Finiteness of Hermitian Levels of Some Algebras

We characterize the hermitian levels of quaternion and octonion algebras and of an 8-dimensional algebra D over the ground field F, constructed using a weak version of the Cayley-Dickson double process. It is shown that all values of the hermitian levels of quaternion algebras with the hat-involution also occur as hermitian levels of D. We give some limits to the levels of the algebra D over some ground field.


Introduction
Lewis [2] define hermitian level of a ring with identity equipped with non-trivial involution.He showed that there exist commutative rings with involutions having any positive integer as hermitian level.He also showed that the hermitian levels of quadratic extensions of fields and the quaternion division algebras, both with standard involutions, are power of two, if they are finite.In this case, any power of two can occur as hermitian level.For a quaternion division algebra equipped with standard and hat-involution, Lewis has obtained results relating the finiteness of the hermitian level with certain sum of squares in the ground field of the algebra.These results were extended to octonion algebras by Pumplün and Unger in [4].
In this paper, we obtain characterizations about finiteness of hermitian levels of quaternion (respectively; octonion) algebras D, with involutions in terms of the hermitian length of one element b ∈ F, or length of −b in a quadratic extension field F ( √ a) (respectively; biquadratic extension field F ( √ c, √ −a)), where F is the ground field of D, and the elements a, b, c ∈ F are related with the generators of D, (Propositions 3.2, 3.3, 3.4).
Based on the weak version of the Cayley-Dickson doubling process, we use the hat-involution over a quaternion algebra Q and obtain a new algebra, which we 70 C. M. dos Santos denote by Q( √ c), equipped with a non-trivial involution which is not scalar.We show that the finiteness of the hermitian level of this algebras and of the octonion algebra has similar behavior (Propositions 3.3 and 3.5).We show that all values of hermitian levels of quaternion algebra occur as hermitian levels of the algebra Q( √ c), (Proposition (4.1)).In addition, we give some limits to the level of the algebra Q( √ c) over some ground field.

Preliminaries
Let D be a ring with an identity (and 2 = 0) equipped with a non-trivial involution, i.e., an anti-automorphism * : D → D of period two.We denote by n× 1 , the hermitian form over (D, * ) represented by the n×n identity matrix, i.e., the form h : Also, for a quadratic form over a field F, q : V → F, (where V is a n-dimensional vectorspace over F ), D Q q := {q(u) ∈ Ḟ , u ∈ V } is the subset of elements of F represented by q.The quadratic form q is said to be isotropic if there exists u ∈ V, u = 0 such that q(u) = 0.The vector u is said to be isotropic.In otherwise q and u are said to be anisotropic.The sum of n quadratic form q : q ⊥ q⊥ • • • ⊥q (n times) is denoted by n × q.In particular n × 1 also denotes the quadratic form If q 1 and q 2 are isometric quadratic forms, we will denote this fact by q 1 ∼ q 2 .The quadratic form a ⊗ q will be denoted by a.q.An n-fold Pfister form over a field F is a quadratic form of the type 1, a 1 ⊗ 1, a 2 ⊗ • • • ⊗ 1, a n , also denoted by a 1 , a 2 , . . ., a n .It is known that if q is an n-fold Pfister form over a field F, then a.q ∼ q, for every a ∈ D Q q.In particular D Q q is a subgroup of Ḟ .These and other basic results on quadratic and hermitian forms can be found in [1] and [5].
Definition 2.1 Let D be a ring equipped with an involution * .An hermitian square in D, (or in (D, * )), is an element of the form x * x for some x ∈ Ḋ, i.e., an element of D H 1 .The hermitian length of c ∈ Ḋ is the least (positive) integer n such that c is a sum of n hermitian squares in D, i.e., c ∈ D H n × 1 .In otherwise the hermitian length is infinite.We denote by l(D, * , c) the hermitian length of c, and by l(D, c) if * is the identity map of D, i.e., l(D, c) is the usual length of c ∈ D.
The hermitian level of (D, * ) is the hermitian length of −1 ∈ D, and it is denoted by s(D, * ).

