平成19年7月9日(月), 理学部C棟3階C３０９ にて
13:00-14:00 Hiroki Abe (University of Tsukuba, Japan) “Noether algebras of finite selfinjective dimension”
14:30-15:30 Colin Ingalls (University of New Brunswick, Canada) “Noncrossing partitions and quiver representations”
Abstract: Joint work with Hugh Thomas. We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. Chief among these is the fact that the finitely-generated, exact abelian, and extension-closed subcategories of the representations of a quiver $Q$ without oriented cycles are in natural bijection with the cluster-tilting objects in the associated cluster category.
16:00-17:00 Daniel Chan (University of New South Wales, Australia) “McKay correspondence for canonical orders”
Abstract: The McKay correspondence is a beautiful theory linking singularities in algebraic geometry to non-commutative algebra and group theory. Classically, it is concerned with a finite subgroup $G$ of $SL_2$ acting on the affine plane $X$. The Kleinian singularity $X/G$ can be studied via both its minimal resolution $Y$ and the skew group ring $k[[x,y]]*G$. The McKay correspondence relates $Y$ to $k[[x,y]]*G$. For example, the number of indecomposable projective $k[[x,y]]*G$-modules is one more than the number of exceptional curves in $Y$. In joint work with Hacking and Ingalls, we generalised the notion of Kleinian singularities to certain non-commutative algebras dubbed canonical orders. They too can be studied via a minimal resolution $Y$, which this time is also non-commutative, as well as a certain cross product algebra $A$. Our version of the McKay correspondence for canonical orders relates this non-commutative $Y$ to $A$. The talk will begin by reviewing the classical case and introducing canonical orders. We will then describe what is known about the McKay correspondence for these orders.