Dialgebra Structure of $$F[x]\otimes F[x]$$, Derivations and Diderivations

Abstract Let F be a field of zero characteristic. We define a dialgebra structure over the polynomial algebra $$F[x] \otimes F[x]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> <mml:mo>⊗</mml:mo> <mml:mi>F</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , and we give a complete classification of the Lie algebra of derivations "Equation missing" and the space of diderivations "Equation missing". We also show that the results of this work are closely related to those obtained in Lin and Zhang (Commun Algebra 38(9):3417–3447, 2010) and Restrepo-Sánchez (Inmersión de dialgebras en dialgebras separables, adición de unidades barra y biderivaciones, Tesis de Maestría, Universidad de Antioquia, Medellin, 2010).