On the Associative Analog of Lie Bialgebras

An infinitesimal bialgebra is at the same time an associative algebra and coalgebra in such a way that the comultiplication is a derivation. This paper continues the basic study of these objects, with emphasis on the connections with the theory of Lie bialgebras. It is shown that non-degenerate antisymmetric solutions of the associative Yang–Baxter equation are in one to one correspondence with non-degenerate cyclic 2-cocycles. The associative and classical Yang–Baxter equations are compared: it is studied when a solution to the first is also a solution to the second. Necessary and sufficient conditions for obtaining a Lie bialgebra from an infinitesimal one are found, in terms of a canonical map that behaves simultaneously as a commutator and a cocommutator. The class of balanced infinitesimal bialgebras is introduced; they have an associated Lie bialgebra. Several well known Lie bialgebras are shown to arise in this way. The construction of Drinfeldʹs double from earlier work by the author (in press, in Contemp. Math., Amer. Math. Soc., Providence) for arbitrary infinitesimal bialgebras is complemented with the construction of the balanced double, for balanced ones. This construction commutes with the passage from balanced infinitesimal bialgebras to Lie bialgebras.