For a party of k-cochains on Hom-Lie algebras, called k-Hom-cochains, there is a series of coboundary operators ; for regular Hom-Lie algebras,  gives a new coboundary operator on k-cochains, and there are many works have been donebythe special coboundary operator [6,7].
where [d.sub.k]: [[conjunction].sup.k] [g.sup.*] [right arrow] [[conjunction].sup.k+1][g.sup.*] is the coboundary operator associated with the trivial representation."
with respect to a coboundary operator [delta] which obeys the graded Leibniz rule
It follows that the Chevalley-Eilenberg differential calculus over a real ring [C.sup.[infinity]](X) is exactly the DGR ([O.sup.*](X),d) of exterior forms on X, where the Chevalley-Eilenberg coboundary operator d (57) coincides with the exterior differential.
Then a Chevalley-Eilenberg coboundary operator d of the complex (83) reads
Accordingly, the graded Chevalley-Eilenberg coboundary operator d (86), termed the graded exterior differential, reads
Let b : [l.sup.2] [([X.sup.1]).sup.-] [right arrow] [l.sup.2] ([X.sup.0]) be the coboundary operator
defined by b[[delta].sub.(x,y)] = [[delta].sub.y] - [[delta].sub.x].
Given the ordering of the vertices of K, we have a coboundary operator
[delta] : [C.sup.j] [right arrow] [C.sup.j+1].
The coboundary operator
[[delta].sup.k]: [C.sup.k]([Alpha]) [right arrow] [C.sup.k+1]([Alpha]) is defined by
These are so-called coboundary operators
or potential like operators .