  We give a construction of a Dirac operator on a quantum group based on any
simple Lie algebra of classical type. The Dirac operator is an element in the
vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor
is a $q$-deformation of the classical Clifford algebra. The tensor space $
U_q(\g) \otimes \mathrm{cl}_q(\g)$ is given a structure of the adjoint module
of the quantum group and the Dirac operator is invariant under this action. The
purpose of this approach is to construct equivariant Fredholm modules and
$K$-homology cycles. This work generalizes the operator introduced by Bibikov
and Kulish in \cite{BK}.
