Deformed enveloping algebras were defined by V. G. Drinfel'd at the International Congress of Mathematicians 1986 in Berkeley [2]. His definition uses a system of generators and relations which is in a sense a deformation of the system of generators and relations that defines the enveloping algebras of semisimple Lie algebras considered by J. P. Serre [15] in 1966 and known since then as Serre's relations. Serre's relations consist of two parts, the first part interrelating the three types of generators and thereby leading to the triangular decomposition, the second, more important one being relations between generators of one type. In 1993, G. Lusztig gave a construction of the deformed enveloping algebras that did not use the second part of Serre's relations [4]. Lusztig's approach was interpreted by P. Schauenburg as a kind of symmetrization process in which the braid group replaces the symmetric group [13]. In this paper, we give a construction of deformed enveloping algebras without referring to generators and relations at all.
The paper is organized as follows: In Section 2, we recall the notion of a Yetter-Drinfel'd bialgebra and review some of their elementary properties that will be needed in the sequel. In Section 3, we carry out the first construction which leads to a Hopf algebra which has a two-sided cosmash product as coalgebra structure. We show that many Hopf algebras with triangular decomposition are of this form. As special cases, we obtain Radford's biproduct and Majid's double crossproduct. In Section 4, we carry out the second construction which applies to a pair of Yetter-Drinfel'd Hopf algebras which are in a sense dual to each other. In Section 5, we explain how Lusztig's algebra ´f which corresponds to the nilpotent part of a semisimple Lie algebra is a Yetter-Drinfel'd Hopf algebra and how the second construction can be used to construct deformed enveloping algebras.