A class of solvable Lie groups and their relation to the canonical formalism

A NNALES DE L ’I. H. P., SECTION A

H ANS T ILGNER

A class of solvable Lie groups and their relation to the canonical formalism

Annales de l’I. H. P., section A, tome 13, n

^o

2 (1970), p. 103-127

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103

A class of solvable Lie groups

and their relation to the canonical formalism

Hans TILGNER

Philipps-Universitat, Marburg, Deutschland.

A.nn. Inst. Henri Poincaré, Vol. XIII, ^n° 2, 1970, j

Section A :

Physique théorique.

ABSTRACT. ^- Some facts

concerning symplectic

^vector spaces, their

automorphism

groups

Spl(2n, R, E),

and derivation Lie

algebras spl(2n, R, E)

^are

given.

For every element R of these Lie

algebras

^{a sol-}

vable Lie _group

exp(RR)

^{x E x R is}

constructed,

^{which is}

nilpotent

iff R is

nilpotent.

^{We calculate the Lie}

algebras

^{f~R (B E of these}

groups, all of which contain the

Heisenberg

^Lie

algebra. Automorphism

groups and derivation Lie

algebras

^{of RR fl3 E fl3}

^R,

and faithful finite dimensional

representations

^{of them}

together

^{with the}

corresponding representations

^of

exp(RR)

^{x E x R are}

given.

In Part II a modifica- tion

weyl(E, cr)

of the universal

enveloping algebra

^{of the}

Heisenberg

^Lie

algebra

is defined. We realize the Lie

algebras

^{E E9 R in this}

algebra. Finally

^some

automorphisms

^and derivations of

are constructed

by

^{means of the}

adjoint representation

^of

weyl(E, ~).

Attention is

given

^{to the case of the harmonic} oscillator and

especially

to the free nonrelativistic

particle

^whose group is

nilpotent.

RESUME.

- Quelques qualites

concernant des _espaces vectoriels

symplec- tiques,

^leurs groupes

d’automorphismes Spl(2n, R, E)

^{et leurs}

algèbres

de Lie des derivations sont discutes. Pour

chaque

^{element R} d’une telle

algèbre

^de

^Lie,

^{on construit un} groupe de Lie

solvable, exp(RR)

^{x E x}

R, qui

^est

nilpotent

^{si et} seulement si R est

nilpotent.

^{On calcule les}

algèbres

de Lie RR © ^E p ^{R de ces} groupes

qui

contiennent tous

1’algebre

^{de Lie}

d’Heisenberg.

^{On donne les} groupes

d’automorphismes

^{et les}

algèbres

^de

ANN. POINCARÉ, ^{w-XIl1-2 8}

Lie des derivations de E Q9

R,

^ainsi _que des

representations

finies fideles ensemble avec les

representations correspondantes

^de

exp(RR)

^{x E x} R. Dans la 2e

partie,

^{on définit une} modification

weyl(E, (1)

de

l’algèbre

universelle

enveloppante appartenant à l’algèbre

^{de Lie d’Hei-}

senberg.

^{Dans cette}

algebre,

^nous realisons les

algebres

^{de Lie RR} 0 ^E 0 IR.

Finalement,

^{on construit}

quelques automorphismes

^et derivations de RR 0 E 3 ^{I~ avec} l’aide de la

representation adjointe

^de

weyl(E, y).

On observe l’oscillateur

harmonique

^et

plus spécialement

^la

particule

^libre

non relativiste

(dont

le groupe ^est

nilpotent).

INTRODUCTION

In §

^{1-1 we collect some facts on the}

symplectic

^matrix group

Spl(2n, R, E)

and its Lie

algebra spl(2n, R, E). In §

^{II-8 we}

give

^an

isomorphism

^R

between

spl(2n, R, E)

and the Lie

algebra

^{of all bilinear}

polynomials

^of

the

position

^{and momentum} operators qi ^and

pi.

^{With the}

help

^{of this}

isomorphism,

^{we define} for every Hamilton operator which is bilinear in the _qs and

pi

^{a 2n +} 2-dimensional solvable group, each

containing

^the

Heisenberg

group ^{as a}

subgroup.

^{Their Lie}

algebras

^are

isomorphic

^to

the Lie

algebras

^formed

by

^the

identity element,

the linear combinations of

the q and pi,

and the chosen Hamilton operator in the infinite dimensional associative

algebra weyl(E, r),

^{which is a} certain modification of the uni- versal

enveloping algebra

^{of the}

(Heisenberg)

^Lie

algebra

^{of the} canonical commutation relations.

These solvable Lie _groups were

suggested (see [1])

^{as «} spectrum generat-

ing

^» groups of the chosen Hamilton operator, ^{i. e. their}

inequivalent

irreducible

unitary representations

should label the

physical

^{states of the}

Hamilton operator,

excluding spin.

