  Classification of finite dimensional representations of the q-deformed
Heisenberg algebra $H_q(3)$ is made by the help of Clifford algebra of
polynomials and generalized Grassmann algebra. Special attention is paid when
$q$ is a primitive $n$ root of unity. As a further application we obtain finite
dimensional representations of $ sl(2)_q$ using its embedding into $H_q(3)$.
