Quantum Group Duality and the Cuntz Algebra

The Cuntz algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra $U_q(g)$, and the structure of a co-module algebra over the quantum group $G_q$ associated with $U_q(g)$. These two algebraic structures are dual to each other via the duality between $G_q$ and $U_q(g)$.