chapter  5
74 Pages

Semisimple Representations

For a Cayley-Hamilton algebra A ∈ [email protected] we have seen that the quotient scheme

trissn A = trepn A/GLn

classifies isomorphism classes of (trace preserving) semisimple n-dimensional representations. A point ξ ∈ trissn A is said to lie in the Cayley-smooth locus of A if trepn A is a smooth variety in the semisimple module Mξ determined by ξ. In this case, the e´tale local structure of A and its central subalgebra tr(A) are determined by a marked quiver setting. We will extend some results on quotient varieties of representations of quiv-

ers to the setting of marked quivers. We will give a computational method to verify whether ξ belongs to the Cayley-smooth locus of A and develop reduction steps for the corresponding marked quiver setting that preserve geometric information, such as the type of singularity. In low dimensions we can give a complete classification of all marked quiver

settings that can arise for a Cayley-smooth order, allowing us to determine the classes in the Brauer group of the function field of a projective smooth surface, which allow a noncommutative smooth model. In the arbitrary (central) dimension we are able to determine the smooth

locus of the center as well as to classify the occurring singularities up to smooth equivalence.