Modified Ringel-Hall algebras, naive lattice algebras and lattice algebras

TL;DR

This paper establishes isomorphisms and epimorphisms between modified Ringel-Hall algebras, naive lattice algebras, and lattice algebras for hereditary abelian categories, and shows invariance under derived equivalences.

Contribution

It demonstrates the isomorphism between modified Ringel-Hall algebras and naive lattice algebras, and describes the kernel of the epimorphism to lattice algebras, extending understanding of their structure.

Findings

Modified Ringel-Hall algebra is isomorphic to naive lattice algebra.

There exists an explicit epimorphism from the modified Ringel-Hall algebra to the lattice algebra.

Naive lattice algebra is invariant under derived equivalences.

Abstract

For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.