Contemporary Mathematics Volume 267, 2000 Infinitesimal Hopf algebras Marcelo Aguiar ABSTRACT. Infinitesimal bialgebras were introduced by Joni and Rota [J-R]. An infinitesimal bialgebra is at the same time an algebra and a coalgebra, in such a way that the comultiplication is a derivation. In this paper we de- fine infinitesimal Hopf algebras, develop their basic theory and present several examples. It turns out that many properties of ordinary Hopf algebras possess an infinitesimal version. We introduce bicrossproducts, quasitriangular infinites- imal bialgebras, the corresponding infinitesimal Yang-Baxter equation and a notion of Drinfeld's double for infinitesimal Hopf algebras. 1. Introduction An infinitesimal bialgebra is a triple (A, m, ~) where (A, m) is an associative algebra, (A,~) is a coassociative coalgebra and for each a, b E A, ~(ab) = Lab10b2 + La1®a2b. Infinitesimal bialgebras were introduced by Joni and Rota [J-R] in order to provide an algebraic framework for the calculus of divided differences. Several new examples are introduced in section 2. In particular, it is shown that the path algebra of an arbitrary quiver admits a canonical structure of infinitesimal bialgebra. In this paper we define the notion of antipode for infinitesimal bialgebras and develop the basic theory of infinitesimal Hopf algebras. Surprisingly, many of the usual properties of ordinary Hopf algebras possess an infinitesimal version. For instance, the antipode satisfies S(xy) = -S(x)S(y) and L S(x1)0S(x2) =- L S(x)l®S(xh, among other properties (section 3). The existence of the antipode is closely related to the possibility of exponenti- ating a certain canonical derivation D : A --4 A that is carried by any t:-bialgebra. This and other related results are discussed in section 4. 2000 Mathematics Subject Classification. Primary: 16W30, 16W25. Key words and phmses. Hopf algebras, derivations, Yang-Baxter equation, Drinfeld's double. Research supported by a Postdoctoral Fellowship at the Centre de Recherches Mathematiques and Institut des Sciences Mathematiques, Montreal, Quebec. © 2000 American Mathematical Society 1 http://dx.doi.org/10.1090/conm/267/04262
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