Derivations of Leavitt path algebra

TL;DR

This paper characterizes the outer derivations of Leavitt path algebras for row-finite graphs, providing explicit formulas and exploring their algebraic structures and extensions to $C^*$-algebras.

Contribution

It offers explicit formulas for outer derivations of Leavitt path algebras and analyzes their Lie algebra structure and extensions.

Findings

Explicit formulas for outer derivations of Leavitt path algebras

Description of the Lie algebra structure of outer derivations

Extension of derivations to $C^*$-algebras

Abstract

In this paper, we describe the $K$-module $HH^1(L_K(\Gamma))$ of outer derivations of the Leavitt path algebra $L_K(\Gamma)$ of a row-finite graph $\Gamma$ with coefficients in an associative commutative ring $K$ with unit. We give an explicit formula for every outer derivation of $L_K(\Gamma)$. We also describe the Lie algebra structure of outer derivations of the Toeplitz algebra and we prove that every derivation of the Leavitt path algebra can be extended to a derivation of the corresponding $C^*$-algebra.