In this paper we apply the method of functional identities to the study of group gradings by an abelian group ▫$G$▫ on simple Jordan algebras, under very mild restrictions on the grading group or the base field of coefficients.
COBISS.SI-ID: 15803225
The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to ▫$\mathbb{C}$▫. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions.
COBISS.SI-ID: 15758681
This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings fromvone non-commutative domain ▫$\mathcal{D}$▫ in say ▫$g$▫ variables to another non-commutative domain ▫$\tilde{\mathcal{D}}$▫ in ▫$\tilde{g}$▫ variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of ▫$\mathcal{D}$▫ into the boundary of ▫$\tilde{\mathcal{D}}$▫. Assuming that both domains contain 0, we show that if ▫$f \colon \mathcal{D} \to \tilde{\mathcal{D}}$▫ is a proper analytic free map, and ▫$f(0)=0$▫, then ▫$f$▫ is one-to-one. Moreover, if also ▫$g = \tilde{g}$▫, then $f$ is invertible and ▫$f^{-1}$▫ is also an analytic free map. These conclusions on the map ▫$f$▫ are the strongest possible without additional assumptions on the domains ▫$\mathcal{D}$▫ and ▫$\tilde{\mathcal{D}}$▫.
COBISS.SI-ID: 15866201