Isotropy of some quadratic forms and its applications on levels and sublevels of algebras

0.15 MB·English

by  Cristina Flaut

Isotropy of some quadratic forms and its applications on
levels and sublevels of algebras 1
2
1 Cristina Flaut
0
2
Faculty of Mathematics and Computer Science,
n
a Ovidius University,
J
Bd. Mamaia 124, 900527, CONSTANT¸A,

8
ROMANIA
2
e-mail: cfl[email protected]; cristina fl[email protected]

]
A http://cristinaflaut.wikispaces.com/
R
.
h
t
a Abstract. Inthispaper,wegivesomepropertiesofthelevelsandsublevelsof
m
algebras obtained by the Cayley-Dickson process. We will emphasize how isotropy
[
ofsomequadraticforms caninfluencethelevels andsublevelsofalgebras obtained

1
by the Cayley-Dickson process.
v
8
7
9 Key Words: Cayley-Dickson process; Division algebra; Level and sublevel of
5
an algebra.
.
1
AMS Classification: 17A35, 17A20, 17A75, 17A45.
0
2
1
:
v
i 0. Introduction
X
r
a
In [18], A. Pfister showed that if a field has a finite level this level is a
power of 2 and any power of 2 could be realised as the level of a field. In the
noncommutative case, the concept of level has many generalisations. The
level of division algebras is defined in the same manner as for the fields.
In this paper, we give some properties of the levels and sublevels of al-
gebras obtained by the Cayley-Dickson process. We will emphasize how
1This paper was partially supported by the grant UNESCO-UNITWIN
OCW/OER Initiative, Handong Global University, South Korea.
1
isotropy of some quadratic forms can influence the levels and sublevels of
algebras obtained by the Cayley-Dickson process.
1. Preliminaries
In this paper, we assume that K is a field and charK 6= 2.
For the basic terminology of quadratic and symmetric bilinear spaces,
the reader is referred to [22] or [11]. In this paper, we assume that all the
quadratic forms are nondegenerate.
A bilinear space (V,b) represents α ∈ K if there is an element x ∈ V,x 6=
0, with b(x,x) = α. The space is called universal if (V,b) represents all
α ∈ K. A vector x ∈ V,x 6= 0 is called isotropic if b(x,x) = 0, otherwise x is
called anisotropic. If the bilinear space (V,b), V 6= {0}, contains an isotropic
vector, then the space is called isotropic. Every isotropic bilinear space V,
V 6= {0}, is universal. (See [22], Lemma 4.11., p.14])
The quadratic form q : V → K is called anisotropic if q(x) = 0 implies
x = 0, otherwise q is called isotropic.
A quadratic form ψ is a subform of the form ϕ if ϕ ≃ ψ ⊥ φ, for some
quadratic form φ. We denote ψ < ϕ.
Let ϕ be a n−dimensional quadratic irreducible form over K, n ∈ N,n >
1, which is not isometric to the hyperbolic plane. We may consider ϕ as a
homogeneous polynomial of degree 2,
ϕ(X) = ϕ(X ,...X ) = a X X ,a ∈ K∗.
1 n ij i j ij
X
The functions field of ϕ,denoted K(ϕ), is the quotient field of the integral
domain
K[X ,...,X ]/ (ϕ(X ,...,X )).
1 n 1 n
Since (X ,...,X ) is a non-trivial zero, ϕ is isotropic over K(ϕ).(See [22])
1 n
For n ∈ N−{0} a n−fold Pfister form over K is a quadratic form of the
type
< 1,a > ⊗...⊗ < 1,a >, a ,...,a ∈ K∗.
1 n 1 n
A Pfister form is denoted by ≪ a ,a ,...,a ≫ . For n ∈ N,n > 1, a Pfister
1 2 n
form ϕ can be written as
< 1,a > ⊗...⊗ < 1,a >=< 1,a ,a ,...,a ,a a ,...,a a a ,...,a a ...a > .
1 n 1 2 n 1 2 1 2 3 1 2 n
2
If ϕ =< 1 >⊥ ϕ′, then ϕ′ is called the pure subform ofϕ. A Pfister form
is hyperbolic if and only if is isotropic. This means that a Pfister form is
isotropic if and only if its pure subform is isotropic.( See [22] )
For the field L it is defined
L∞ = L∪{∞},
where x+∞ = x, for x ∈ K, x∞ = ∞ for x ∈ K∗,∞∞ = ∞, 1 = 0, 1 = ∞.
∞ 0
An L−place of the field K is a map
λ : K → L∞
with the properties:
λ(x+y) = λ(x)+λ(y),λ(xy) = λ(x)λ(y),
whenever the right sides are defined.
