PAIRS OF CLIFFORD ALGEBRAS OF THE HURWITZ TYPE

GENERALIZATIONS OF COMPLEX ANALYSIS BANACH CENTER PUBLICATIONS, VOLUME 37

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1996

PAIRS OF CLIFFORD ALGEBRAS OF THE HURWITZ TYPE

W I E S l A W K R ´ O L I K O W S K I

Institute of Mathematics, Polish Academy of Sciences, L´ od´ z Branch Narutowicza 56, 90-136 L´ od´ z, Poland

Abstract. For a given Hurwitz pair [S(Q

_S

), V (Q

_V

), ◦] the existence of a bilinear mapping

? : C(Q

_S

) × C(Q

_V

) → C(Q

_V

) (where C(Q

_S

) and C(Q

_V

) denote the Clifford algebras of the quadratic forms Q

_S

and Q

_V

, respectively) generated by the Hurwitz multiplication “◦” is proved and the counterpart of the Hurwitz condition on the Clifford algebra level is found. Moreover, a necessary and sufficient condition for “?” to be generated by the Hurwitz multiplication is shown.

1. Introduction. The general Hurwitz problem was studied e.g. by Lawrynowicz and Rembieli´ nski [2-4]. They introduced the notions of “Hurwitz pairs” and “pseudo- Hurwitz pairs” and gave their systematic classification according to the relationship with real Clifford algebras. In the present work we show the existence of a bilinear mapping

?: C(Q

_S

) × C(Q

_V

) → C(Q

_V

), where (S, V , ◦) is a given Hurwitz pair which makes the following diagram:

(1)

S × V V V

C(Q

S

) × C(Q

V

) C(Q

V

)

◦ (Hurwitz multiplication)

//

i_S×i_V



^iV



_?

//

commutative.

Moreover, we prove that if such a mapping exists and satisfies the following “algebraic Hurwitz condition”: N (x

_S

? y

_V

) = N (x

_S

)N (y

_V

) for any x

_S

∈ Γ

_S

and y

_V

∈ Γ

_V

, where Γ denotes the Clifford group of the Clifford algebra C(Q) and N is a spinor norm, then ? is generated by the Hurwitz multiplication, i.e. ?

_|S×V

= ◦. An example of a mapping ? which does not satisfy the N -norm condition is given. Since in the meantime the detailed proofs have appeared in [1], they are only sketched here.

1991 Mathematics Subject Classification: Primary 15A66, Secondary 30G35.

Research supported by the KBN grant PB 2 1140 91 01.

The paper is in final form and no version of it will be published elsewhere.

[327]

328

W. KR ´OLIKOWSKI

2. Product of Clifford algebras generated by the Hurwitz multiplication.

Let (S, V , ◦) be a Hurwitz pair. Suppose that the vector spaces S and V are equipped with non-degenerate quadratic forms Q

_S

and Q

_V

, respectively. We will only consider the elliptic and hyperbolic cases (see, e.g. [2-4]). In S and V we choose some bases (

α

) and (e

_j

) with α = 1, . . . , p = dim S; j = 1, . . . , n = dim V . Assume that p ≤ n.

Let C(Q

S

) (resp. C(Q

V

)) denote the Clifford algebra of (S, Q

S

) (resp. (V, Q

V

)).

There are canonical injections i

_S

: S → C(Q

_S

) and i

_V

: V → C(Q

_V

). Then we get the diagram (2). It would be interesting to complete the diagram (2) by the suitable mapping C(Q

S

) × C(Q

V

) → C(Q

V

). Define the following mapping ? : C(Q

S

) × C(Q

V

) → C(Q

V

) by:

(2)



 

 

 

 

 

 

1

_S

? y

_V

:= y

_V

,

(

_i1

. . . 

_ir

) ? (e

_j1

. . . e

_jk

) :=



 

 

e

j_k

. . . e

j_r+1

(

i_r

◦ e

j_r

) . . . (

i₁

◦ e

j₁

), r < k, (

i_r

◦ e

j_r

) . . . (

i₁

◦ e

j₁

), r = k,



i_r

◦ [

i_r−1

◦ [. . . ◦ [

i_k+1

◦[(

i_k

◦ e

j_k

) . . . (

i₁

◦ e

j₁

)] . . .], r > k, (

i₁

. . . 

i_r

) ? 1

V

:= k

i₁

k . . . k

i_r

k1

V

for 1 ≤ r ≤ p, 1 ≤ i

1

< . . . < i

r

≤ p; 1 ≤ k ≤ n, 1 ≤ j

1

< . . . < j

k

≤ n. Then, the required mapping ? : C(Q

S

) × C(Q

V

) → C(Q

V

) is defined by the bilinear extension of (2).

R e m a r k. If (S, Q

S

) is a Euclidean vector space then all k

i

k

²

> 0. In this case the Clifford algebras C(Q

_S

) and C(Q

_V

) are considered to be real. But, if (S, Q

_S

) is a pseudo-Euclidean vector space then there are some 

i₁

, . . . , 

i_r

, 1 ≤ r ≤ p, such that k

i_s

k

²

< 0, 1 ≤ s ≤ r. This time the Clifford algebras have to be treated as complex ones.

