Erik Darpö

Normal forms for the G2-action on the real symmetric 7 × 7-matrices by conjugation

The exceptional Lie group G2 ⊂ O7(R) acts on the set of real symmetric 7 × 7-matrices by conjugation. We solve the normal form problem for this group action. In view of the earlier results [1], [3] and [4], this gives rise to a classification of all real flexible division algebras. By a classification is meant a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. We also give a parametrisation of the set of real symmetric matrices, based on eigenvalues. Mathematics Subject Classification 2000: 15A21, 17A20, 17A35, 17A36, 17A45.

On the representation ring of the polynomial algebra over a perfect field

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

Vector product algebras

{\tilde{\mathbb{A}}}

Classification of Pairs of Rotations in Finite-Dimensional Euclidean Space

is provided.

On the classification of the real flexible division algebras

We study algebras that are isotopic to Hurwitz algebras. Isomorphism classes of such algebras are shown to correspond to orbits of a certain group action. An explicit, geometrically intutive description of the category of isotopes of Hamilton’s quaternions is given. As an application, some known results concerning the classification of finite-dimensional composition algebras are deduced.

Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.

In which dimensions does a division algebra over a given ground field exists

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimension 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-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterised.

The double sign of a real division algebra of finite dimension greater than one

A rotation in a Euclidean space V is an orthogonal map δ ∈ O(V) which acts locally as a plane rotation with some fixed angle a(δ) ∈ [0,π]. We give a classification of all finite-dimensional representations of the real algebra