12 research outputs found
Variety of idempotents in nonassociative algebras
In this paper, we study the variety of all nonassociative (NA) algebras from
the idempotent point of view. We are interested, in particular, in the spectral
properties of idempotents when algebra is generic, i.e. idempotents are in
general position. Our main result states that in this case, there exist at
least nontrivial obstructions (syzygies) on the Peirce spectrum of a
generic NA algebra of dimension . We also discuss the exceptionality of the
eigenvalue which appears in the spectrum of idempotents in
many classical examples of NA algebras and characterize its extremal properties
in metrised algebras.Comment: 27 pages, 1 figure, submitte
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 A is provided.Comment: 17 page