He describes Lie groups and their representations, with a focus on the quantum mechanically relevant Heisenberg group
H3 and special unitary group SU(2).
This group is realized as a central extension of the standard Heisenberg group
[mathematical expression not reproducible].
On an eigenspace of [M.sub.[mu]v] and F of integral eigenvalues, as a representation space, the set of [mathematical expression not reproducible] behaves like the commuting set of U and V of the "Heisenberg group
" discussed in the modular picture of .).
Capogna, "Regularity of quasi-linear equations in the Heisenberg group
," Communications on Pure and Applied Mathematics, vol.
The definition says that any prime p to the power 1 + 2n, that is, [p.sup.1+2n], sustains the twofold properties: (i) Heisenberg group
and semidirect product of cyclic group order and/or (ii) dihedral order 8 and quaternion group.
Homogeneous groups include the Euclidean space, the Heisenberg group
, and the Carnot group; see [20, 21].
New methods for constructing a canal surface surrounding a biharmonic curve in the Lorentzian Heisenberg group
[Heis.sup.3] were given, (KORPINAR; TURHAN, 2010; 2011; 2012).
The general Heisenberg group
[H.sub.n](q) (sometimes also written as [H.sub.n] if we do not want to specify q) of dimension 2n + 1 over [F.sub.q], with n a positive integer, is the group of square (n + 2) x (n + 2)-matrices with entries in[F.sub.q], of the following form (and with the usual matrix multiplication):
Quasiconformal mappings on the Heisenberg group
were introduced by Mostow  in his study of rigidity theorem.
We will be interested in deformations of the discrete Heisenberg group
as a group acting properly discontinuously and cocompactly on a space X.
But many groups in physics such as the Heisenberg group
and also many applicable groups in engineering such as Motion groups are non-abelian and so that the standard STFT theory in abelian case fails.