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Enveloping algebras via motivic Hall algebras

Xinyi Feng, Fan Xu

Abstract

We give a geometric realization of the whole universal enveloping algebra of the Borcherds-Bozec algebra using quiver with loops via the motivic semi-derived Hall algebra approach. In particular, using acyclic quivers, we give a geometric realization of the whole universal enveloping algebra of a certain generalized Kac-Moody algebra using the motivic Bridgeland's Hall algebra given in [12].

Enveloping algebras via motivic Hall algebras

Abstract

We give a geometric realization of the whole universal enveloping algebra of the Borcherds-Bozec algebra using quiver with loops via the motivic semi-derived Hall algebra approach. In particular, using acyclic quivers, we give a geometric realization of the whole universal enveloping algebra of a certain generalized Kac-Moody algebra using the motivic Bridgeland's Hall algebra given in [12].
Paper Structure (22 sections, 40 theorems, 251 equations)

This paper contains 22 sections, 40 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Realization of enveloping algebras via acyclic quivers
  4. Quivers with loops
  5. Main result
  6. Structure of the paper
  7. Acknowledgment
  8. Motivic form of semi-derived Ringel-Hall algebras
  9. Moduli stacks of objects and filtrations in C2A
  10. Motivic Hall algebra for C2A
  11. Fibers of the convolution map
  12. Localization and reduced quotient
  13. Triangular decomposition
  14. Realization of enveloping algebras of Borcherds-Bozec algebras
  15. Borcherds-Bozec algebras and Quantum Borcherds-Bozec algebras
  16. ...and 7 more sections

Key Result

Theorem 1.1

There is an injective homomorphism of $\mathbb{C}$-algebras

Theorems & Definitions (80)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Corollary 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Remark 2.6
  • ...and 70 more