1. Preliminaries
The Heisenberg group (of order
),
is a noncommutative nilpotent Lie group whose underlying manifold is
with coordinates
and group law given by
![](https://www.scirp.org/html/11-5300231\e947b15d-8366-4a8c-bf12-254ffd725fa2.jpg)
Setting
, then
forms a real coordinate system for
. In this coordinate system, we define the following vector fields:
![](https://www.scirp.org/html/11-5300231\6ca8023a-3781-48a9-af00-f342976339a9.jpg)
It is clear from [1] that
is a basis for the left invariant vector fields on
These vector fields span the Lie algebra
of
and the following commutation relations hold:
![](https://www.scirp.org/html/11-5300231\38b92408-1e28-45a1-ba16-8bd3a578d333.jpg)
Similarly, we obtain the complex vector fields by setting
(1)
In the complex coordinate, we also have the commutation relations
![](https://www.scirp.org/html/11-5300231\fe09fdc3-e889-4a7f-940a-74d0128736ab.jpg)
The Haar measure on
is the Lebesgue measure
on
[2]. In particular, for
, we obtain the 3-dimensional Heisenberg group
(since
). Hence
may also be referred to as (2n + 1)-dimensional Heisenberg group.
One significant structure that accompanies the Heisenberg group is the family of dilations
![](https://www.scirp.org/html/11-5300231\00b335b5-a10b-4eba-8219-a1d059fe6103.jpg)
This family is an automorphism of
. Now, if
is an automorphism, there exists an induced automorphism,
such that
![](https://www.scirp.org/html/11-5300231\e3d1bbee-e274-4fbe-a43e-d8defb7983b5.jpg)
For simplicity, assume that
and
coincide. Thus we may simply assume that if
we have ![](https://www.scirp.org/html/11-5300231\7ecef1c3-0726-4352-ab1d-a2ba3fdf977d.jpg)
2. Heisenberg Laplacian
An operator that occurs as an analogue (for the Heisenberg group) of the Laplacian
on
is denoted by
where ![](https://www.scirp.org/html/11-5300231\c8ab6922-bd88-4a99-bbcf-6b8aa2628742.jpg)
is a parameter and defined by
![](https://www.scirp.org/html/11-5300231\d426217b-ff5d-4b38-bcc5-2ce139291abb.jpg)
where
are as defined in (1) so that
can be written as
(2)
is called the sublaplacian.
satisfies symmetry properties analogous to those of
on
. Indeed, we have that ![](https://www.scirp.org/html/11-5300231\8659bba9-208c-48ca-b657-2634d3505a59.jpg)
1) is left-invariant on
;
2) has degree 2 with respect to the dilation automorphism of
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 [3-5].
The Heisenberg-Laplacian is a subelliptic differential operator defined for
as
on
and denoted by
. It is obtained from the usual vector fields as
(3)
By a technique in [6], the operator
is factorized into two quasi-linear first order operators on
as:
![](https://www.scirp.org/html/11-5300231\0a70e8b3-682b-4b46-a65c-3e91449e9423.jpg)
and
![](https://www.scirp.org/html/11-5300231\dd01836b-366a-4ef4-b891-c0a0b3396a99.jpg)
so that
![](https://www.scirp.org/html/11-5300231\6c791cd5-efee-4c39-bc97-df6af49d2bc9.jpg)
Introducing the Lie algebra structure, we have
![](https://www.scirp.org/html/11-5300231\21ab39fe-0631-4d66-b16f-f41498b14742.jpg)
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
is the Heisenberg algebra) [5] consistent with that of Hans Lewy [7]. In [2], it has been shown that none of the factors of
,
or
is solvable and as such,
is not solvable.
In this paper, we shall prove that
only possesses a trivial group-invariant solution and for
a compact subgroup of
we have that
the K-invariant universal enveloping algebra of the Heisenberg group is generated by
and
.
Now, by a solution of a factor
say, we shall mean that if
are independent real variables, and
such that
has a solution
in the neighbourhood
of the point
, with
then
is analytic at
.