^{We find the hard work} of

classifying

the

unitary representations

easier for the above solvable _groups than for the

corresponding

^« invariance » groups, since their dimensions are in

general

smaller and their

algebraic

structures easier to handle. Besides

this,

ⁱⁿ

general

^{it is not known} which of the local but not

global isomorphic covering

groups is the invariance group; for

instance,

it is hard to say which

covering

group fo the

infinitely

connected group

U(n, C)

is the inva- riance group of the ⁿ dimensional harmonic oscillator.

There is another

advantage

ⁱⁿ

using

these solvable _groups instead of invariance _{groups :}

having

classified their

representations,

^we will have no

A CLASS OF SOLVABLE LIE GROUPS 105

trouble with the construction of

position

operators ^since

they

^{are included}

from the

beginning.

We denote the direct ^vector _space ^sum of two vector spaces A and B

by

^A ae

B,

^the direct Lie

algebra

^{sum of two Lie}

algebras

^{A and B}

by

^A

+) ^B,

their semidirect sum, B

being

^the

ideal, by B,

^{and the semidirect}

product

^{of two} groups A and

B,

^B

being

^{the normal}

subgroup, by ^{A 8 B~}

their direct

product by A 0

^B.

PART I

A CLASS OF SOLVABLE LIE GROUPS

§

^I-I:

Symplectic

^Vector

Spaces

and the

Symplectic Group.

A

pair (E, u)

^{of a real vector} space E

(in

^the

following

^{we consider}

only

finite dimensional

ones)

^{and a}

nondegenerate antisymmettic

bilinear form ^~:

E x is called a

symplectic

^vector space. Between the

(antisymme- tric)

bilinear forms J and the

(antisymmetric)

^{matrices A} exists the

bijec-

tion

n

where x

= 03BEiei with 03BEi

^{e R and} ei ^E E is the

general

element of

E,

tT being

^{the row vector}

(ç 1,

^...,

çn)

^{of x}

and 03BE

^the

corresponding

^column

vector. The bilinear form 03C3 is

nondegenerate

^{iff det}

(A) #

0. Because of det

(A) = (- 1)dim

^{(E) det}

(A)

^{this is}

possible only

^for evendimensional E.

We write dim

(E) = :

^2n.

(1)

^LEMMA

[2;

p.

10].

^{- In}

(E, a)

^{we can introduce a basis} el, ... , e~

/1, ...,/"

^{so that for all}

i, k

⁼

1,

^{..., n}

In the

following

^{the elements of}

(E, 0)

^{are written x = +}

çifi),

and the basis of E is chosen such that

u(x, y) =

^{where 3 is the 2n x 2n}

matrix

(- .. ^{id" .}

^The

automorphism

group of

(E, r)

^is called the

symplectic

group

Spl (2n, ^{R, E).}

^{It consists of} all invertible 2n x 2n matrices which invariant the bilinear form a :

The

defining

condition is in matrix form

STJS

⁼ J. The

(abstract) symplectic

group is a

n(2n

⁺

I)-dimensional,

noncompact,

simple, infinitely connected,

connected Lie _group; with the

help

^{of the}

exponential mapping

we

get

^{its Lie}

algebra

which is the derivation Lie

algebra

^of

(E, 0)

Herein the

defining

relation is in matrix form

RTJ

+ 3R ⁼ 0.

Every

^{matrix of}

Spl (2n, IR, E)

^is

multiplicatively generated by

^{3 and}

symplectic

matrices of the

type . ),

^{where B is a}

symmetric n

^{x n}

matrix

[3 ;

p.

140].

^From this follows det

(S)

^{= + 1 for all S ~}

Spl (2~ ~ E)

and center

(Spl (2n, IR, E» = {

:t

id2n}.

^{The set of all}

matrices ( VU),

where U and V are real n x n matrices with

UVT

⁼

YUT

and +

yTy

⁼ is a

subgroup

^of

Spl (2n, R, E).

^The

correspondence

to the

unitary

^matrix group in n dimensions is

given by

B ^{. - /}

[4;

p.

350].

^Here

(U

⁺

iV)

^is

unitary

^{iff U E}

Spl (2n, E).

^{It is} easy to see that this

correspondence

^{is a} Lie group

isomorphism;

^{we call this}

2n-dimensional

representation

^{U of the}

unitary

group in n dimensions

U(n, R, E).

Similar results hold for the Lie

algebras :

^{the matrices}

with L a real

antisymmetric,

^{K a real}

symmetric n

^{x n}

^matrix,

^{form a}

Lie

algebra u(n, R, E),

^the

correspondence

^{to the}

unitary

^{matrix Lie}

algebra

in n dimensions

being given by 4).