Theorem([8], Theorem 3.3. ) Let F be a field of characteristic 6= 2, ϕ be
a quadratic form over F and K,L extensions field of F. If ϕ is isotropic,
K
then there exist an F−place from F (ϕ) to K.
An algebra A over K is called quadratic if A is a unitary algebra and, for
all x ∈ A, there are a,b ∈ K such that x2 = ax+b1, a,b ∈ K. The subset
2
A = {x ∈ A−K | x ∈ K1}
0
is a linear subspace of A and
A = K ·1⊕A .
0
A composition algebra is an algebra A with a non-degenerate quadratic
form q : A → K, such that q is multiplicative, i.e.
q(xy) = q(x)q(y),∀x,y ∈ A.
A unitary composition algebra is called a Hurwitz algebra. Hurwitz algebras
have dimensions 1,2,4,8.
Since over fields, the classical Cayley-Dickson process generates all possi-
ble Hurwitz algebras, in the following, we briefly present the Cayley-Dickson
process and the properties of the algebras obtained.
3
Let A be a finite dimensional unitary algebra over a field K,with a scalar
involution
: A → A,a → a,
i.e. a linear map satisfying the following relations:
ab = ba, a = a
and
a+a,aa ∈ K ·1,
for all a,b ∈ A. The element a is called the conjugate of the element a.
The linear form
t : A → K, t(a) = a+a
is called the trace of the element a and the quadratic form
n : A → K, n(a) = aa
is called the norm of the element a. It results that an algebra A with a
scalar involution is quadratic. If the quadratic form n is anisotropic, then
the algebra A is called anisotropic, otherwise A is isotropic.
Let γ ∈ K be a fixed non-zero element. We define the following algebra
multiplication on the vector space A⊕A.
(a ,a )(b ,b ) = a b +γb a ,a b +b a . (1.1.)
1 2 1 2 1 1 2 2 2 1 2 1
We obtain an algebra structure ov(cid:0)er A⊕A. This algebra, d(cid:1)enoted by (A,γ),
is called the algebra obtained from A by the Cayley-Dickson process. A is
canonically isomorphic with the algebra
A′ = {(a,0) ∈ A⊕A | a ∈ A},
where A′ is a subalgebra of the algebra (A,γ). We denote (1,0) by 1,
where (1,0) is the identity in (A,γ). Taking
2
u = (0,1) ∈ A⊕A, u = γ ·1 ∈ K ·1,
it results that (A,γ) = A⊕Au. We have dim(A,γ) = 2dimA.
Let x ∈ (A,γ),x = (a ,a ). The map
1 2
: (A,γ) → (A,γ) , x → x¯ = (a ,−a ),
1 2
4
is a scalar involution of the algebra (A,γ), extending the involution of the
algebra A, therefore the algebra (A,γ) is quadratic. For x ∈ (A,γ),x =
(a ,a ), we denote
1 2
t(x)·1 = x+x = t(a )·1 ∈ K ·1,
1
n(x)·1 = xx = (a a −γa a )·1 = (n(a )−γn(a ))·1 ∈ K ·1
1 1 2 2 1 2
and the scalars
t(x) = t(a ), n(x) = n(a )−γn(a )
1 1 2
arecalledthetrace andthenorm oftheelementx ∈(A,γ),respectively. It follows that
2
x −t(x)x+n(x) = 0, ∀x ∈ (A,γ).
If we take A = K and apply this process t times, t ≥ 1, we obtain an
algebra over K, A = K{α ,...,α }. By induction, in this algebra we find a
t 1 t
basis {1,f ,...,f },q = 2t, satisfying the properties:
2 q
2
f = α 1, α ∈ K,α 6= 0, i = 2,...,q.
i i i i
f f = −f f = β f , β ∈ K, β 6= 0,i 6= j,i,j = 2,...q, (1.2.)
i j j i ij k ij ij
β and f being uniquely determined by f and f .
ij k i j
If
q
x ∈ A ,x = x 1+ x f ,
t 1 i i
i=2
X
then
q
x¯ = x 1− x f
1 i i
i=2
X
and
q
2 2
t(x) = 2x ,n(x) = x − α x .
1 1 i i
i=2
X
In the above decomposition of x, we call x the scalar part of x and x′′ =
1
q
x f the pure part of x. If we compute
i i
i=2
P
x2 = x2 +x′′2 +2x x′′ =
1 1
5
t
= x2 +α x2 +α x2 −α α x2 +α x2 −...−(−1)t( α )x2 +2x x′′,
1 1 2 2 3 1 2 4 3 5 i q 1
i=1
Y
the scalar part of x2 is represented by the quadratic form
t
t
T =< 1,α ,α ,−α α ,α ,...,(−1) ( α ) >=< 1,β ,...,β > (1.3.)