Definition 2.0. Let
be any open subset of
, and
a number such that
A function
on
satisfying
![](https://www.scirp.org/html/11-5300231\2744ea31-f213-4d29-9816-072d2c616152.jpg)
is said to be uniformly Holder continuous with Holder exponent
if
when
they are called uniformly Lipschitz continuous. When
they are simply continuous and bounded. A function is said to be in
-space if its first partial derivatives satisfy a Holder condition with positive exponent, provided the distance of the points involved does not exceed 1.
Theorem 2.1. Let
be a periodic real
-function which is analytic in no t-interval. Then there exists a
-function
determined by the derivative
of
such that
![](https://www.scirp.org/html/11-5300231\799c94b0-a2de-43ff-8f03-c2fb9e0c04d5.jpg)
has no
-solution,(no matter what open
-set taken as domain of existence).
For Proof, see [8].
Theorem 2.2. The Heisenberg Laplacian,
defined in (3) has no non-trivial group invariant solution.
Proof. Let
be a group-invariant solution of (3). We wish to show that
To do this, let
be a map generated by the group of automorphisms, dilations
where
determines the growth or decay rate. If
is defined by
![](https://www.scirp.org/html/11-5300231\5006dce3-6464-4392-b9ed-bc1610f7bd2d.jpg)
then obtaining the first and second order derivatives of
with respect to the independent variables we have
![](https://www.scirp.org/html/11-5300231\76a37970-c52f-4fdb-a633-8d3b9e7ba368.jpg)
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
. □
Theorem 2.3. Let
be a compact subgroup of
, then
the
-invariant universal enveloping algebra of the Heisenberg group is generated by
and
.
Proof. Let
be the algebra of
-invariant differential operators on
and let
be the symmetric algebra generated by the set
![](https://www.scirp.org/html/11-5300231\c9b0f6a7-c429-4ea9-90f8-02a020bb782c.jpg)
We note that the derived action of
on
is given by
![](https://www.scirp.org/html/11-5300231\2dd3e622-e57f-423b-84dc-2ae6309c76a3.jpg)
and
acts on
via
![](https://www.scirp.org/html/11-5300231\f4edb24a-b30d-4bba-8bed-5cc5693582b7.jpg)
and on
the
-valued polynimial functions on
-vector space
via
![](https://www.scirp.org/html/11-5300231\72c4c560-1c99-40f2-8f28-888173a3c792.jpg)
Now, if we identify
with the complexified symmetric algebra
then the symmetric product
of
becomes the polynomial
given by
![](https://www.scirp.org/html/11-5300231\e23f08fd-e82a-4575-901f-a16e1adf8bba.jpg)
Now, define a symmetrization map by
![](https://www.scirp.org/html/11-5300231\ae5d7c8c-5d62-4aa1-9efb-681621b1ea22.jpg)
with
![](https://www.scirp.org/html/11-5300231\dacb5609-cd75-49f8-9297-97c119637150.jpg)
Now since
acts on
and
by automorphism and
defined by
![](https://www.scirp.org/html/11-5300231\03c24978-48cc-4267-8a0b-588a93373a31.jpg)
induces an algebra map on the associated graded algebras and by induction [10, p. 282] the eigenfunctions of ![](https://www.scirp.org/html/11-5300231\47c3550a-4698-458e-99fb-bc9e748ecdb4.jpg)
and
are eigenfunctions of any element in ![](https://www.scirp.org/html/11-5300231\6c71f1f0-f529-4bd4-b7e4-ae5a89264334.jpg)
we have that the following diagram is commutative.
![](https://www.scirp.org/html/11-5300231\d92f182b-36fc-4dac-be48-e10e98d70c58.jpg)
for
Since
is a linear isomorphismit maps
onto
Since the action of
preserves degree on
, and by [11], if
generates
then,
generates
If
then
![](https://www.scirp.org/html/11-5300231\ef345cd3-11c9-48bc-8c6d-86ff29951d6f.jpg)
where the sum is finite and each
is a polynomial which is
-invariant. Thus, the result follows by the fact that the eigenfunctions of
and
are the eigenfunctions of
[12]. □