Representations of Tame Quivers and Affine Canonical Bases

Abstract

There are several different ways to construct affine canonical bases, in addition to approaches by Lusztig and Kashiwara. In this paper we present a different approach to canonical bases via Hall algebras and representations of tame quivers over finite fields. The main idea is to tensor together integral bases constructed for cyclic quivers and Kronecker quivers with those from the preinjective and preprojective parts of tame quiver representations. Several different bases: a PBW type basis, a monomial basis, and a barinvariant basis are constructed and their relations to the canonical basis are discussed. The result also answers a question by Nakajima.

A parameterization of the canonical bases of affine modified quantized enveloping algebras

“…In this subsection, we shall review the definition of Ringel-Hall algebras (see [5,10,18]). A quiver Q = (I, H, s, t) consists of a vertex set I, an arrow set H, and two maps s, t : H → I such that an arrow ρ ∈ H starts at s(ρ) and terminates at t(ρ).…”

“…Inspired by the method of Peng and Xiao, we want to study the relations between the canonical basisḂ and the corresponding root category R. In this paper, first we associate a set M to R. In [10], Lin, Xiao and Zhang associated a set M to a hereditary category and the definition of M is based on that of M. However, M is independent of the embedding of the hereditary category to R. Fixing an embedding of the hereditary category to R, we can get a bijection between M and the canonical basisḂ λ ofU1 λ for every λ ∈ P . Hence we say that the set M provides a parameterization of the canonical basisḂ.…”

“…The geometric construction of B was generalized to the cases of all types in [12]. In the case of affine type, Lin, Xiao and Zhang in [10] provided a process to construct a PBW-type basis of U + and the canonical basis B by using the Ringel-Hall algebra approach.…”

“…In Section 4, we study the case of affine type. We first review the construction of the PBW-type basis of U + in [10]. Then we define a set M depending on the corresponding root category R and a PBW-type basis ofU1 λ with M as an index.…”

Geometric realizations of Lusztig's symmetries

“…Hall algebra approach to f. In this subsection, we shall review the Hall algebra approach to f ( [18,15,8,9]). …”