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Many fundamental results of pluripotential theory on the quaternionic space$\mathbb{H}^n$ are extended to the Heisenberg group. We introduce notions of aplurisubharmonic function, the quaternionic Monge-Amp\`{e}re operator,differential operators $d_0$ and $d_1$ and a closed positive current on theHeisenberg group. The quaternionic Monge-Amp\`{e}re operator is the coefficientof $ (d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, theexistence of quaternionic Monge-Amp\`{e}re measure for a continuousquaternionic plurisubharmonic function and the minimum principle for thequaternionic Monge-Amp\`{e}re operator. Unlike the tangential Cauchy-Riemannoperator $ \overline{\partial}_b $ on the Heisenberg group which behaves badlyas $ \partial_b\overline{\partial}_b\neq -\overline{\partial}_b\partial_b $,the quaternionic counterpart $d_0$ and $d_1$ satisfy $ d_0d_1=-d_1d_0 $. Thisis the main reason that we have a better theory for the quaternionicMonge-Amp\`{e}re operator than $ (\partial_b\overline{\partial}_b)^n$.

The quaternionic Monge-Amp\`{e}re operator and plurisubharmonic functions on the Heisenberg group