LetA be a finite-dimensional commutative algebra over the complex numbers with nonzero multiplicative identity element e. Let M denote the set of homomorphisms from A onto the complex numbers. Thus we get a homomorphism from A into the algebra of complex-valued functions on M, with respect to pointwise addition and multiplication of functions, by sending an element x of A to the function on M w...

LetA be a finite-dimensional commutative algebra over the complex numbers, with identity element e. Thus A is a finite-dimensional complex vector space equipped with an additional binary operation of multiplication which satisfies the usual rules of associativity, commutativity, and distributivity, and e is a nonzero element of A such that e x = x for all x ∈ A. As a basic class of examples, on...

One proves that the Mazur-Ulam theorem can be extended in the framework of metric spaces as long as a well behaved concept of midpoint is available. This leads to the new concept of Mazur-Ulam space. Besides the classical case of real normed spaces, other examples such as Sym++(n,R), the space of all n × n dimensional positive definite matrices, appear as cones attached to suitable Euclidean Jo...

Reframing certain well-known particle models in terms of normed division algebras leads to two new results for BSM physics. (1) We identify a sequence complex structures which induces cascade breaking symmetries: Spin(10) $\mapsto$ Pati-Salam Left-Right symmetric Standard model + B-L (both pre- and post-Higgs-mechanism). These derive from the octonions, then quaternions, numbers. (2) provide, a...