On the Associative Analog of Lie Bialgebras

✍ Scribed by Marcelo Aguiar

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 233 KB
• Volume: 244
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8877

✦ Synopsis

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 Ž construction of Drinfeld's 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.