Orderings and *-orderings on cocommutative Hopf algebras

Our aim is to construct new examples of totally ordered and *-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings kG and smash products U(L)#(phi)kG. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of *-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for k = R and L finite-dimensional, we characterize the orderability of U(L)#(phi)kG; for k = C, we give a necessary and a sufficient condition for *-orderability of kG (G orderable, respectively, G residually 'torsion-free nilpotent'). Moreover, for k = C and L finite-dimensional, we reduce the problem of characterizing the *-orderability of U(L)#(phi)kG to the problem of characterizing the *-orderability of kG. The latter remains open.