Elliptic Quantum Group Uq,p(sl2)

Abstract

In this chapter, the elliptic dynamical quantum group Uq,p(sl2) is defined by generators and relations. The generators are the Drinfeld type, i.e. an analogue of the loop generators of the affine Lie algebra sl2. We call their generating functions the elliptic currents. The dynamical nature of Uq,p(sl2) is realized by introducing the dynamical parameter P and considering a copy H= CP of the Cartan subalgebra h-=Ch. We take the field MH* of meromorphic functions on H∗ as the basic coefficient field and make it not commutative to the other generators of Uq,p(sl2). We also introduce the half currents and construct the L+-operator L+(z) in the Gauss decomposed form by taking the half currents as its Gauss coordinates. It is then shown that the L+(z) satisfies the dynamical RLL-relation. In addition, following the quasi-Hopf formulation Bq,λ(sl2), we introduce the L−-operator and show that the difference between the + and the − half currents gives the elliptic currents of Uq,p(sl2). Furthermore a connection to Felder’s formulation is shown by introducing the dynamical L-operators.