The Heisenberg group and the Schrödinger representation

We consider the Heisenberg group N as being the simply connected Lie group of Lie algebra η. Via the exponential map exp, N is identified with the 2n + 1 vector space V + ℝE with the multiplication law: $$ \exp \left( {v + tE} \right) \cdot \exp \left( {v' + t'E} \right) = \exp \left( {v + v' + \left( {t + t' + \frac{{B\left( {v,v'} \right)}}{2}} \right)E} \right) $$ where v,v′ Є V, t,t′ Є ℝ