How to obtain division algebras from a generalized Cayley-Dickson doubling process

New families of eight-dimensional real division algebras with large derivation algebra are presented: We generalize the classical Cayley-Dickson doubling process starting with a unital algebra with involution over a field F by allowing the scalar in the doubling to be an invertible element in the algebra. The resulting unital algebras are neither power-associative nor quadratic. Starting with a quaternion division algebra D, we obtain division algebras A for all invertible scalars chosen in D outside of F. This is independent on where the scalar is placed inside the product and three pairwise non-isomorphic families of eight-dimensional division algebras are obtained. Over the reals, the derivation algebra of each such algebra A is isomorphic to $su(2)\oplus F$ and the decomposition of A into irreducible su(2)-modules has the form 1+1+3+3 (denoting an irreducible su(2)-module by its dimension). Their opposite algebras yield more classes of pairwise non-isomorphic families of division algebras of the same type. We thus give an affirmative answer to a question posed by Benkart and Osborn in 1981.Comment: 23 pages; extended versio

Albert's twisted field construction using division algebras with a multiplicative norm

Albert's classical construction of twisted fields generates unital division algebras out of cyclic field extensions. We apply a generalized version to unital division algebras with a multiplicative norm and give conditions for the resulting twisted algebras to be division. Four- and eight-dimensional real unital division algebras with large derivation algebras are constructed out of Hamilton's quaternion and Cayley's octonion algebra.Comment: 19 page

Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes

Let S be a unital ring, S[t; σ, δ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ε S[t; σ, δ] has degree m and an invertible leading coefficient. Using right division by f to define the multiplication, we obtain unital nonassociative algebras Sf on the set of skew polynomials in S[t; σ, δ] of degree less than m. We study the structure of these algebras. When S is a Galois ring and f base irreducible, these algebras yield families of finite unital nonassociative rings A, whose set of (left or right) zero divisors has the form pA for some prime p. For reducible f, the Sf can be employed both to design linear (f, σ, δ)-codes over unital rings and to study their behaviour

The eigenspaces of twisted polynomials over cyclic field extensions

Let $K$ be a field and $\sigma$ an automorphism of $K$ of order $n$. We study the eigenspace of a bounded skew polynomial $f\in K[t;\sigma]$, with emphasis on the case of a cyclic field extension $K/F$ of degree $n$, where $\sigma$ generates the Galois group. We obtain lower bounds on its dimension, and compute it in special cases.Comment: Rewritten and streamlined new version, some results are improve