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Hall Algebras and Edge Contractions with Loops

Adhish Rele

TL;DR

This paper addresses how Hall algebras behave under edge contractions in graphs with multiple edges and loops. It develops a broad framework, extending Li's approach, by formulating a generalized Cartan datum $(I,\cdot,\phi_1,\phi_2)$, a corresponding root datum, and Weyl groups, then building quiver representation spaces with loops and Lusztig diagrams. The main achievement is proving an embedding $\psi = j_! \circ \mu^\star$ of the contracted Hall algebra ${\bf H}_{\widehat{\Omega}}$ into the original ${\bf H}_{\Omega}$, preserving multiplication and Hopf structure, along with a PBW-basis compatibility and a split short exact sequence showing ${\bf H}_{\widehat{\Omega}}$ as a split subquotient of ${\bf H}_{\Omega}$. These results generalize Li's edge-contraction embeddings to settings with loops and multiple edges, with potential implications for categorification and connections to quantum groups.

Abstract

We extend the study of Hall algebras and edge contractions by generalizing Yiqiang Li's work to contraction along vertices with multiple edges. Using the edge contractions, we establish new embeddings among Hall algebras in this broader setting. Our results demonstrate that these embeddings preserve key algebraic structures, including Hopf algebra operations.

Hall Algebras and Edge Contractions with Loops

TL;DR

This paper addresses how Hall algebras behave under edge contractions in graphs with multiple edges and loops. It develops a broad framework, extending Li's approach, by formulating a generalized Cartan datum , a corresponding root datum, and Weyl groups, then building quiver representation spaces with loops and Lusztig diagrams. The main achievement is proving an embedding of the contracted Hall algebra into the original , preserving multiplication and Hopf structure, along with a PBW-basis compatibility and a split short exact sequence showing as a split subquotient of . These results generalize Li's edge-contraction embeddings to settings with loops and multiple edges, with potential implications for categorification and connections to quantum groups.

Abstract

We extend the study of Hall algebras and edge contractions by generalizing Yiqiang Li's work to contraction along vertices with multiple edges. Using the edge contractions, we establish new embeddings among Hall algebras in this broader setting. Our results demonstrate that these embeddings preserve key algebraic structures, including Hopf algebra operations.

Paper Structure

This paper contains 11 sections, 15 theorems, 56 equations, 3 figures.

Table of Contents

  1. Foundations
  2. Generalized Cartan data
  3. Graph Structures
  4. Root datum
  5. Weyl Groups
  6. Embedding among Hall Algebras associated to Quivers with Loops
  7. Quiver Representation Spaces
  8. Lusztig's Induction and Restriction Diagram
  9. Embedding among Hall Algebras
  10. Compatibility of $j_!\mu_\nu^*$ with the PBW basis
  11. Split short exact sequence of Hall Algebras

Key Result

Theorem A

For a quiver with multiple edges between two vertices along which the contraction takes place, the edge contraction operation induces an embedding of the Hall algebra of the contracted quiver into the associated Hall algebra prior to the edge contraction. This embedding is compatible with the multip

Figures (3)

  • Figure 1: An example of a graph with 6 vertices and 3 a-orbits drawn as ellipses. The red edges are loops and the blue edges are the between different vertices
  • Figure 2: The first graph is before the edge contraction and the second graph is after the edge contraction. The red edges denote the edges along which the contraction takes place. The blue edges in the first graph form loops after the contraction.
  • Figure 3: The first graph is before the edge contraction and the second graph is after the edge contraction. The red edges denote the edges along which the contraction takes place. The blue edges in the first graph form loops after the contraction.

Theorems & Definitions (32)

  • Theorem A: Theorem \ref{['main embedding']}
  • Theorem B: Theorem \ref{['split subquotient']}
  • Example 1.1.1
  • Lemma 1.1.2
  • proof
  • Lemma 1.2.1
  • proof
  • Proposition 1.2.2
  • proof
  • Definition 1.2.3
  • ...and 22 more