Communications of the
Korean Mathematical Society CKMS

Recently, Bre\v{s}ar's Jordan $\{g,h\}$-derivations have been investigated on triangular algebras. As a first aim of this paper, we extend this study to an interesting general context. Namely, we introduce the notion of Jordan $\mathcal{G}_n$-derivations, with $n \ge 2$, which is a natural generalization of Jordan $\{g,h\}$-derivations. Then, we study this notion on path algebras. We prove that, when $n > 2$, every Jordan $\mathcal{G}_n$-derivation on a path algebra is a $\{g,h\}$-derivation. However, when $n = 2$, we give an example showing that this implication does not hold true in general. So, we characterize when it holds. As a second aim, we give a positive answer to a variant of Lvov-Kaplansky conjecture on path algebras. Namely, we show that the set of values of a multi-linear polynomial on a path algebra $KE$ is either $\{0\}$, $KE$ or the space spanned by paths of a length greater than or equal to $1$.

Keywords: Lvov-Kaplansky conjecture, Jordan $\mathcal{G}_n$-derivations, Jordan $\{g,h\}$-derivations, derivations, path algebras, quivers

MSC numbers: Primary 16W25; Secondary 15A78