^{The maximal compact}

subgroup

^of

Spl (2n, ^R, E)

^is

E).

^This follows from

[4 ;

^Lemma

4.3,

p.

345]

^and

Every

^{R E}

spl (2n, I~, E)

^{can be}

decomposed uniquely

107

A CLASS OF SOLVABLE LIE GROUPS

where

K,

^{A and B are}

symmetric n

^{x n}

matrices,

^{and L is}

antisymmetric.

The first part ^{is in}

u(n, R, E),

the second not.

According

^{to this}

decompo-

sition we have

spl(2n, R, E)

⁼

u(n, R, E)

^Q3 p where the vector space p is the intersection of

spl(2n, R, E)

^{with the}

(Jordan algebra of) symmetric

matrices in 2n dimensions. It is not a Lie

algebra

^{but a} so-called Lie

triple

system

[4 ;

p.

1 89, 5 ;

p.

78]. Actually

^{it is the}

eigenspace

^of

eigenvalue -

¹

of the involutive

automorphism

^{JRJT of}

spl(2n, R, E).

^{Since for}

R E

spl(2n, R, E)

^{we have}

(RJ)T

⁼

^R3,

^the

mapping

R H R3 defines

a

bijection

^of

spl(2n, R, E)

^{onto the}

n(2n

⁺

1)-dimensional

^Jordan

algebra

of

symmetric

^{2n x 2n matrices.}

By

^{means of this}

bijection

^William-

son

[6;

p.

911] ] has proved

^that every S ^E

Spl(2n, R, E)

^can be written uni-

quely

^{in the form S =}

exp(aR)

exp with

R,

^{R’ E}

spl(2n, R, E)

^and

E R. A one parameter

subgroup exp(aR)

^of

Spl(2n, R, E)

^is compact iff R E

u(n, R, E).

^{The one} parameter

subgroup

is

isomorphic

^{as a Lie} group ^to the one-dimensional torus in the usual

normtopology

^on

U(2n, E), given by

§

^{1-2: The} Oscillator

Group

^{and the}

Heisenberg Group.

Let with a E IR and Re

gl(2n, I~, E)

^{be a one} parameter

subgroup

of

Gl(2n, R, E).

^The

topological

^manifold

^eDfR

^{x E x R} becomes a Lie group if we define

and iff R E

spl(2n, R, E).

^The

identity

element is

0, 0),

^{the inverse}

of

(e"R,

^x,

~)

^is

(e - "R, - 6T~ - fJ).

^{The Lie} group

given by

^{R = 3 is}

called oscillator _group

Osz(2n).

^{Its dimension is 2n} + 2. We have the

subgroups 0, 0) ~ eRR,

^and

0, 0) (id2n, ^0, ~) ~ ^e~R

⁰

R,

^{and the} normal

subgroups (id2n, ^0, M) ~

^R

(the center),

^and

(id2n, ^E, R).

^The

latter we call

Heisenberg

group

Heis(2n).

^{It is a 2n +} 1-dimensional Lie.

group ^on the manifold E x

I~,

^{connected and}

simply

connected with

composition

Every

group

eRR

x E R is the semidirect

product

^of

^eRR

^and

Heis(2n).

In the

product topology

^{on x E}

x R, given by

^{the usual}

topology

on Gl

(2n, il~, E),

^the

normtopology

⁼

x))1~2

^on

^E,

^{and the}

usual

topology

^on

R,

^the

groups eRR

^{x E R and Heis}

(2n)

^are noncompact and connected. The commutant of two elements is

Calculating

the successive commutants we see that x E ^x R is sol-

vable,

^{and even}

nilpotent

^{iff R is}

nilpotent.

^Heis

(2n)

^is

nilpotent

^{but not} commutative.

The manifold R x E x IR can be made a Lie _group if we define

and iff R E

spl(2n, R, E).

^These groups have the same

algebraic properties

as the

corresponding

groups

above,

^{but are}

simply

^connected for all R.

Since

they

^{have the same Lie}

algebras (see below) they

^are the universal

covering

groups of the

original

^{ones. For R = 5 we} get ^{an infinite}

covering

of the

infinitely

^connected

Osz(2n).

^{If the}

projection

^{of R onto}

u(n, I~, E)

vanishes the _groups are

homeomorphic [4;

^lemma

4.3,

p.

345].

On the manifold x E we define a Lie _group

by

with ⁼

e-CXRx)

^and

identity

^element

(id2n, 0).

^{If E}

denotes the commutative ^additive _group of elements of

E,

^{we have for this}

group E. Its center

is { 0) ~ ;

it is solvable

and

^for

nilpotent

^R

even

nilpotent.

