My research interests have two main components: nonassociative algebra and incidence geometry.  In nonassociative algebras, I have been primarily concerned with Jordan theory (Jordan algebras, Jordan pairs, etc.) and with structurable algebras and the related concept of a Kantor pair.  There are two direct ways that nonassociative structures are connected with geometries.  First, the geometry can have a nonassociative ring as "coordinates" much as the Euclidean plane has the real numbers as coordinates.  Secondly, an algebra structure can be used to describe certain subobjects which have a geometric interpretation.  For example, the nontrivial left ideals in the ring of n by n matrices over a field forms a projective geometry.  There are also indirect connections between nonassociative structures and geometries via groups and even Lie algebras.

Jordan Pairs and Hopf Algebras

A (quadratic) Jordan pair is constructed from a Z-graded Hopf algebra having divided power
sequences over all primitive elements and with three terms in the Z-grading of the primitive
elements. The notion of a divided power representation of a Jordan pair is introduced and the
universal object is shown to be a suitable Hopf algebra. This serves as a replacement for the
Tits-Kantor-Koecher construction.

      Link to this paper and other Jordan theory papers