On Some Properties of the Heisenberg Laplacian

Let n IH be the -dimensional Heisenberg group and let   2 1 n    n and T be the sublaplacian and central element of the Lie algebra of H respectively. For = 0,  denote by the Heisenberg Laplacian and let 0 := ,    n  K Aut H  be a compact subgroup of Automorphism of n H . In this paper, we give some properties of the Heisenberg Laplacian and prove that and generate the -invariant universal enveloping algebra,  T   K K n h 


Preliminaries
The Heisenberg group (of order ), n n H is a noncommutative nilpotent Lie group whose underlying manifold is , , x x x  y y y t  forms a real coordinate system for n H .In this coordinate system, we define the following vector fields: . is a basis for the left invariant vector fields on n H These vector fields span the Lie algebra n of h n H and the following commutation relations hold: Similarly, we obtain the complex vector fields by setting In the complex coordinate, we also have the commutation relations The Haar measure on n IH is the Lebesgue measure .[2].In particular, for , we obtain the 3-dimensional Heisenberg group Hence n H may also be referred to as (2n + 1)-dimensional Heisenberg group.
One significant structure that accompanies the Heisenberg group is the family of dilations For simplicity, assume that  and  coincide.
Thus we may simply assume that if

Heisenberg Laplacian
An operator that occurs as an analogue (for the Heisenberg group) of the Laplacian where and j j Z Z are as defined in (1) so that   can be written as satisfies symmetry properties analogous to those of on .Indeed, we have that   1) is left-invariant on n H ; 2) has degree 2 with respect to the dilation automorphism of n H and 3) is invariant under unitary rotations.Several methods for the determination of solutions, fundamental solutions of (2) and conditions for local solvability are well known [3][4][5].
The Heisenberg-Laplacian is a subelliptic differential operator defined for = 0  as . It is obtained from the usual vector fields as By a technique in [6], the operator is factorized into two quasi-linear first order operators on H as: indicating that the Heisenberg algebra is noncommutative and is hypoelliptic [4].We thus obtain an operator (which is a homogeneous element of  , the universal enveloping algebra of the Heisenberg group when n is the Heisenberg algebra) [5] consistent with that of Hans Lewy [7].In [2], it has been shown that none of the factors of ,

solution,(no matter what open has no
x y t  -set taken as domain of existence).
Proof.Let  be a group-invariant solution of (3).
We wish to show that 0.

 
To do this, let   with respect to the independent variables we have Substituting these into (3), we obtain a trivial equation.But by Group-invariant method, we should obtain a system of ordinary differential equations of lower order (see [9] p. 185).Thus, there exists no non-trivial groupinvariant solution for .
We note that the derived action of on is given by and K acts on and on n the C -valued polynimial functions on -vector space via Now, if we identify n with the complexified symmetric algebra IC S h then the symmetric product becomes the polynomial given by Now since acts on and   where  is a parameter and defined by Copyright © 2012 SciRes.APM generated by the group of automorphisms, dilations r where r determines the growth or decay rate.If

2 ,
, = , , , r x y t rx ry r t   then obtaining the first and second order derivatives of Copyright © 2012 SciRes.APM