Configurations in abelian categories. II. Ringel-Hall algebras

This is the second in a series math.AG/0312190, math.AG/0410267, math.AG/0410268 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration (\sigma,\iota,\pi) is a finite collection of objects \sigma(J) and morphisms \iota(J,K) or \pi(J,K) : \sigma(J) --> \sigma(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces of (I,<)-configurations in A, using the theory of Artin stacks. It proved well-behaved moduli stacks Obj_A, M(I,<)_A of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective K-scheme P, or mod-KQ of representations of a quiver Q. Write CF(Obj_A) for the vector space of constructible functions on Obj_A. Motivated by Ringel-Hall algebras, we define an associative multiplication * on CF(Obj_A) using pushforwords and pullbacks along 1-morphisms between the M(I,<)_A, making CF(Obj_A) into an algebra. We also study representations of CF(Obj_A), the Lie subalgebra CF^ind(Obj_A) of functions supported on indecomposables, and other algebraic structures on CF(Obj_A). Then we generalize these ideas to stack functions SF(Obj_A), a universal generalization of constructible functions on stacks introduced in math.AG/0509722, containing more information. Under extra conditions on A we can define (Lie) algebra morphisms from SF(Obj_A) to some explicit (Lie) algebras, which will be important in the sequels on invariants counting t-(semi)stable objects in A.Comment: 66 pages, LaTeX. (v4) Minor changes, now in final for

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Invariants for Weighted Digraphs under One-Sided State Splittings

Using Matrix-Forest theorem and Matrix-Tree theorem, we present some invariants for weighted digraphs under state in-splittings or out-splittings