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In this article, we explore new aspects in the classification of non-associative division algebras. By a detailed study of the representations of the Lie group of autotopies of real division algebras we show that, if the group of autotopies has a sufficiently rich structure, then the algebra is isotopic to one of the classical real division algebras. This turns out to be the case for large classes of real division algebras, including many that are defined by identities. In several cases, a classification up to isomorphism can be worked out from this information.

While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensorproducts of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral 2-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.

We give a new formula for the decomposition of a tensor product of indecomposable modules of cyclic two-groups. This formula is also shown to describe thedecomposition of tensor products of an important class of modules of dihedral two-groups

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a∈ A{set minus}{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category of all n-dimensional realdivision algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of.

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative realdivision algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensionalpower-commutative real division algebras that have a unique non-zero idempotent are characterized.

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramirez Alvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent. 

We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type à is provided.

The study of real division algebras was initiated by the construction of the quaternion and the octonion algebras in the mid-19th century. In spite of its long history, the problem, of classifying all finite-dimensional real division algebras is still unsolved. We review the theory of this problem, with focus on recent contributions.