The notions of totally indefinite and weakly isotropic algebras with involution are introduced and a proof is given of the fact that a field satisfies the Effective Diagonalization Property (ED) if and only if it satisfies the following weak Hasse principle: every totally indefinite central simple algebra with involution of the first kind over the given field is weakly isotropic. This generaliz...

The level of a ring R with 1 6= 0 is the smallest positive integer s such that −1 can be written as a sum of s squares in R, provided −1 is a sum of squares at all. D.W. Lewis showed that any value of type 2n or 2n + 1 can be realized as level of a quaternion algebra, and he asked whether there exist quaternion algebras whose levels are not of that form. Using function fields of quadratic forms...

Let D be a positive definite quaternion algebra over a totally real number field K, F (X, Y ) a hermitian form in 2N variables over D, and Z a right D-vector space which is isotropic with respect to F . We prove the existence of a small-height basis for Z over D, such that F (X, X) vanishes at each of the basis vectors. This constitutes a non-commutative analogue of a theorem of Vaaler [19], an...

Let B be the indefinite quaternion algebra over Q of discriminant D. Let O be a maximal order in B. Let XD be the Shimura curve over Q attached to O, whose set of complex points is given by XD(C) = (Ô×\B̂× × P)/B×, where P = C\R. As it is well known, suchXD is equipped with a natural group of involutions called the Atkin-Lehner group W (D), where each involution ωn is indexed by the divisors n |...

We study polynomial identities of algebras with involution nonassociative over a field characteristic zero. prove that the growth sequence ⁎-codimensions finite-dimensional algebra is exponentially bounded. construct series fractional ⁎-PI-exponent. also family infinite-dimensional Cα such exp⁎(Cα) does not exist.

We determine the possible cohomology algebra of orbit space of any free involution on a mod-2 cohomology lens space X using the Leray spectral sequence associated to the Borel fibration X →֒ XZ2 −→ BZ2 . As an application we show that if X is a mod-2 cohomology lens space of dimension 2m − 1 where 4 ∤ m, then there does not exist any Z2-equivariant map S n → X for n ≥ 2, where S is equipped with...

Using the concept of the λ-lattice introduced recently by V. Snášel we define λ-lattices with antitone involutions. For them we establish a correspondence to ring-like structures similarly as it was done for ortholattices and pseudorings, for Boolean algebras and Boolean rings or for lattices with an antitone involution and the so-called Boolean quasirings.

Let G be a connected reductive algebraic group defined over an algebraically closed field F of characteristic not 2. Denote the Lie algebra of G by 9. In this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of G. This is shown to be independent of the characteristic of F and can be applied to describe all semisimple locally symmetric spac...

For scenarios of coherent underwater sources at low signal-to-noise ratio (SNR), a novel quaternion-based DOA algorithm without eigendecomposition is proposed using linear vector-hydrophone array. We construct four quaternion models by judiciously arranging the received data to fully utilize statistical information incident signals. To avoid high computational complexity caused eigenvalue decom...

In this paper algebras with identity, over ordered fields, which possess a (weak) submultiplicative positive bilinear form are studied. They are called (weak) subdecomposition algebras. It is proved that every weak subdecomposition algebra is a simple quadratic algebra with no nontrivial nilpotents or idempotents. If a weak subdecomposition algebra is also flexible, then it is a noncommutative ...