On Lie algebras associated with representation directed algebras

TL;DR

This paper proves that for any representation-directed algebra over the complex numbers, the Lie algebra defined by Riedtmann is isomorphic to the one defined by Ringel, establishing a fundamental equivalence.

Contribution

The paper demonstrates the isomorphism between two Lie algebras associated with representation-directed algebras, clarifying their relationship.

Findings

$L(B)$ and $ ext{CK}(B)$ are isomorphic for any representation-directed algebra $B$

Establishes a fundamental link between Riedtmann's and Ringel's Lie algebras

Provides a unified understanding of Lie algebras in the context of representation-directed algebras

Abstract

Let $B$ be a representation-finite $\mathbb{C}$-algebra. The $\mathbb{Z}$-Lie algebra $L(B)$ associated with $B$ has been defined by Ch. Riedtmann. If $B$ is representation-directed there is another $\mathbb{Z}$-Lie algebra associated with $B$ defined by C. M. Ringel and denoted by $\CK(B)$. We prove that the Lie algebras $L(B)$ and $\CK(B)$ are isomorphic for any representation-directed $\mathbb{C}$-algebra $B$.