Composition algebra of a weighted projective line

TL;DR

This paper explores the algebraic structure of the Hall algebra of coherent sheaves on weighted projective lines, revealing new realizations and embeddings of quantized enveloping algebras related to affine and toroidal Lie algebras.

Contribution

It introduces a novel realization of quantized enveloping algebras of affine Lie algebras and establishes new embeddings, advancing the understanding of their structure and bases.

Findings

New realization of quantized affine Lie algebras

Embeddings between different algebra classes

Construction of PBW-type bases

Abstract

In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of quantized enveloping algebras of affine Lie algebras of simply-laced type and get new embeddings between such algebras. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.