On n-ary Hom-Nambu and Hom-Nambu-Lie algebras

It is observed that the category of n-ary Hom-Nambu(-Lie) algebras is closed under twisting by self-weak morphisms. Constructions of ternary Hom-Nambu algebras from Hom-associative algebras, Hom-Lie algebras, ternary totally Hom-associative algebras, and Hom-Jordan triple systems are given. Every multiplicative n-ary Hom-Nambu algebra gives rise to a sequence of Hom-Nambu algebras of exponentially higher arities. Under some conditions, an n-ary Hom-Nambu(-Lie) algebra gives rise to an (n-1)-ary Hom-Nambu(-Lie) algebra.Comment: 23 page

Jordan triple product homomorphisms on Hermitian matrices of dimension two

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$\Phi(ABA) = \Phi(A)\Phi(B)\Phi(A)$$ on the set of all Hermitian $2 \times 2$ complex matrices.Comment: 34 page

Jordan triple product homomorphisms on Hermitian matrices to and from dimension one

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$\Phi(ABA) = \Phi(A)\Phi(B)\Phi(A)$$ from the set of all Hermitian $n \times n$ complex matrices to the field of complex numbers. Further we characterise all Jordan triple product homomorphisms from the field of complex or real numbers or the set of all nonnegative real numbers to the set of all Hermitian $n \times n$ complex matrices.Comment: 10 page

Multiplicative Cauchy functional equation on symmetric cones

We solve the multiplicative Cauchy functional equation on symmetric cones with respect to two different multiplication algorithms. We impose no regularity assumptions on respective functions.Comment: 15 page

Continuous Jordan triple endomorphisms of P2\mathbb{P}_2

We describe the structure of all continuous Jordan triple endomorphisms of the set $\mathbb{P}_2$ of all positive definite $2\times 2$ matrices thus completing a recent result of ours. We also mention an application concerning sorts of surjective generalized isometries on $\mathbb{P}_2$ and, as second application, we complete another former result of ours on the structure of sequential endomorphisms of finite dimensional effect algebras

The Octonions

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction