International Journal of Scientific & Engineering Research, Volume 1, Issue 2, November-2010 1

ISSN 2229-5518

Ideals in Group algebra of Heisenberg Group

M. L. Joshi

Abstract— In spectral theory ideals are very important. We derive the relation between non commutative and commutative algebra by a transformation which is associated to the semi-direct product of groups. We obtain and classify the ideal in

L1 -algebra of Heisenberg group.

Index Terms— Heisenberg group, Ideals in L -algebra of the Heisenberg group, Semi-direct product.

—————————— • ——————————

1 INTRODUCTION

WE recall some definitions.

Definition 1.1: The Heisenberg group is the group of
= ((a, b) + (a’+b’c ,b’); c + c’)
= ((a+a’+b’c , b+b’); c + c’) (1.4)
Where c (a’,b’) = p (c) (a’,b’)

( 1 a b

By mean of group isomorphism lf' : G � H

defined by

3 X 3 upper triangular matrices of the form 0 1 c

( 1 a b

/ lf'((c, b); a) = / 0

I c the group H n

with the

/ 0 0 1 / n

)

Definition 1.2: For a E O n , b E O n , c E O and I ( Iden- tity matrix of order n), the Heisenberg group of dimension

2n +1 is the group of upper triangular matrices of the

( 1 a b

/ 0 0 1

group G can be identified.
Definition 1.3: Let L1 (M) be the Banach algebra that con- sists of all complex valued functions on the group M - an unimodular Lie group, which are integrable with respect

form / 0 In c

(1.1)

to the Harr measure of M and multiplication is defined by
Let

n +1

/ 0 0 1

Aut(O n +1 ) is the group of all automorphisms of

convolution on M.
Let us denote the restriction of subgroup N of M by

L1 (M) on any

L1 (M)|N. Then

O then for any

a = (a1 , a2 , ...an ) E O

L1 ( M ) | =

{F | : F E L1 ( M )}

where

F |N

is the re-

b = (b1 , b2 , ...bn ) E O , c E O and

ab = I ai bi , de-

striction of the function F on N.
fine

G = O n +1 p

i =1

O n be the group of semi direct

2 PRILIMINARIES.

product of

n +1

and

O by the group homomorphism

Let J is real vector group which is direct product of O

p : O n � Aut(O n +1 ) which is defined by,

and

O and K is real vector group which is direct prod-

p (a) (b,c)= (b + ac , c) (1.2)

uct of G and O
then the group G can be identified with
The inverse of an element in G is defined by
the closed subgroup G x {0} of K and J can be identified
For X= ((a, b); c) E G ,
with the closed subgroup O

p {0}x O of K.
X-1 = ((a, b); c)-1
= (-c(-a,-b); -c)
Let L =

n +1

O x O be the group of the direct

n n

= ((-a + bc,-b); -c) (1.3)
Where -c(-a,-b) = p (-c) (-a,-b)
The multiplication of two elements X and Y in G is de- fined by
For X = ((a, b); c) , Y = ((a’, b’); c’) E G ,
X · Y = ((a, b); c) ((a’, b’); c’)
= ((a, b) + c(a’, b’); c + c’)
product of O , O and O .
The inverse of element X in L is defined by
For X = ((a, b); c,d) E L , X-1 = ((a, b); c, d)-1
= (-d (-a,-b); -c, -d)
= (-a+bd, -b); -c, -d) (2.1)
Where -d (-a,-b)= p (-d) (-a,-b)

International Journal of Scientific & Engineering Research, Volume 1, Issue 2, November -2010 2

ISSN 2229-5518

The multiplication of two elements X and Y in L is de- fined by
fined by

il(f * | )((a, b), 0, c) = f * | (c(a, b), 0, c)

For X = ((a, b); c,d) , Y = ((a’, b’); c’,d’) E L ,
X · Y = ((a, b); c ,d) ((a’, b’); c’,d’)
= ((a, b)+d(a’, b’); c+c’ ,d+d’)
= ((a, b)+(a’+b’d, b’) ; c+c’ ,d+d’)
= ((a+a’+b’d, b+b’) ; c+c’ ,d+d’) (2.2)
is a topological isomorphism.
Proof follows from the fact that mapping il is

(

is also continuous.
Where d(a’, b’) = p (d)(a’, b’).
In such a case the group G can be identified with the

Corollary 2.3 The mapping fined by

Q : L1 ( J ) � L1 (G)

de-
closed subgroup

n +1

p {0}x O

of L and J can be

Q(f * | )((a, b), 0, c) = f * | (c(a, b), 0, c)

identified with the closed subgroup of L.