(12)

THEOREM. - For R E

spl(2n, ^R, E)

^the group ^x E x R is the central extension of

e"~ g)

^E

by ^0, !?),

^{i. e. the} sequence

is exact, the

homomorphism

cp

being

^the restriction.

For the notation see

[7].

^The

proof

^is

straightforward.

^Without

difficulty

^{we can define a class of}

locally isomorphis

Lie groups ^on

x E x Tor if we substitute the torus for In the same way

we get

bigger

Lie groups

by inserting

the whole group

Spl(2n,

R,

E)

^instead

.of

eR,

which remain solvable if we restrict

Spl(2n, !?, E)

^{to a} commutative

subgroup.

A CLASS OF SOLVABLE LIE GROUPS 109

§ 1-3:

^{The Lie}

Algebras

^of

Osz(2n)

^and

Heis(2n).

(13)

PROPOSITION. ^- The Lie

algebra

^{of the} group ^x E x R is the vector space RR (B ^E ~ R

together

with the Lie bracket

Proof -

^{The Lie}

algebra

^{is the}

tangential

space of

eRR

^{x e} x R in

0, 0) [4 ;

p.

88~].

^We calculate its elements with the

help

^{of the two}

one parameter

subgroups 0,

^{0’: x E x I~ defined}

by

^9:

u H o,

0)

^and

()’:

~u H

,u~3).

^From

follows the first statement. To get ^{the Lie brackets} consider the commutant of the two one parameter

subgroups

^{0 and}

^0’,

^{which is}

(id2n, (id2n - a(y,

^from

(9).

^{From the curve} segment

we get

[4 ;

p.

97]

^the tangent ^vector

By

^{the same line of}

reasoning

^we get ^{from the commutant of two different} elements of 0’ the element

(id2n, 0, 2~(x, y)),

and from this the tangent

vector

These Lie brackets are

just

^{those of the}

Heisenberg subgroup.

^The

isomorphic

^Lie

algebras

of the groups I~ ^x E x R are found in a similar way. We call the Lie

algebra

^{M 0 E} oscillator Lie

algebra osz(2n) [8]

and the

subalgebra

^{E 0 1I~}

Heisenberg

^Lie

algebra heis(2n).

^The

algebraic

facts

of §

^1-2

immediately

carry over ^to the Lie

algebras ; especially

Inserting

the basis from lemma

1)

in the Lie

algebra

^RR (B ^E Q)

R, writing - 2(R, 0, 0) = : HR, (0,

ei,

0) = : (0, 0) = :

^and

2(0, 0, 1) = :

^{c we} get for the elements

°

with

’R’

^the

general

Lie bracket relations

which

specializes

^{in the case of}

osz(2n)

^to

(rest zero).

^{The Lie}

algebra

^{of the} group

(11)

is calculated

by

^{the same}

way. Its Lie bracket relations are

It will be shown

in §

^1-6 that this Lie

algebra

^RR

+)

^E

^{is just}

^{the «}

^{adjoint »}

Lie

algebra

^{of RR} Q3 ^E ae ^R.

§ 1-4: Automorphisms

^and Derivations of

Heis(2n).

Let G be a 2n x 2n

matrix,

^{a E}

R, bT

^{a 2n row} vector and a another 2n column vector. Then the matrix

(~ )

^{= : A is an}

automorphism

^of

heis(2n)

^iff

det(A) #

^{0 and the}

defining

relation for

automorphisms

holds. From this we get ^the

automorphism

group of

heis(2n)

With the matrix

Ei:

⁼

( )

the matrix A becomes

A CLASS OF SOLVABLE LIE GROUPS 111

with S E

E).

^{Here we} used that because of

E~3E~ = - 3

every

antisymplectic

^{matrix F}

(i.

^e.

^pTJF

^{= -}

j)

^{can be written}

£iS,

^where

now S ^E

Spl (2n ; R, E). Proof ;

^Given

FTjF = - 3 ;

^{it follows}

that

is 03A31F

^{= :}

S ~ Spl(2n, R, E)

^{and F =}

E 1 S.

^{We have}

det

(A) = I (X In+ ^1.

^Aut

(heis (2n))

^{is a}

n(2n

⁺

1)

^{+ 2n +} 1-dimensional matrix group which

decomposes

^{in two} nonconnected

pieces.

^{We write}

its

general

^element

A(S, b, oc !, sign a).

^{For the}

identity component

^we

have

the inverse is in the both components

respectively

We set the

following subgroups

^{of the}

identity component

Note that the elements of

~’(2n)

^{do not act as} translations ^on

E,

^{as would} do the

matrices (id2n J

^for

nonvanishing a.