C 1 2 1 2 3 i 2 q
i=1
Y
and, since
t
x′′2 = α x2 +α x2 −α α x2 +α x2 −...−(−1)t( α )x2 ∈ K,
1 2 2 3 1 2 4 3 5 i q
i=1
Y
it is represented by the quadratic form T = T | : A → K,
P C A0 0
t
t
T =< α ,α ,−α α ,α ,...,(−1) ( α ) >=< β ,...,β > . (1.4.)
P 1 2 1 2 3 i 2 q
i=1
Y
The quadratic form T is called the trace form, and T the pure trace form
C P
of the algebra A . We remark that T =< 1 >⊥ T , and the norm
t C P
n = n =< 1 >⊥ −T , resulting that
C P
t
t+1
n =< 1,−α ,−α ,α α ,α ,...,(−1) ( α ) >=< 1,−β ,...,−β > .
C 1 2 1 2 3 i 2 q
i=1
Y
The norm form n has the form
C
n =< 1,−α > ⊗...⊗ < 1,−α >
C 1 t
and it is a Pfister form.
Since the scalar part of any element y ∈ A is 1t(y), it follows that
t 2
t(x2)
T (x) = .
C
2
Brown’s construction of division algebras
In 1967, R. B. Brown constructed, for every t, a division algebra A of
t
dimension 2t over the power-series field K{X ,X ,...,X }. We will briefly
1 2 t
6
demonstrate this construction, using polynomial rings over K and their field
of fractions (the rational function field) instead of power-series fields over K
(as it is done by R.B. Brown).
First of all, we remark that if an algebra A is finite-dimensional, then it is
adivisionalgebraifandonlyifAdoesnotcontainzerodivisors(See[20]). For
every t we construct a division algebra A over a field F . Let X ,X ,...,X
t t 1 2 t
be t algebraically independent indeterminates over the field K and F =
t
K(X ,X ,...,X ) be the rational function field. For i = 1,...,t, we construct
1 2 t
the algebra A over the rational function field K(X ,X ,...,X ) by setting
i 1 2 i
α = X for j = 1,2,..., i. Let A = K. By induction over i, assuming
j j 0
that A is a division algebra over the field F = K(X ,X ,...,X ),
i−1 i−1 1 2 i−1
we may prove that the algebra A is a division algebra over the field F =
i i
K(X ,X ,...,X ).
1 2 i
Let
Ai−1 = F ⊗ A .
Fi i Fi−1 i−1
For α = X we apply the Cayley-Dickson process to algebra Ai−1. The ob-
i i Fi
tained algebra, denoted A , is an algebra over the field F and has dimension
i i
2i.
Let
x = a+bv , y = c+dv ,
i i
be nonzero elements in A such that xy = 0, where v2 = α . Since
i i i
¯
xy = ac+X db+(bc¯+da)v = 0,
i i
we obtain
¯
ac+X db = 0 (2.1)
i
and
bc¯+da = 0. (2.2.)
But, the elements a,b,c,d ∈ Ai−1 are non zero elements. Indeed, we have:
Fi
i) If a = 0 and b 6= 0, then c = d = 0 ⇒ y = 0, false;
ii)If b = 0 and a 6= 0, then d = c = 0 ⇒ y = 0, false;
iii) If c = 0 and d 6= 0, then a = b = 0 ⇒ x = 0, false;
iv) Ifd = 0and c 6= 0, then a = b = 0 ⇒ x = 0,false.
This implies that b 6= 0,a 6= 0, d 6= 0, c 6= 0. If {1,f2,...,f2i−1} is a basis
in A ,then
i−1
2i−1 2i−1
a = g (1⊗f ) = g f ,g ∈ F ,
j j j j j i
j=1 j=1
X X
7
g′
g = j,g′,g′′ ∈ K[X ,...,X ], g′′ 6= 0, j = 1,2,...2i−1,
j g′′ j j 1 i j
j
where K[X ,...,X ] is the polynomial ring. Let a be the less common
1 t 2
multiple of g′′,....g′′ , then we can write a = a1, where a ∈ Ai−1,a 6= 0.
1 2i−1 a2 1 Fi 1
Analogously, b = b1,c = c1,d = d1,b ,c ,d ∈ Ai−1 −{0} and a ,b ,c ,d ∈
b2 c2 d2 1 1 1 Fi 2 2 2 2
K[X ,...,X ]−{0}.
1 t
If we replace in relations (2.1.) and (2.2.), we obtain
¯
a c d b +X d b a c = 0 (2.3.)
1 1 2 2 i 1 1 2 2
and
b c¯ d a +d a b c = 0. (2.4.)
1 1 2 2 1 1 2 2
If we denote a = a b ,b = b a ,c = c d ,d = d c , a ,b ,c ,d ∈
3 1 2 3 1 2 3 1 2 3 1 2 3 3 3 3
Ai−1 −{0}, relations (2.3.) and (2.4.) become
Fi
¯
a c +X d b = 0 (2.5.)
