Unitary representations of three dimensional Lie groups revisited: A short tutorial via harmonic functions

Abstract

The representation theory of three dimensional real and complex Lie groups is reviewed from the perspective of harmonic functions defined over certain appropriate manifolds. An explicit construction of all unitary representations is given. The realisations obtained are shown to be related with each other by either natural operations as real forms or Inönü-Wigner contractions.

Infinite degeneracy of Landau levels from the Euclidean symmetry with central extension revisited

“…as a semidirect product of T(2) and O(2) Lie groups (written T( 2) O( 2)), which are twodimensional translation and orthogonal groups respectively. (Refer to [16] for a good description of this group.) The E(2) group has a set of three Hermitian generators, { X1 , X2 , X3 }.…”

Infinite degeneracy of Landau levels from the Euclidean symmetry with central extension revisited

“…It defines 2 translation and 1 rotation symmetries. 6 Mathematically, the group E(2) is defined as a semidirect product of T (2) and O(2) Lie groups (written T (2) ⋊ O(2)), which are twodimensional translation and orthogonal groups respectively (Refer to [16] for a good description of this group.). The E(2) group has a set of three Hermitian generators, { X1 , X2 , X3 }.…”