^We write the discrete _group

{id2n, ~1} =: ~2-

^{Then we have}

(30)

^Aut

(heis (2n))

⁼

Z~

3

(Spl (2n, ^R, E)

^{(8) Dil}

(2n)))

where

~(2n)

^{is the} group of inner

automorphisms. ~2 interchanges

^the

basiselements _{qi and}

/?’

^{‘ of E. Therefore it is a}

^good

candidate for the

Legendre

transformations known from classical mechanics.

Spl(2n,

^I~,

E)

can be

interpreted

^{as the} group of canonical transformations of ^a linear

phase

space E.

D

e gl(2n

⁺ 1, R, ^{E 0}

R)

^{is a} derivation of

heis(2n)

^iff

The derivations of

heis(2n)

^{form a Lie}

algebra

^of linear transformations which is

given by

^{the set of}

matrices (), ^{where p}

^{is an}

^arbitrary

^{2n vec-}

tor, fJ

^{e R and N is a 2n x 2n matrix}

subject

^{to the condition}

Therefore

where

A(2n)

is the Lie

algebra

^of

~(2n)

^{and dil}

(2n)

that of Dil

(2n).

§ 1-5: Automorphisms

and Derivations of the Oscillator Lie

Algebras.

Given 0 # R E

spl (2n, I~, E),

^the centralizer

of

^{R in}

spl (2n, R, E)

^is

and the centralizer

of

^{R in}

Spl (2n, IR, E)

^U

E1 Spl (2n, ^IR, E)

^is

Let

3K

denote the vector space of all V E

spl (2n, R, E) subject

^to

where _Vv ⁼ 0

only

^{for V =}

0,

^and

lR

the manifold of all

with

where

ðo

^{= 1}

only

^{for G = Then we have}

(35)

LEMMA. - The set of elements V ^e

spl (2n, R, E)

^with

is just

^the matrix Lie

algebra

^zent

(R)

^(B

3~.

A CLASS OF SOLVABLE LIE GROUPS 113

The set of elements G E

Spl(2n, R, E)

^u

Ei Spl(2n, R, E)

^with

is

just

the matrix _group

Zent(R)

^x

lR.

In this Lie

algebra (resp.

^Lie

group) zent(R) (resp. Zent(R)) ;s

^{an ideal}

(resp.

^normal

subgroup).

Proof. Every

^{element V with}

(36)

^{can be}

decomposed uniquely

^into

V ⁼

Vo

⁺

V~

^{where now}

Vo

^{E zent}

(R)

^{and V E}

3R. By

definition from V E

zent(R)

ⁿ

3R

^{follows V =}

^0,

^{i. e. the vector space sum of}

zent(R)

and

3R

is direct. From the Jacobi

identity

follows that zent

(R)

^{is an ideal.}

The

proof

for the group theoretical statement is similar..

For the

physical interesting

^{R the vector} space

3R

^is one-dimensional

or zero. Then the matrix Lie

algebra

^{of the V E}

spl(2n, E)

^with

(36)

is

zent(R) O ^3R

^or

^{only zent(R).}

^The

corresponding

facts hold for the groups.

(38)

THEOREM. - Given 0 # R E

spl(2n, R, E), R 2 #= 0,

^an element of Gl

(2n

⁺

2, I~,

^R 3 ^E 0

R)

^{is in Aut}

(RR x E x R) iff it

has the form

where we have

GRG-1

⁼

5R,

^{i. e.}

GeZent(R)

^x

lR’

^and

S E

Spl (2n, R, E).

/~ ^dT

y

Proof. -

^We insert the

matrix

^{c G}

^a

^{into the}

automorphism

^condi-

iion

(19)

of the Lie

algebra

^relations

(16).

^This

gives

^for

03BER ~ 0, y ~ 0,

rest zero

for x # 0, y ~ 0,

^{rest zero}

and

0,

110 #

0,

^{rest zero}

From

(f)

^{we have det}

(G)

⁼

an;

^since every

automorphism

^{must be inver-}

tible from

(g)

it follows that a ⁼ 0 iff G invertible. Let G be not invertible : i. 0 and a ⁼ 0. From

(e)

^we have for x ⁼ Rz

and y

^{= Rv with the}

help

^of

(a)

^{for all v, z E E : 0 =}

Rv)a

^{= -}

R2v)a,

^{i. e.}

0,

which is not true. So G must be invertible. From

(b)

^we get ^{with the}

help

^of

(a)

^for y ⁼ Rz and all z E E : 0 #

GR2z

⁼

03B4RGRz,

^{i. e.} 03B4 ~ ⁰ and RGR ~ 0. For x ⁼ Rz we get therefore from

(e)

^{for all} y, ^{z E}

E, y)RGR

⁼

0,

^{i. e. d = 0.} The result for the vector c follows from

(c)

and

(f),

^{and the rest of the} theorem from lemma

35)

^and

(21) ff. 1 (40)

^{THEOREM. - Given 0 # R E}

spl(2n, R, E), R2 1= 0,

^{an element of}

gl (2n

⁺

2, R,

^R 3 ^E (B

R)

^{is in E 0}

R)

iff it has the form

where

[V, R] - = vR,

^{i. e. V E zent}

(R) ? 3R c: spl (2n, IR, E).

