  Recently B. Hartwig and the second author found a presentation for the
three-point $sl_2$ loop algebra via generators and relations. To obtain this
presentation they defined an algebra $\boxtimes$ by generators and relations,
and displayed an isomorphism from $\boxtimes$ to the three-point $sl_2$ loop
algebra. We introduce a quantum analog of $\boxtimes$ which we call
$\boxtimes_q$. We define $\boxtimes_q$ via generators and relations. We show
how $\boxtimes_q$ is related to the quantum group $U_q(sl_2)$, the $U_q(sl_2)$
loop algebra, and the positive part of $U_q(\hat{sl_2})$. We describe the
finite dimensional irreducible $\boxtimes_q$-modules under the assumption that
$q$ is not a root of 1, and the underlying field is algebraically closed.
