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Heisenberg Group

The Heisenberg group $H^n$ in $n$ Complex variables is the Group of all $(z,t)$ with $z\in\Bbb{C}^n$ and $t\in\Bbb{R}$ having multiplication

\begin{displaymath}
(w,t)(z,t')=(w+z,t+t'+\Im[w^{\rm T} z])
\end{displaymath}

where $w^{\rm T}$ is the conjugate transpose. The Heisenberg group is Isomorphic to the group of Matrices

\begin{displaymath}
\left[{\matrix{1 & z^{\rm T} & {\textstyle{1\over 2}}\vert z\vert^2+it\cr 0 & 1 & z\cr 0 & 0 & 1\cr}}\right],
\end{displaymath}

and satisfies

\begin{displaymath}
(z,t)^{-1}=(-z,-t).
\end{displaymath}

Every finite-dimensional unitary representation is trivial on $Z$ and therefore factors to a Representation of the quotient $\Bbb{C}^n$.

See also Nil Geometry


References

Knapp, A. W. ``Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.




© 1996-9 Eric W. Weisstein
1999-05-25