This paper discusses quaternion L geometric weighting averaging working on the multiplicative Lie group of nonzero quaternionsH∗, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian L center of mass of a set of points, with 1 ≤ p ≤ ∞ (i.e., median, mean, L barycenter and minimax center), are particularized to the case of H∗. Two different approaches ...

We prove that the abelian K-surfaces whose endomorphism algebra is an indefinite rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions. To cite this article: X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I

We consider the string, like point particles and branes, to be an irreducible representation of semi-direct product Cartan involution invariant subalgebra [Formula: see text] its vector representation. show that preserves string charges, little algebra, is essentially Borel text]. also known physical states carry a parts this algebra.

We determine the norm Euclidean orders in a positive definite quaternion algebra over Q. Lagrange (1770) proved the four square theorem via Euler’s four square identity and a descent argument. Hurwitz [4] gave a quaternionic proof using the order Λ(2) with Z-basis: 1, i, j, 1 2 (1 + i + j + k). Here i = j = −1 and ij = −ji = k, the standard basis of the quaternions. The key property of Λ(2) is ...

Let (A,σ) be an Azumaya algebra with involution over a regular ring R. We prove that the Gersten–Witt complex of defined by Gille is isomorphic to Bayer-Fluckiger, Parimala and author. Advantages both constructions are used show exact when dim⁡R≤3, indA≤2 σ orthogonal or symplectic. This means Grothendieck–Serre conjecture holds for group R-scheme σ-unitary elements in A under same hypotheses; ...

Jordan C*-algebras go back to Kaplansky, see [20]. Let J be a complex Banach Jordan algebra, that is, a complex Banach space with commutative bilinear product x◦y satisfying x◦(x2◦y) = x2◦(x◦y) as well as ||x◦y|| ≤ ||x||·||y||, Bounded symmetric domains and generalized operator algebras 51 and suppose that on J is given a (conjugate linear) isometric algebra involution x 7→ x∗. Then J is called...