The

proof

^{is similar to the}

proof

^{above. For the}

special

^{case R = 3}

(the

^harmonic

oscillator)

^we get

(41)

COROLLARY. -

Aut(osz(2n))

^is

given by

^all matrices of the form

where G is

subject

^to

(39)

^{and S E Zent}

(3)

⁼

U(n, R, E).

A CLASS OF SOLVABLE LIE GROUPS 115

der(osz(2n))

^is

given by

all matrices of the form

Proof.

^{- From}

[V, J] _

^{= v3 and}

^yTJ

^{+ 3V = 0 follows V +}

^VT

^{= v}

id2n.

Since

spl(2n, I~, E)

^{we have v = 0. For the} group ^we have

(sign oc) id2n,

from which we have 5 ⁼

sign

^{a, since}

^STS

^E

Spl (2n, E)

is

positive

definite. The rest follows from

(5)..

§

^{1-6: « Self}

»-representations

^and

adjoint representations.

The invertible matrices

(~ being

the column vector

corresponding

^to

x)

are the elements of the _group

Spl (2n, IR, E) 3

^Heis

(2n)

^{with the}

composi-

tion law

iff

S,

^{U E}

Spl (2n, R, E).

^{For S}

^{= e°‘R}

^{and U = we} get ^{a 2n +} 2-dimen- sional faithful

representation

^of

eRR

x E x

R,

^{for R = 0 of Heis}

(2n).

The lie

algebra

^{of this} matrix group is

given by

the matrices

which have the commutation relations

iff V,

^{Z E}

spl (2n, IR, E).

^{For V =}

çRR

^{and Z = we have the commuta-}

tion relations of

proposition 13).

^So

(44) gives

^faithful

representations

of the Lie

algebras

^RR 0 ^E 3

R,

^{for R = 3 of osz}

(2n),

^{and for R = 0}

of heis

(2n).

The

adjoint representation

^{ad : RR} EÐ ^E ~ R ~

inder(RR

EÐ ^E EÐ

R)

onto the inner derivations ad

(RR

^{EÐ E EÐ the} kernel of which is the center

(0, 0,

^is

given by

These matrices form a faithful 2n + 2-dimensional

representation

^{of the}

Lie

algebra (18); they

^{are an ideal in}

der(RR

^{0153 E} EE&#x3E;

R).

^{The exact}

sequence from theorem

12),

^{read in the Lie}

algebraic form,

^{is thus}

nothing

else than the well known exact sequence of the

adjoint algebra

^{of a Lie}

algebra.

The

adjoint representation

^Ad: exp

(RR)

^{x E x R - Int} 3 ^E (B

R)

onto the inner

automorphisms

^{of RR ED E 0153}

R,

the kernel of which is the

center

(0, 0,

^is

given by

These matrices form a faithful 2n + 2-dimensional

representation

^of

the Lie _group

(11); they

^{are a normal}

subgroup

^{in (B E 3}

Thus theorem

12) gives

^{the exact} sequence of the

adjoint group e~R

^{Q9 E}

of exp

^{x E x R}

[4 ;

p.

116].

^The restriction to heis

(2n)

^{of the above}

matrices

gives

^{for rx = 0 the}

adjoint representation

^{of heis}

(2n)

^{and Heis}

(2n),

which

correspond

^to

A(2n)

ⁱⁿ

(32)

^{and to}

.~ (2n)

ⁱⁿ

(30).

^Since

the determinant of the matrices

(47) equals

^{1; therefore} the groups

x E x R are unimodular

[4;

p.

366].

PART II

THE WEYL ALGEBRA

§ II-7:

Definition of the

Weyl Algebra.

Let

ten(heis(2n))

^{be the tensor}

algebra

^{of the vector space E E9 ~c}

of

heis(2n),

^{0 the tensor}

multiplication,

^and

([a, b]- - a))

A CLASS OF SOLVABLE LIE GROUPS 117

the two-sided ideal of

ten(heis(2n)),

^{which is}

generated by

^{all elements}

of the form

[~]- 2014 (a

^{0 b - b 0}

a) with a,

^{b E heis}

(2n) c

^ten

(heis (2n)).

Then the infinite dimensional associative

algebra

is called universal

enveloping algebra

^{of heis}

(2n).

