ORDERINGS AND ∗-ORDERINGS ON COCOMMUTATIVE HOPF ALGEBRAS

Abstract. 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)#ϕkG. 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)#ϕkG; for k = C, we give a necessary and a sufficient condition for ∗-orderability of kG (G orderable, resp., G residually ‘torsion-free nilpotent’). Moreover, for k = C and L finite-dimensional, we reduce the problem of character-izing the ∗-orderability of U(L)#ϕkG to the problem of characterizing the ∗-orderability of kG. The latter remains open. 1