I am attempting to use 1-cocycles to construct examples of twists for group algebras using the method described in the paper of Etingof and Gelaki, in Proposition 2.2 in their paper ``A method of construction of finite dimensional triangular Hopf algebras'' Mathematical Research Letters 5 (1998), 191-197.
It is known that if $N$ is a finite nilpotent ring then there is a 1-cocycle to the additive group of $N$, denoted as $(N,+)$ from the adjoint group of $N$ , denoted as $(N, \circ )$, where $n\circ m=nm+n+m$ for $m,n\in N$. This cocycle is given by the identity map $\pi $.
I am just learning Hopf algebras by myself, so I am not sure if I understand things correctly, any comment would be gratefully appreciated:
it seems that for the trivial twist we get the following result: if G is the additive group of a nilpotent ring N and R is a universal R-matrix for the group algebra C[G] then R is a universal R-matrix for the group algebra C[H] where H is the adjoint group of the ring N. Is it correct? Are there any easy, ready to use, examples of universal R matrices for group algebras C[G] where G is abelian? Especially interesting is the case when G is an abelian p-group in which every element has order p (p is prime) as it is the additive group of any nilpotent F-algebra over the field of p-elements.
In the same notation: let J be a twist for the algebra C[G], what does it mean that J is H invariant? (this assumption appears in the aforementioned Proposition 2.2).
Would triangular Hopf algebras constructed in a such way have any applications (for example as an input for testing conjectures)?