Commutative quantum current operators, semi-infinite construction and functional models

Abstract

We construct the commutative current operator x ¯ + ( z ) \bar x^+(z) inside U q ( s l ^ ( 2 ) ) U_q(\hat {\mathfrak {sl}}(2)) . With this operator and the condition of quantum integrability on the quantum currents of U q ( s l ^ ( 2 ) ) U_q(\hat {\mathfrak {sl}}(2)) , we derive the quantization of the semi-infinite construction of integrable modules of s l ^ ( 2 ) \hat {\mathfrak {sl}}(2) which has been previously obtained by means of the current operator e ( z ) e(z) of s l ^ ( 2 ) \hat {\mathfrak {sl}}(2) . The quantization of the functional models for s l ^ ( 2 ) \hat {\mathfrak {sl}}(2) is also given.