On Drinfeld Realization of Quantum Affine Algebras

In 1987 Drinfeld [Dr2] gave an extremely important realization of quantum affine algebras [Dr1][Jb]. This new realization has lead to numerous applications such as the vertex representations [FJ][J]. The proof of this realization was not in print until Beck’s braid group interpretation for the untwisted types [B]. Some of lower rank cases were also studied in [D][S]. All these work started from the quantum group towards the quantum loop realization, and were based on Lusztig’s theory of braid group action on the quantum enveloping algebras [L]. However, Drinfeld did give the exact isomorphism between two definitions of quantum affine algebras in [Dr2]. In this paper we give another proof directly from the Drinfeld isomorphism. Our proof is self-contained and elementary and works from the opposite direction from the quantum loop algebras towards the quantum groups. In doing this, we discovered that there are rich structures held by the q-loop algebra realization. We directly deform the argument used by Kac [K] to identify the affine Lie algebras and the Kac-Moody algebra defined by generators and relations. Here we must admit that the q-arguments are much more complicated than the classical analog, where we have the root space structure avaliable. It is nontrivial to properly deform usual brackets by q-brackets. As we have shown here in many cases there are strong indications for us to follow. The key are the following identities: