  We discuss $w_\infty$ and $sl_q(2)$ symmetries in multiple Chern-Simons
theory on a torus. It is shown that these algebraic structures arise from the
dynamics of the non-integrable phases of the Chern-Simons fields. The
generators of these algebras are constructed from the Wilson line operators
corresponding to these phases. The vacuum states form the basis of cyclic
representation of $sl_q (2)$.