O x {0} p O

is a topological isomorphism.
Proof follows from the fact that mapping Q is conti-
Let the subspace of all complex valued functions on L
nuous and its inverse

Q-1

defined by,
is denoted by

L1 (L) such that

L1 (L)|G =

L1 (G) and

Q-1

(f * | )((a, b), c, 0) = f *| ((-c(a, b)), c, 0)

L1 (L)|J =

L1 (J).

is also continuous.

Definition 2.1 For every f E

L1 (L) , define a function

Remark: For

I EL1

( L ) , I * is the image of I under

f * as follows. For all ((a, b); c, d) E L ,

mapping * . Let E = I * |G, then I |G = I * |G = E
f * ((a, b); c, d) = f (c(a, b); 0, d + c) (2.3)

Theorem 1. Let I be a subset of

L E ( L)

, then

E = I |G

is a

left ideal in the algebra

L1 ( J )

if and only if

I ' = I | is

It is noted that for all ((a, b); c, d) E L and k E
function f * is invariant because,

O the

an ideal in the algebra L1 ( J ) .

f * (k(a,b); c - k, d + k) = f * ((a, b); c, d) (2.4)
Proof: First suppose

L1 ( J )

I ' = I |J

is an ideal in the algebra
Further it should be noted that restricted functions
.
By considering the group

K = G x O n and the

f * |G E

L1 (G) and f * |J E

L1 (J).

mapping f

� f *

which is defined by,

Definition 2.2: For every v E

L1 (G) or v E L1

(J) and

( ) ( )

for any F E

L1 (L) two convolutions product on the

f * ( a, b ; c, d) = f

(-c a, b ; 0, c + d )

group L are defined by,
(i) v * F ((a,b) ; c, d)
it is easy to show that

L1 ( J ) .

I ' = I |J

is an ideal in the algebra

= f F[((x, y); z)-1 ((a, b); c, d)]u((x, y); z)dxdydz

Conversely suppose that

bra L1 ( J ) .

I ' = I |J

is an ideal in the alge-

= f F[-z(a-x, b-y); c, d-z)]u((x, y); z)dxdydz

(2.5)

We know that

v *c

I ' I '

and

v *c

I I

(ii) v * c F ((a, b) ; c, d)
for any

v E L ( J ) where

= f F[(a-x, b-y); c-z, d)]u((x, y); z)dxdydz

(2.6)

v *c I ' = {v *c (F |J ), F E I } and

J v *

I = {v *

F , F E I } .

where dx dy dz is the lebesgue measure on group G. c c

Corollary 2.1: For each v E (G) , F E

L1 (L) and for all

If I
is the image of I under the mapping * , then we
((a, b); c, d) E L

E have,

v * I * = v *

I * I

v * F((a, b); c, d) = v * c F((a, b); c, d)
and by taking the restriction on the group G,

proof of this lemma is easily given by the help of (2.4),
we have

v * I

|G = v * I |G I |G

(2.5), (2.6).

So that v * E E .

Corollary 2.2 The mapping

il : L1 (G) � L1 (G)

de-
This proves that

E = I |G

is a left ideal in the algebra

International Journal of Scientific & Engineering Research, Volume 1, Issue 2, November -2010 3

ISSN 2229-5518

L1 ( J ) .

3. RESULTS

From the above theorem the following results can be veri- fied easily.
Ann. Math., 68, pp. 709–712.
(i) If r be a subspace of the space

L1 ( K ) such that

I =r

| is an ideal in L1 ( J ) , then (i) I =r

|J is a max-

imal ideal in the algebra

L1 ( J ) if and only if

E =r |G is

a left maximal ideal in the algebra L1 (G) .

* 1

(ii) I =r

|J is a closed ideal in the algebra

L ( J ) if and

only if

L1 (G) .

E = r |G is a left closed ideal in the algebra

(iii) I =r

|J is a dense ideal in the algebra

L1 ( J ) if and

only if

L1 (G) .

E = r |G

is a left dense ideal in the algebra

ACKNOWLEDGMENT

We are thankful to Prof. L. N. Joshi, Retd. Prof in mathe- matics, D.K.V. Science College, Jamnagar and Prof. J. N. Chauhan, Mathematics Department, M. & N. Virani Science College, Rajkot for their cooperation in the prepa- ration of this paper. We are also thankful to the numerous referees for their helpful and valuable comments.

REFERENCES

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