The Cayley-Dickson doubling process
Now, we consider a finite dimensional F -algebra A equipped with a scalar involution * , that is, the trace of x : x * +x and the norm of x : x * x belong to F.
The Cayley-Dickson doubling process is a well-known way to construct new algebra with scalar involution from a given algebra with scalar involution.If A is an F -algebra with involution * and c ∈ Ḟ , then the F -vectorspace A × A becomes an F -algebra D with the multiplication This new algebra is denoted by Cay(A, c), and we say that is obtained from A by the Cayley-Dickson doubling process.
Since A is F -isomorphic to A × {0}, if we denote (0, 1) by e, we can denote the algebra A × A by A ⊕ Ae.Thus, the multiplication in A ⊕ Ae is and the involution is: The involution * is a scalar involution on A ⊕ Ae if and only if * is a scalar involution on A. In this case, In this case, we write where where u r = α r + β r i + γ r j + δ r k ∈ Q.
In the third case, we consider an octonion algebra O obtained from the quaternion algebra Q and c ∈ Ḟ by Cayley-Dickson doubling process.If Q = (a, b)/F , then the octonion algebra Cay(Q, c) is also denoted by O := (a, b, c)/F .Since Q is noncommutative and i(je) = (ij)e we see that O is noncommutative and nonassociative algebra.For more details, see ( [3], §2) and ( [4], §2.2).
The standard involution on O is given by u + ve = u − ve, for every u, v ∈ Q.This involution is scalar and the norm form of O is Lewis [2] has defined and studied the hat-involution on quaternion algebras and Pumplüm and Unger [4], have generalized the hat-involution to octonion algebras.
If u = α + βi + γj + δk then u is defined by u = α − βi + γj + δk.Thus, the hermitian square uu is the following element in In the last case we consider the following weak version of the Cayley-Dickson doubling process: From the hat-involution on Q and (*) we get: where u, u 1 , v, v 1 ∈ Q.This yields a new algebra and it is easy to prove that u+ve = u + ve is an involution on this new algebra.From now on, we denote this algebra by Q( √ c).(x 4r y 1r + x 1r y 4r + x 3r y 2r + x 2r y 3r ) = 0.   (i) Zero is a non-trivial sum of hermitian squares.
(ii) −1 is a sum of hermitian squares.
(iii) Each symmetric element of D, that is, x ∈ D | x * = x, is a sum of hermitian squares.

Main Results
The following propositions give relations between levels and lengths of some elements.So, when we deal with the levels of the algebras Q and O equipped with the standard involutions, it is suffices to consider division algebras.But, this is not true for quaternion algebra and the algebra Q( √ c) equipped both with involution .For instance, take Q 1 = (−2, 3)/Q and Q 1 ( √ 3).From equations ( 4) and (5), it is clear that s is not a division algebra with the multiplication given by (**), because (−j + e)(j + e) = 0. We will prove in the example 3.
for some α r , β r , γ r , δ r ∈ F. Multiplying both sides of this equality by d we get (ii) Take representations of −a, −b and ab as sums of m squares in F. Replacing −a, −b and ab in the equation ( 2) by these sums, it yields a representation of −1 as a sum of n + 3mn squares in F. Thus s(F ) ≤ n(3m + 1).
(iii) (a) =⇒ (b) We will consider Proof: The proofs of (i) and (ii) are similar to the proof of (i) and (ii) of the Proposition 3.2, respectively.

5 )
Since D can be noncommutative and nonassociative ring, we need of the following set Assoc(D, * ) =: {B ∈ D such that B is invertible and (B −1 ) * (X * X) B −1 = (XB −1 ) * (XB −1 ), for every X ∈ D}.If − is the standard involution on O, then Assoc(O, −) = {B ∈ O | B is inverti− ble}, because XX = N (X) ∈ F for every X ∈ O, and N is multiplicative.In particular if O is a division algebra then Assoc(O,−) = Ȯ.Definition 2.2 The hermitian form n × 1 over (D, * ) is isotropic if there exist x 1 , x 2 , . . ., x n ∈ D, at least one x i ∈ Assoc(D, * ), such that n i=1 x * i x = 0. Lemma 2.1 Let Q( √ c) be the algebra given as earlier with multiplication ( * * ).
ie, je, ke} if, and only if, A ∈ F ∪F i∪F e.A straightforward calculation shows it.The next two lemmas have the same proof of the ( [2], Lemma 1.1 and Lemma 1.2) and ( [4], Corollary 3.3 and Lemma 3.4).Lemma 2.2 Let D a ring equipped with a non-trivial involution * such that Assoc(D, * ) is non-empty.Then s D, * = n, if and only if, the hermitian form n × 1 is anisotropic but the hermitian form (n + 1) × 1 is isotropic.

Lemma 2 . 3
Let D a ring equipped with a non-trivial involution * such that Assoc(D, * ) is non-empty.The following statements are equivalent: