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
These rules define the addition of a Deligne coboundary
to a Deligne cocycle.
with respect to a coboundary
operator [delta] which obeys the graded Leibniz rule
These are so-called coboundary
operators or potential like operators .
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].
Our results are most cleanly expressed in terms of the (arithmetic) coboundary
polynomial, which is the following simple transformation of the (arithmetic) Tutte polynomial:
This Lie bialgebra is also called a coboundary
Lie bialgebra because the cobracket [delta] is a 1-cocycle.
Furthermore, for two distinct connections [omega], [omega], on a bundle, the difference of the characteristic forms can be written as a coboundary
p([omega]) - p([omega]) = dT where T = T([omega],[omega]') is also canonically expressed in terms of the connections.
Moreover, it is not a coboundary
since [THETA]([e.sub.2], [e.sub.3]) = 1 [not equal to] 0 = f([[e.sub.2], [e.sub.3]]), for every linear map f, f : se(2) [right arrow] R.
Next the Lie bialgebra contraction analysis of the r-matrix (26) and Lie bialgebra (27) shows that there exists a unique quantum Inonu-Wigner contraction (a coboundary
Lie bialgebra contraction) that ensures the convergence of both (26) and (27).
If [omega] is not a coboundary
, then there is no ample division algebra in C(G,[omega], k).