Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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JORDAN 𝒢n-DERIVATIONS ON PATH ALGEBRAS

  • Received : 2021.10.18
  • Accepted : 2021.12.08
  • Published : 2022.10.01
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Abstract

Recently, Brešar'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 𝒢n-derivations, with n ≥ 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 𝒢n-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.

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