Proposition. ? is a well defined bilinear mapping. Moreover , ?

|S×V

= ◦, the Hurwitz multiplication, i.e. the diagram (1) is commutative.

Lemma. Let x

^S

∈ Γ

S

and y

V

∈ Γ

V

, where Γ

S

(resp. Γ

V

) denotes the Clifford group in C(Q

_S

) (resp. C(Q

_V

)) and let N

_S

, N

_V

be the spinor norms in C(Q

_S

) and C(Q

_V

), respectively. Then

(3) N

_V

(x

_S

? y

_V

) = N

_S

(x

_S

)N

_V

(y

_V

).

Theorem. Let S and V be real vector spaces equipped with non-degenerate quadratic forms Q

S

and Q

V

, respectively. Denote by C

^C

(Q

S

) (resp. C

^C

(Q

V

)) the complex Clifford algebras of (S, Q

S

) (resp. (V ,Q

V

)). Suppose that there is a bilinear mapping ? : C

^C

(Q

S

)×

C

^C

(Q

V

) → C

^C

(Q

V

) satisfying the condition (3). Then ? is generated by the Hurwitz multiplication, i.e. ?

_|S×V

= ◦, where ◦ : S × V → V is a bilinear mapping such that ks ◦ vk

_V

= ksk

_S

kvk

_V

for all s ∈ S and v ∈ V .

P r o o f. Let s ∈ S ⊂ Γ

_S

and v ∈ V ⊂ Γ

_V

. By definition of N we have (4) N

V

(s ? v) = N

S

(s)N

V

(v) = ksk

²_S

kvk

²_V

∈ R.

Let (e

₁

, . . . , e

_n

) be an orthogonal base in V . Suppose s ? v = a

₀

+

n

X

i=1

a

ⁱ

e

_i

+

n

X

l=2

X

i1<...<i_l

a

_l^i1...il

e

_i1

. . . e

_il

.

PAIRS OF CLIFFORD ALGEBRAS OF THE HURWITZ TYPE

329

Then

N (s ? v) = a

²₀

+

n

X

i=1

(a

ⁱ

)

²

Q

V

(e

i

) +

n

X

l=2

X

i₁<..<i_l

(a

ⁱ_l^1..il

)

²

Q

V

(e

i₁

) . . . Q

V

(e

i_l

) + R(e

1

, . . . , e

n

), where

R(e

1

, . . . , e

n

) = X

b

ⁱ

e

i

+ X

i<j

b

^ij

e

i

e

j

+ . . . + X

i₁<...<i_m

b

^i1...im

e

i1

. . . e

im

+ be

1

. . . e

n

. Since N (s ? v) is a scalar then R(e

₁

, . . . , e

_n

) must vanish. The multiplication ? is bilinear so the coefficients a

0

, a

ⁱ

and a

ⁱ_l^1...il

are bilinear functions in s and v. Thus N (s ? v) should be separated into two parts, first depending only on s and second only on v. Then we can write

a

²₀

+

n

X

i=1

(a

ⁱ

)

²

Q

V

(e

i

) +

n

X

l=2

X

i1<...<il

(a

ⁱ_l^1...il

)

²

Q

V

(e

i₁

) . . . Q

V

(e

i_l

)

= ksk

²_S

kvk

²_V

= ksk

²_S

[c

²₀

+

n

X

i=1

(c

ⁱ

)

²

Q

V

(e

i

) +

n

X

l=2

X

i₁<...<i_l

(c

ⁱ_l^1...il2

Q

V

(e

i1

) . . . Q

V

(e

i_l

)]

Thus, the following equality has to be satisfied:

c

²₀

+

n

X

i=1

(c

ⁱ

)

²

Q

V

(e

i

) +

n

X

l=2

X

i1<...<il

(c

ⁱ_l^1...il

)

²

Q

V

(e

i₁

) . . . Q

V

(e

i_l

) =

n

X

i=1

(v

ⁱ

)

²

Q

V

(e

i

).

The coefficients c

0

, c

ⁱ

, c

ⁱ_l^1...il

are linear in v so, by continuity, we can write c

₀

(v) = c

_0j

v

^j

, c

ⁱ

(v) = c

ⁱ_j

v

^j

, c

ⁱ_l^1...il

(v) = c

ⁱ_lj^1...il

v

^j

. Thus, for any 1 ≤ j, k ≤ n we get the identity

c

0j

c

0k

+

n

X

i=1

(c

ⁱ_j

c

ⁱ_k

− δ

_jⁱ

δ

_kⁱ

)Q

V

(e

i

) +

n

X

l=2

X

i₁<...<i_l

c

ⁱ_lj^1...il

c

ⁱ_lk^1...il

Q

V

(e

i₁

) . . . Q

V

(e

i_l

) ≡ 0.

