Ringel-Hall Algebras of Duplicated Tame Hereditary Algebras

TL;DR

This paper studies the structure of Ringel-Hall and composition algebras of duplicated tame hereditary algebras over finite fields, proving the existence of Hall polynomials and identifying related Lie subalgebras.

Contribution

It establishes the structure of these algebras for duplicated tame hereditary algebras and proves Hall polynomial existence for modules, revealing new algebraic and Lie subalgebra structures.

Findings

Existence of Hall polynomials for modules over duplicated tame hereditary algebras.

Structural description of Ringel-Hall and composition algebras for these algebras.

Identification of Lie subalgebras induced by the duplicated algebra.

Abstract

Let $A$ be a tame hereditary algebra over a finite field $k$ with $q$ elements, and ${\bar{A}}$ be the duplicated algebra of $A$. In this paper, we investigate the structure of Ringel-Hall algebra $\mathscr{H} (\bar{A})$ and of the corresponding composition algebra $\mathscr{C} (\bar{A})$. As an application, we prove the existence of Hall polynomials $g_{XY}^M$ for any $\bar{A}$-modules $M, X$ and $Y$ with $X$ and $Y$ indecomposable if $A$ is a tame quiver $k$-algebra, then we also obtain some Lie subalgebras induced by $\bar{A}$.