The hermitian level of composition algebras

Abstract

Abstract. The hermitian level of composition algebras with involution over a ring is studied. In particular, it is shown that the hermitian level of a composition algebra with standard involution over a semilocal ring, where two is invertible, is always a power of two when finite. Furthermore, any power of two can occur as the hermitian level of a composition algebra with nonstandard involution. Some bounds are obtained for the hermitian level of a composition algebra with involution of the second kind.

Finiteness of Hermitian Levels of some Algebras

“…Hence s(Q( √ c), ) ≤ 7.2 i−3 . But, we see that, if there exists a solution for equation (1), we will take γ r = δ r = 0 to get a solution for (4). Similarly, a solution of (4) can be completed to a solution of (5), by taking x 1r = α r , x 2r = β r , x 3r = γ r , x 4r = δ r and y ir = 0, for all i and r. It is easy to relate the hermitian level of (Q( √ c), ) to the u-invariant of the base field F. The u-invariant of F, u(F ) is the maximal dimension of an anisotropic quadratic form over F. See [1] and [5], for more on the u-invariant.…”

“…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].…”

“…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.…”

“…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).…”