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On the Ringel--Hall algebra of the gentle one-cycle algebra $Λ(n-1,1,1)$

Hui Chen, Dong Yang

Abstract

It is shown that the gentle one-cycle algebra $Λ(n-1,1,1)$ has Hall polynomials. The Hall polynomials are explicitly given for all triples of indecomposable modules, and as a consequence, the Ringel--Hall Lie algebra of $Λ(n-1,1,1)$ is shown to be isomorphic to its Riedtmann Lie algebra.

On the Ringel--Hall algebra of the gentle one-cycle algebra $Λ(n-1,1,1)$

Abstract

It is shown that the gentle one-cycle algebra has Hall polynomials. The Hall polynomials are explicitly given for all triples of indecomposable modules, and as a consequence, the Ringel--Hall Lie algebra of is shown to be isomorphic to its Riedtmann Lie algebra.
Paper Structure (8 sections, 13 theorems, 20 equations)

This paper contains 8 sections, 13 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ringel--Hall algebras
  4. Representations of the algebra $\Lambda(n-1,1,1)$
  5. The Ringel--Hall algebra of $\Lambda(n-1,1,1)$
  6. The existence of Hall polynomials
  7. Hall polynomials for indecomposable modules
  8. The Ringel--Hall Lie algebra

Key Result

Theorem 1.1

$A$ has Hall polynomials.

Theorems & Definitions (17)

  • Theorem 1.1: Theorem \ref{['Hall-exist-for-A']}
  • Theorem 1.2: Theorem \ref{['thm:iso-between-g(A)-and-L(A)']}
  • Proposition 2.1: Ringel90
  • Lemma 2.2
  • Definition 2.3: Ringel92
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6: Nasr12
  • Theorem 2.7
  • ...and 7 more