ , n, n) ⊗ · · · ⊗ C(Pk′ , n, n), where pi is the number of variables in Mi . Namely one sends any variable x of f ′ appearing in Pi′ to yi ⊗ 1 ⊗ · · · ⊗ x ⊗ 1 ⊗ · · · ⊗ 1, where the x appears in the i + 1 position and yi are the 3

variables in S. So in our example x4 would be sent to y3 ⊗ 1 ⊗ 1 ⊗ x4 ⊗ 1.

Given this homomorphism the existence of a finite dimensional representation reduces immediately to the sum and product case. The advantages is that one does not have to introduce explicit matrices until the end of the process. However, let us briefly recall how the S and P polynomials can be linearized, and how their L-algebra is defined. It has

been proved [4,10-15] that the S-polynomial is linearized using generators Σi (i = 1 · · · p) obeying the relations

   2iπ   Σi Σj = exp Σj Σi n   n Σi = 1

i 2, one obtains different Clifford and linearizing algebras. We would like to acknowledge Ph. Revoy and M. Slupinski for helpful discussions. REFERENCES [1] N. Roby — Alg`ebres de Clifford des formes polynomes, C.R. Acad. Sc. Paris, 268 (1969) 484-486 [2] Ph. Revoy — Nouvelles alg`ebres de Clifford, C.R. Acad. Sc. Paris, 284 (1977) 985-988 [3] L.N. Childs — Linearizing of n-ic Forms and Generalized Clifford Algebras, Lin. and Mult Alg., 5 (1978) 267-278 [4] Ph. Revoy — Sur certaines alg`ebres de Clifford, Commun. in Alg., 11 (1983) 1877-1891 [5] D.E. Haile — On the Clifford Algebra of a binary Cubic Form, Amer. J. Math., 106 (1984) 1269-1280 [6] M. Van den Bergh — Linearization of Binary and Ternary Forms, J. Alg., 109 (1987) 172-183 6

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