THE POSITIVE PART OF THE QUANTIZED UNIVERSAL ENVELOPING ALGEBRA OF TYPE AnAS A BRAIDED QUANTUM GROUP

The aim of this paper is to explain the bialgebra structure of the positive part of the quantized universal enveloping algebra (Drinfeld-Jimbo quantum group) of type Anusing the Lie algebra theory concepts. Recently has been introduced a generalization of Lie algebras, the basic T -Lie algebras [1]. Using the T -Lie algebra concept some new (we think) quantum groups of type Ancan be constructed. Such quantum groups arise as universal enveloping algebras of certain deformations as generalized Lie algebras of the Lie algebras form by upper triangular matrices sl n+1. Let us explain, embedded in the positive (negative) parts Uq(sln+1) of the Drinfeld-Jimbo quantum groups of type Anthere are some generalized Lie algebras (sl n+1)q called T -Lie algebras, [1]. Such T -Lie algebras satisfy not only a generalized antisymmetry and a generalized Jacobi identity, but an additional condition called multiplicativity. Through these T -Lie algebras the Poincaré-Birkhoff-Witt theorem for U q (sln+1) can be explained. The Poincaré-Birkhoff-Witt theorem is a general property for the universal enveloping algebras of adequate T -Lie algebras. In order to keep the proof of such theorem closest to the classical one [2] a Gurevich’s condition of multiplicativity is needed, [1], [4]. On the other hand, the next natural step in the study of (sl n+1)q as a T -Lie algebra is to give to its universal enveloping algebra a structure of Hopf algebra. Now a structure as a generalized Hopf algebra (braided quantum group) of the universal enveloping algebra U q (sln+1) of (sl + n+1)q is presented. As a matter of fact, U q (sln+1) has the usual algebra structure: generators x1, . . . , xn and relations xixj − xjxi = 0, if |i− j| > 1 (1.1)