  We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the
Virasoro algebra with values in a tensor product of two braided vertex operator
algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is
constructed from the free Fock space realization of the Virasoro algebra with
an additional q-deformed harmonic oscillator degree of freedom. The braided VOA
structure arises from the theory of local systems over configuration spaces and
it yields an associative algebra structure on the cohomology. We explicitly
provide the four cohomology classes that serve as the generators of $SL_q(2)$
and verify their relations. We also discuss the possible extensions of our
construction and its connection to the Liouville model and minimal string
theory.
