  We study the structure of abelian extensions of the group $L_qG$ of
$q$-differentiable loops (in the Sobolev sense), generalizing from the case of
central extension of the smooth loop group. This is motivated by the aim of
understanding the problems with current algebras in higher dimensions. Highest
weight modules are constructed for the Lie algebra. The construction is
extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An
application to the twisted K-theory on $G$ is discussed.