^For the notion of uni- versal

enveloping algebras

^{and the}

following

statements see

[4 ;

p.

90],

[9 ;

p.

151 ], [l o ;

p.

26]

^and

[11 ; expose

^no

1] :

(49)

^{LEMMA. -}

heis(2n)

is imbedded

injectively

^{in u}

(heis (2n)).

(50)

LEMMA. 2014 We have

inj (heis (2n))

^{n R1}

= { 0);

^{here 1 is the}

identity

element of ^u

(heis (2n)),

^and

inj (heis (2n))

^{is the}

isomorphic image

^{of heis}

(2n)

in u

(heis (2n)).

(51)

LEMMA. - A basis of u

(heis (2n))

^is

given by

^the

identity

^element

and the standard monomials of the basis elements of

inj (heis (2n)).

Because of lemma

(49)

^we

identify

^heis

(2n)

^and

inj (heis (2n)).

^Because

of lemma

(50)

^{we cannot}

identify

^{the element}

inj (c)

^of

inj (heis (2n))

^with

the

identity

element of ^u

(heis (2n)).

^But

actually

^this

always

is done in

physical applications,

for instance in the Poisson bracket Lie

algebra

^of

position

^{and momentum} variables in classical

mechanics,

^{and the commuta-}

tor Lie

algebra

^of

position

^{and momentum} operators ⁱⁿ quantum ^mechanics.

Therefore we consider instead of u

(heis (2n))

^{a different} noncommutative associative infinite dimensional

algebra

which identifies ^c and the

identity

element. Let

(c - 1)

^{be the} two-sided ideal of u

(heis (2n))

^{which is} gene- rated

by

the elements ^{c -} 1 E u

(heis (2n)).

^{Then the}

algebra

is identical with the

algebra

where ten

(E)

^{is the tensor}

algebra

^{over the vector} space

E,

^{1 the}

identity

element of

ten(E), and x, y

^{E E}

ten(E).

^{We call this}

algebra Weyl algebra [l2;

p.

148).

^heis

(2n)

is embedded

injectively

ⁱⁿ

(1).

A basis of

weyl(E, ~)

^is

given by

the standard monomials of the basis elements of

weyl(E, a)

^{and 1. One should} compare

(53)

^{with the}

definition of the Clifford

algebra

^{over an}

orthogonal

^vector space

[12 ;

p

148], [13,

p.

367].

For the

following

^{we need two} other universal

algebras.

^Let

sym

(heis (2n))

^be the universal

enveloping algebra

^{of the} trivial Lie

algebra

on the vector space E E9 Rc of heis

(2n),

^we get

by demanding

every Lie bracket to

vanish,

^{i. e. for} x, heis

(2n) c:

^ten

(heis (2n))

It is clear that the universal

enveloping algebra

^{of the} commutative trivial Lie

algebra

^on

E,

^{i. e.}

is

isomorphic

^{as an} associative

algebra

^{to the}

algebra

Let us denote the inclusion _sym

(E)

^- sym

(heis (2n)) by j,

^the

projection u(heis (2n))

^-

weyl (E, u) by 03C0

^{and the vector} space in ^u

(heis (2n)) generated linearly by

^{all monomials} x1x2 ^... xi

(for all xk ~

^heis

^{(2n) ~}

^u

(heis (2n))) by

^u

;(heis(2n))

^{with uo}

(heis(2n)) :

^{= Rl. The vector} space

is

generated linearly by

the monomials _xix2 ... x;

with ~

^{E E c:}

weyl (E, 6).

We have

Wo

⁼ [Rl. In the

following

^{we denote the}

product

^{of X and Y}

in _sym

(heis (2n)) by X. Y,

^and

by

^{A’ the}

mapping

^{which we} get

by

^linear

continuation from

(here

y~ denotes the

symmetric permutation

group of k

objects).

(58)

^{LEMMA. - The}

mapping

is a vector space

isomorphism (i.

^e.

bijective

^and

linear).

This lemma is due ^to Harish-Chandra

[4 ;

^p.

98,

p.

392].

^Lemma

(51),

the

Birkhoff-Witt-theorem,

^shows that the total

symmetrized

^standard

monomials form a basis of the commutative

algebra

sym

(heis (2n)),

^and

from lemma

(58)

^{we have the same basis for u}

(heis (2n)).

^{We define a}

mapping

^{A :} sym

(E) -weyl(E, 0’) by

^{1B: = x o A’}

o j,

^{which is linear} and

bijective.

^It follows that the total

symmetrized

standard monomials

A CLASS OF SOLVABLE LIE GROUPS 119

of the basis elements ^{of E E}

weyl(E, cr)

^{and the}

identity

^{element 1 form}

a basis of

weyl(E, 0’).

