A left (right) corepresentation is a synonym for a left (right) coaction of a coalgebra (comonoid) in one of the monoidal categories which have linear, and possibly, functional connotation; e.g. in the context of topological vector spaces.
There is however also another meaning of a corepresentation for a Leibniz algebra.
Both a representation and a corepresentation of a right Leibniz -algebra involve a -module and two -linear maps “actions” and with 3 axioms.
[m, [x, y]] = [[m, x], y] - [[m, y], x]
[x, [a, y]] = [[x, m], y] - [[x, y], m]
[x, [y, m]] = [[x, y], m] - [[x, m], y]
for and for .
[[x, y], m] = [x, [y, m]] - [y, [x, m]]
[y, [a, x]] = [[y, m], x] - [m, [x, y]]
[[m, x], y] = [m, [x, y]] - [[y, m], x].
If the two “actions” are symmetric, i.e. for all , then all the 6 axioms of representation or corepresentation are equivalent. If is underlying a Leibniz algebra then an action of on is by definition symmetric, hence all the 6 equivalent conditions hold.