Gabriel Gustavo Restrepo-Sánchez

Let F be a field of zero characteristic. We define a dialgebra structure over the polynomial algebra $$F[x] \otimes F[x]$$ F [ x ] ⊗ F [ x ] , 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).

The aim of this paper is to establish a correspondence between the categories given by the structures of racks and g-digroups, by defining three functors, denoted similarly to those in the case of rack-groups, $$Conj: \mathbf {g-dig}\rightarrow {\textbf {Rack}}$$C o n j : g - dig → Rack , $$As: {\textbf {Rack}}\rightarrow \mathbf {g-dig}$$A s : Rack → g - dig , and $$Inn: {\textbf {Rack}}\rightarrow \mathbf {g-dig}$$I n n : Rack → g - dig . Additionally, we establish certain properties of these functors and prove, as in the case of rack-groups, that the functor As is left adjoint to the functor Conj.