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Hall algebras and quantum symmetric pairs of Kac-Moody type

Ming Lu, Weiqiang Wang

Abstract

We extend our $\imath$Hall algebra construction from acyclic to arbitrary $\imath$quivers, where the $\imath$quiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal $\imath$quantum group of Kac-Moody type arising from quantum symmetric pairs to the $\imath$Hall algebra associated to a virtually acyclic $\imath$quiver.

Hall algebras and quantum symmetric pairs of Kac-Moody type

Abstract

We extend our Hall algebra construction from acyclic to arbitrary quivers, where the quiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal quantum group of Kac-Moody type arising from quantum symmetric pairs to the Hall algebra associated to a virtually acyclic quiver.

Paper Structure

This paper contains 41 sections, 50 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. $\imath$Quiver algebras and homological properties
  3. Notations
  4. The $\imath$quiver algebras
  5. $\Lambda^\imath$ as a $1$-Gorenstein algebra
  6. Modules of finite projective dimensions
  7. Singularity categories
  8. The $\imath$Hall algebras
  9. Euler forms
  10. Semi-derived Hall algebras for $\imath$quiver algebras
  11. An $\imath$Hall basis
  12. $\imath$Hall algebras for $\imath$subquivers
  13. An $\imath$Hall multiplication formula
  14. Quantum symmetric pairs and $\imath$quantum groups
  15. Quantum groups
  16. ...and 26 more sections

Key Result

Lemma \oldthetheorem

LW19a A $\Lambda^{\imath}$-module $X=(X_i,X(\alpha), X(\varepsilon_i))_{i\in Q_0,\alpha\in Q_1}$ is isomorphic to an indecomposable projective $\Lambda^\imath$-module $\Lambda^\imath e_j$ if and only if for some $j\in Q_0$; see eqn:rigt adjoint. In particular, we have a short exact sequence in $\operatorname{mod^{\rm nil}}\nolimits(\Lambda^\imath):$

Theorems & Definitions (93)

  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 83 more