For two diierent integral forms K of the exceptional Jordan algebra we show that Aut K is generated by octave reeections. These providègeometric' examples of discrete reeection groups acting with nite covolume on the octave (or Cayley) hyperbolic plane O H 2 , the exceptional rank one symmetric space. (The isometry group of the plane is the exceptional Lie group F 4(?20) .) Our groups are deene...

In this article, we study geometric unitaries of JB-algebras (in particular, self-adjoint parts of C∗-algebras). We will show that the geometric unitaries of a JB algebra are precisely those central algebraic unitaries (or central symmetries). We will also show that the group of surjective isometries on a JB-algebra A (with point-norm topology) is the semi-direct product of the canonical action...

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives functions. We propose simple logarithmically homogeneous barriers we derive efficient, num...

We show that, if there exists a realization of a Hopf algebra H in a H-module algebra A, then one can split their cross-product into the tensor product algebra of A itself with a subalgebra isomorphic to H and commuting with A. This result applies in particular to the algebra underlying inhomogeneous quantum groups like the Euclidean ones, which are obtained as cross-products of the quantum Euc...

Velásquez and Felipe recently introduced quasi-Jordan algebras based on the product a / b = 1 2 (a a b + b ` a) in an associative dialgebra with operations a and `. We determine the polynomial identities of degree ≤ 4 satisfied by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity. Loday [1, 2] defined an (associa...