The role of the triply extended group of translations of , as the defining group of two-dimensional noncommutative quantum mechanics (NCQM), has been studied in Chowdhury and Ali (2013 J. Math. Phys. 54 032101). In this paper, we revisit the coadjoint orbit structure and various irreducible representations of the group associated with them. The two irreducible representations corresponding to the Landau and symmetric gauges are found to arise from the underlying defining group. The group structure of the transformations, preserving the commutation relations of NCQM, has been studied along with specific examples. Finally, the relationship of a certain family of unitary irreducible representations of the underlying defining group with a family of deformed complex Hermite polynomials has been explored.

We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in (2+1)-space-time dimensions and the two-fold extension of the group of translations of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4$\end{document}R4. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.

This paper is devoted to the construction and analysis of the Wigner functions for noncommutative quantum mechanics, their marginal distributions, and star-products, following a technique developed earlier, viz, using the unitary irreducible representations of the group GNC, which is the three fold central extension of the Abelian group of ℝ4. These representations have been exhaustively studied in earlier papers. The group GNC is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom. The Wigner functions studied here reflect different levels of non-commutativity—both the operators of position and those of momentum not commuting, the position operators not commuting and finally, the case of standard quantum mechanics, obeying the canonical commutation relations only.

In this paper, we obtain an infinite dimensional Lie algebra of exotic gauge invariant observables that is closed under Goldman-type bracket associated with monodromy matrices of flat connections on a compact Riemann surface for G 2 gauge group. As a byproduct, we give an alternative derivation of known Goldman bracket for classical gauge groups GL(n, R), SL(n, R), U(n), SU(n), Sp(2n, R) and SO(n).