Geometric methods in the theory of Hall algebras.

摘要

Let A be an associative C -algebra. Given three A-modules M 1, M2, and M3, which are finite dimensional over C , consider an algebraic variety NM3M1 M2 of all submodules X of M3 such that X is isomorphic to M1, and M3/X is isomorphic to M2. The varieties NM3M1 M2 can be used to define structure constants of a new associative algebra called the Hall algebra of A. The purpose of this manuscript is to study geometry of the varieties NM3M1 M2 and the structure of the Hall algebra using interrelations between them.;First let A = C [[x]]. Finite dimensional C [[x]]-modules are labeled by partitions, and a theorem of P. Hall implies that the number of irreducible components of NMgM aMb is equal to the Littlewood-Richardson coefficient cgab which appears in the tensor product decomposition for representations of GL(N). The first chapter of this manuscript explains the role of the varieties NMgM aMb in the tensor product, and thus provides a direct proof of the Hall theorem. It employs new "tensor product" varieties, M. Kashiwara's theory of crystals, and a geometric construction of representations of GL(N) due to V. Ginzburg. As a generalization a new family of varieties is introduced and used to describe tensor products of representations of simple simply laced Lie algebras. These varieties are related to quiver varieties of H. Nakajima.;In the second chapter A is the path algebra C Q of a quiver Q. C. M. Ringel proved that if Q is of Dynkin type then the corresponding Hall algebra is isomorphic to the universal enveloping algebra of the nilpotent radical of a Borel subalgebra of the simple Lie algebra associated to Q. The second chapter contains a generalization of the Ringel's result to the affine case. The proofs use a new technique based on functorial properties of the Hall algebra with respect to maps of quivers. In particular, a simple proof of the original Ringel's theorem for a Dynkin quiver is given. The applications provided include a canonical integral form of an affine Lie algebra, and some properties of affine root systems.