[Paper Review] Jordan Derivations and Lie derivations on Path Algebras

Yanbo Li, Feng Wei|arXiv (Cornell University)|Mar 22, 2012

Advanced Topics in Algebra18 references3 citations

TL;DR

This paper establishes that every Jordan derivation on a path algebra of a quiver without oriented cycles is a derivation, and every Lie derivation on such algebras is of the standard form, without requiring the faithful module assumption. The results are proven via structural analysis of path algebras and decomposition theorems for derivations and Lie derivations, extending prior work that relied on faithfulness.

ABSTRACT

Without the faithful assumption, we prove that every Jordan derivation on a class of path algebras of quivers without oriented cycles is a derivation and that every Lie derivation on such kinds of algebras is of the standard form.

Motivation & Objective

• To investigate Jordan derivations and Lie derivations on path algebras of quivers without oriented cycles.
• To determine whether these derivations are of standard form without assuming the module is faithful.
• To extend existing results on matrix algebras and triangular algebras to path algebras under weaker assumptions.
• To characterize the structure of Lie derivations and Jordan derivations in the absence of faithfulness.

Proposed method

• Analyzes path algebras of quivers without oriented cycles as one-point extensions of simpler algebras.
• Uses the structure of path algebras to decompose derivations and Lie derivations into components involving derivations and central maps.
• Applies Lemma 4.1 to decompose Lie derivations into a derivation and a central-valued linear map.
• Employs Theorem 4.4 to show that Lie derivations on a subalgebra are of standard form.
• Uses Lemma 4.6 to characterize derivations on path algebras via coefficient conditions on paths.
• Applies Lemma 4.7 to show that the central part of a Lie derivation must act as a scalar on each trivial path.

Experimental results

Research questions

• RQ1 Are Jordan derivations on path algebras of quivers without oriented cycles necessarily derivations without the faithful assumption?
• RQ2 Can every Lie derivation on such path algebras be expressed in the standard form (derivation plus central map) without faithfulness?
• RQ3 How do the structures of Jordan and Lie derivations change when the faithful module condition is removed?
• RQ4 What conditions on the quiver or path algebra ensure that derivations are determined by their action on paths and vertices?

Key findings

• Every Jordan derivation on a path algebra of a quiver without oriented cycles is a derivation, even without the faithful assumption.
• Every Lie derivation on such a path algebra is of the standard form, i.e., it decomposes into a derivation and a central-valued linear map.
• The central part of a Lie derivation must act as a scalar on each trivial path, and these scalars are equal across the quiver due to connectivity.
• The structure of derivations on path algebras is completely determined by coefficient conditions on paths, as formalized in Lemma 4.6.
• The standard form of Lie derivations is preserved under restriction to subalgebras, as shown via Theorem 4.4 and Lemma 4.1.
• An explicit example shows a Lie derivation that is not a derivation when the central part is non-scalar, confirming the necessity of the standard form decomposition.