Matrix exponential via Clifford algebras

We use isomorphism $\varphi$ between matrix algebras and simple orthogonal Clifford algebras \cl(Q) to compute matrix exponential ${e}^{A}$ of a real, complex, and quaternionic matrix A. The isomorphic image $p=\varphi(A)$ in \cl(Q), where the quadratic form $Q$ has a suitable signature $(p,q),$ is exponentiated modulo a minimal polynomial of $p$ using Clifford exponential. Elements of \cl(Q) are treated as symbolic multivariate polynomials in Grassmann monomials. Computations in \cl(Q) are performed with a Maple package `CLIFFORD'. Three examples of matrix exponentiation are given

The arithmetic of QM-abelian surfaces through their Galois representations

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.Comment: To appear in J. Algebr

A Cauchy kernel for slice regular functions

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results

On the Kauffman bracket skein module of the quaternionic manifold

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, we determine that they are linearly independent.Comment: corrected summation signs in figures 14, 15, 17. Other minor change

Solution of a quadratic quaternion equation with mixed coefficients

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space $\mathbb{H}$. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space $\mathbb{H}$.Comment: 19 pages, to appear in the Journal of Symbolic Computatio