Minimal generators of Hall algebras of 1-cyclic perfect complexes

TL;DR

This paper constructs a minimal generating set and PBW-basis for the Hall algebra of 1-cyclic complexes over Dynkin quiver path algebras, revealing its structure as a universal enveloping algebra and deriving quantum Serre relations.

Contribution

It provides the first explicit minimal generators and PBW-basis for the Hall algebra of 1-cyclic complexes, connecting it to Lie algebras and quantum groups.

Findings

Hall algebra is the universal enveloping algebra of a Lie algebra generated by indecomposables

Derived quantum Serre relations in a twisted Hall algebra setting

Established relations between degenerate, twisted, and original Hall algebras

Abstract

Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathscr{P})$ be the category of 1-cyclic complexes of projective $A$-modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra $\H(C_1(\mathscr{P}))$ of $C_1(\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations in the generators in $\H(C_1(\mathscr{P}))$, and obtain quantum Serre relations in a quotient of certain twisted version of $\H(C_1(\mathscr{P}))$. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of $A$ and those of $C_1(\mathscr{P})$, respectively.