Take an orthogonal transformation R ∈ O(Q

_V

). In a new base e

⁰

= Re we have c

_0j

c

_0k

+

n

X

i=1

( e c

ⁱ_j

e c

ⁱ_k

− δ

ⁱ_j

δ

ⁱ_k

)Q

_V

(Re

_i

) +

n

X

l=2

X

i₁<...<i_l

e c

ⁱ_lj^1...il

e c

ⁱ_lk^1...il

Q

_V

(Re

_i1

) . . . Q

_V

(Re

_il

) ≡ 0.

But Q

V

(Re

i

) = Q

V

(e

i

). Then the new coefficients e c

j

and e c

ⁱ_lj^1...il

, obtained by the changing of the base, satisfy the same identity as the previous ones. This is possible if and only if

c

0j

≡ 0 for j = 1, . . . , n,

c

ⁱ_j

c

ⁱ_k

− δ

_jⁱ

δ

_kⁱ

≡ 0 for 1 ≤ i, j, k ≤ n,

c

ⁱ_lj^1...il

≡ 0 for l = 2, . . . , n; 1 ≤ i

1

< . . . < i

l

≤ n; j = 1, . . . , n.

Thus, we get s ? v = ksk

_S

P

n

i=1,j=1

c

ⁱ_j

v

^j

e

_i

∈ V and ksk

²_S

kvk

²_V

= N

_V

(s ? v) = ks ? vk

²_V

, so

?

_|S×V

satisfies the Hurwitz condition, as required.

Example. We now construct a bilinear map  ^{: C}

^C

^(Q

S

) × C

^C

(Q

_V

) → C

^C

(Q

_V

)

which does not satisfy the condition (4). Choose some bases (

α

) and (e

j

) in S and V ,

330

W. KR ´OLIKOWSKI

respectively. Define

(5)



 

 

 

 

 

 

 

 

 

 

1

S

 ¹

^V

^{:= e}

¹

^{. . . e}

ⁿ

^, 1

S

 ^(e

ⁱ1

. . . e

i_k

) := e

i₁

d . . . e

i_k

,

1

S

 ^(e

¹

^{. . . e}

ⁿ

^{) := 1}

^V

^,

(

_j1

. . . 

_jr

)  ^(e

i1

. . . e

_ik

) := k

_j1

k . . . k

jr

k e

_i1

d . . . e

_ik

, (

_j1

. . . 

_jr

)  ¹

V

:= k

_j1

k . . . k

jr

ke

1

. . . e

_n

, (

_j1

. . . 

_jr

 ^(e

1

. . . e

_n

) := k

_j1

k . . . k

_jr

k1

_V

, where “ b .” is defined by

e

i₁

d . . . e

i_r

:= e

j₁

. . . e

j_s

with j

1

< . . . < j

s

and (i

1

, . . . , i

r

, j

1

, . . . , j

s

) = (1, . . . , n).

The map  ^{: C}

^C

^(Q

^S

^{) × C}

^C

^(Q

^V

^{) → C}

^C

^(Q

^V

) is defined by the bilinear extension of (5).

It is easy to see that  does not satisfy the condition (4). Indeed, take s ∈ S and v ∈ V . We have

s  ^{v = s}

^α

^v

ⁱ

^

α

 ^e

i

= s

^α

v

ⁱ

k

_α

k

_S

e

₁

. . . b e

_i

. . . e

_n

6∈ V.

and

N

V

(s  ^{v) = s}

^α

^s

^β

^k

^α

^k

S

k

β

k

_S

(v

ⁱ

)

²

Q

V

(e

1

) . . . Q d

V

(e

i

) . . . O

V

(e

n

).

Suppose that N

V

(s  ^{v) = N}

^S

^(s)N

^V

(v). Then we get X

α,β

s

^α

s

^β

k

α

k

_S

k

β

k

_S

= X

α

(s

^α

)

²

k

α

k

²_S

, X

i

(v

ⁱ

)

²

Q

V

(e

1

) . . . Q d

V

(e

i

) . . . Q

V

(e

n

) = X

i

(v

ⁱ

)

²

Q

V

(e

i

).

The above condition is equivalent to

k

α

k

_S

= 0 and ke

₁

k

²_V

. . . k e b

_i

k

²_V

. . . ke

_n

k

²_V

= ke

_i

k

²_V

, but this is impossible.

References

[1] W. K r ´ o l i k o w s k i, On Fueter-Hurwitz regular mappings, Dissertationes Math. 353 (1996).

[2] J. L a w r y n o w i c z and J. R e m b i e l i ´ n s k i, Pseudo-Euclidean Hurwitz pairs and generalized Fueter equations, in: Clifford algebras and their applications in mathematical physics, Proc., Canterbury 1985, J. S. R. Chisholm and A. K. Common (eds.), Reidel, Dordrecht, 1986, 39–48.

[3] –, —, On the composition of nondegenerate quadratic forms with an arbitrary index , Ann.

Fac. Sci. Toulouse 10 (1989), 141–168.

[4] —, —, Pseudo-Euclidean Hurwitz pairs and the Ka lu˙za–Klein theories, J. Phys. A Math.

Gen. 20 (1987), 5831–5848.