3 3 i 3 3
and
b c¯ +d a = 0. (2.6.)
3 3 3 3
Since the algebra Ai−1 = F ⊗ A is an algebra over F with basis
Fi i Fi−1 i−1 i−1
Xi ⊗f , i ∈ N and j = 1,2,...2i−1, we can write a ,b ,c ,d under the form
j 3 3 3 3
j
a = x X ,
3 j i
j≥m
X
j
b = y X ,
3 j i
j≥n
X
j
c = z X ,
3 j i
j≥p
X
j
d = w X ,
3 j i
j≥r
X
where x ,y ,z ,w ∈ A ,x ,y ,z ,w 6= 0. Since A is a division algebra,
j j j j i−1 m n p r i−1
we have x z 6= 0,w y 6= 0, y z 6= 0,w x 6= 0. Using relations (2.5.) and
m p r n n p r m
(2.6.), we have that 2m+p+r = 2n+p+r +1, which is false. Therefore,
the algebra A is a division algebra over the field F = K(X ,X ,...,X ) of
i i 1 2 i
dimension 2i.
8
3. Main results
The level of the algebra A, denoted by s(A), is the least integer n such
that −1 is a sum of n squares in A. The sublevel of the algebra A, denoted
by s(A), is the least integer n such that 0 is a sum of n+1 nonzero squares
of elements in A. If these numbers do not exist, then the level and sublevel
are infinite. Obviously, s(A) ≤ s(A).
Let Abe a division algebra over a field K obtained by the Cayley-Dickson
process, of dimension q = 2t,T , T and n be its trace, pure trace and norm
C P C
forms.
Proposition 3.1. With above notations, we have:
i) If s(A) ≤ n then −1 is represented by the quadratic form n×T .
C
ii) −1 is a sum of n squares of pure elements in A if and only if the
quadratic form n×T represents −1.
P
iii) For n ∈ N−{0}, if the quadratic form < 1 >⊥ n×T is isotropic
P
over K, then s(A) ≤ n.
Proof. i) Let y ∈ A,y = x + x f + ... + x f , x ∈ K, for all i ∈
1 2 2 q q i
{1,2,...,q}. Using the notations given in the Introduction, we get
y2 = x2 +β x2 +...+β x2 +2x y′′,
1 2 2 q q 1
where
y′′ = x f +...+x f .
2 2 q q
If −1 is a sum of n squares in A, then
2 2
−1 = y +...+y =
1 n
= x2 +β x2 +...+β x2 +2x y′′ +...+ x2 +β x2 +...+β x2 +2x y′′ .
11 2 12 q 1q 11 1 n1 2 n2 q nq n1 n
Then(cid:0) we have (cid:1) (cid:0) (cid:1)
n n n
2 2 2
−1 = x +β x +...+β x
i1 2 i2 q iq
i=1 i=1 i=1
X X X
and
n n n
x x = x x = ... = x x = 0,
i1 i2 i1 i3 i1 in
i=1 i=1 i=1
X X X
9
then n×T represents −1.
C
ii) With the same notations, if −1 is a sum of n squares of pure elements
in A, then
2 2
−1 = y +...+y =
1 n
= β x2 +...+β x2 +2x y′′ +...+ β x2 +...+β x2 +2x y′′ .
2 12 q 1q 11 1 2 n2 q nq n1 n
We have(cid:0) (cid:1) (cid:0) (cid:1)
n n
2 2
−1 = β x +...+β x .
2 i2 q iq
i=1 i=1
X X
Therefore n×T represents −1. Reciprocally, if n×T represents −1,
P P
then
n n
2 2
−1 = β x +...+β x .
2 i2 q iq
i=1 i=1
X X
Let
2
u = x f +...+x f .
i i2 2 iq q
It results t(u ) = 0 and
i
2 2 2
u = −n(u ) = β x +...+β x ,
i i 2 i2 q iq
for all i ∈ {1,2,...,n}. We obtain
2 2
−1 = u +...+u .
1 n
iii) Case 1. If −1 ∈ K∗2, then s(A) = 1.
Case 2. −1 ∈/ K∗2. Since the quadratic form < 1 >⊥ n×T is isotropic
P
then it is universal. It results that < 1 >⊥ n×T represent −1. Then, we
P
have the elements α ∈ K and p ∈ Skew(A), i = 1,...,n, such that
i
n n
2 2 2
−1 = α +β p +...+β p ,
2 i2 q iq
i=1 i=1
X X
and not all of them are zero.
i) If α = 0, then
n n
2 2
−1 = β p +...+β p .
2 i2 q iq
i=1 i=1
X X
10