We extend the Siegel-Weil formula to all quaternion dual pairs. Applications include the classification problem of skew hermitian forms over a quaternion algebra over a number field and a product formula for the weighted average of the representation numbers of a skew hermitian form by another skew hermitian form.

We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex quaternion matrices, and so on. AMS Mathematics Subject Classification: 15A06; 15A24; 15A33

Constructions of quaternion and octonion algebras, suggested to have new level and sublevel values, are proposed and justified. In particular, octonion algebras of level and sublevel 6 and 7 are constructed. In addition, Hoffmann’s proof of the existence of infinitely many new values for the level of a quaternion algebra is generalised and adapted.

We prove that the non-CM Q-abelian surfaces whose endomorphism algebra is a quaternion algebra are parametrized, up to isogeny, by the rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions.

Let F be a Henselian valued field with char(F ) 6= 2, and let S be an inertially split F -central division algebra with involution σ∗ that is trivial on an inertial lift in S of the field Z(S). We prove necessary and sufficient conditions for S to contain a σ∗stable quaternion F -subalgebra, and for (S, σ∗) to decompose into a tensor product of quaternion algebras. These conditions are in terms...

Inspired by Stark’s analytic proof of the finiteness of the class number of a ring of integers in an algebraic number field, we give a new proof of the finiteness of the number of classes of ideals in a maximal order of a quaternion division algebra over a totally real number field. Previous proofs of this well-known result have used adeles or geometry of numbers, while our proof uses the class...

We classify the quadratic extensions K = Q[ √ d] and the finite groups G for which the group ring oK [G] of G over the ring oK of integers of K has the property that the group U1(oK [G]) of units of augmentation 1 is hyperbolic. We also construct units in the Z-order H(oK) of the quaternion algebra H(K) = ` −1, −1 K ́ , when it is a division algebra. Mathematics Subject Classification. Primary [...

For a family of local algebras of semidihedral type over an algebraically closed field of characteristic 2, the Hochschild cohomology algebra is described in terms of generators and relations. The calculations are based on the construction of a bimodule resolution for the algebras in question. As a consequence, the Hochschild cohomology algebra is described for the group algebras of semidihedra...

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra ...

We introduce a generalization of the affine moment invariants for color textures recognition using Quaternion Algebra and present results of experiments on a challenging dataset. The advantage of the proposed method is that it can process color image directly, without losing color information. It is shown that the QAMIs can be obtained from the usual AMIs. Experiments show that color texture de...