^{Let us} write in the

following

The

decomposition weyl(E, o-) ===

^{!R1 +}

W 1

⁺

w 2

^{+ ... of}

weyl(E, (1)

is not

direct,

^{since the vector} spaces

Wi

^{are not}

disjoint

for different

indexes;

for instance the element _{xy - yx} is in Rl and in

W2.

^We get ^{a direct}

decomposition

^of

weyl(E, 0)

^{if we} consider the vector spaces

AW,,

^defined

as

linearly generated by

^the

symmetrized

standard monomials of

degree i

instead of the

W~

To prove this we

apply

the linear transformation ad

(pl)il

^{... ad}

(p"/"

^{ad ... ad}

(qn)kn

^of

weyl(E, cr)

^{on the}

equation

showing that ç

^{= 0.}

Continuing

^{this process causes all} coefficients to

vanish,

^{which is}

equivalent

^to the directness of the

decomposition (60).

Even in this

decomposition weyl(E, r)

^{is not}

graded,

^{i. e.}

for instance we have

= A~ + . ^~)1.

^{° Instead we}

tion on

weyl(E, cr)

from which we can construct a

graduation by

^standard

procedure [9]

^and

[11 ; expose

^no

1];

but since the

resulting algebra

^{is com-}

mutative

(in

^{fact it is}

isomorphic

^to sym

(E),

^cf.

[77]),

^{it does not seem to be}

interesting

^for

physical applications.

^{It is} easy ^to prove the formula

with the

help

^{of which we} get

by

^induction

From this we have

from this and

(60)

follows that the center of

weyl(E, 0)

is trivial

[12,

p.

148].

ANN. INST. POINCARÉ, A-X III-2 9

§

^{11-8: Some Lie}

Algebras

ⁱⁿ

weyl (E, 0).

We now look for Lie

algebras

^{which are formed}

by

linear and bilinear

polynomials

^of

elements xk ~

^E

weyl(E, r).

^{We have for}

Wi

^E9

AW2

For x _{= y} and v ⁼ z the first

equation

^{reduces to}

which is

equivalent

^{to the}

original,

^since

by linearizing

^{it twice}

(i.

^e.

by substituting x F+ x

+ y

and

^{z H- v +}

z)

^we

regain

^the

original

^one.

(For commuting x and y

^{we can}

drop

^{A in}

Axy).

(68)

LEMMA. - The normalizer

of

^{E in}

weyl(E, a) (i.

^{e. the set}

equals AW2.

The

proof

^{follows from}

(60), (63), (64)

^and

(66).

^{We have}

t,x

which suggests ^to introduce in

AW2

^the

n(2n

⁺

1~

basis elements

They

have the commutation relations

[14 ;

p.

1203]

We prove that the elements of

A W 2

span the Lie

algebra spl(2n, R).

First note that from the Jacobi

identity

ⁱⁿ

weyl(E, a)

^{we have}

121

A CLASS OF SOLVABLE LIE GROUPS

i. e. the 2n x 2n matrices ad

(ad

restricted to

E)

^{form a Lie}

algebra,

^{which is a}

subalgebra

^of

spl (2n, I~, E)

^{because of}

It is

straightforward

^to prove that the kernel of the Lie

algebra

^homo-

morphism

^R:

spl (2n, R, E)

^defined

by

is zero. From a dimensional arguement ^now follows that this homo-

morphism

^{is a Lie}

algebra isomorphism.

^It follows that the Lie

algebra

Rl p

Wi

^ae

AW2

^is

isomorphic

^to

heis(2n) ~+ ^{spl (2n,} M). Choosing

^now

a

special

element of

AW2,

say

HR,

^we get the commutation relations

(16)

of

heis(2n)

^{where now}

This is for instance for the Hamilton operator ^{of the} harmonic oscillator

In the

following

^{we list some}

s ubalgebras of AW2

^{and their}

images

^under

the action of the

isomorphism

^R:

The n2

elements

N:

⁼

Aqipk

^{form a}

subalgebra

^of

AW2

^with

which under the action of R is

gl(n, R, E) given by

and, using

the matrix

exponential mapping,

^we get

where N is some linear combination of the

N~.

The 1 2 ^{n(n - 1)}

^{elements =}

^{^qkpi}

^{form a}

subalgebra

^of

this Lie aleebra with

which under the action of R is

isomorphic

^to

so(n, E) given by

^{the above}

matrices

(77)

^for

antisymmetric

^{G. The}

corresponding

matrix Lie _group is

generated by

the matrices

(78)

^for

orthogonal ^eG. The 1 2 ⁿ⁽ⁿ

⁺

¹⁾

^ele-

ments

Hik:

⁼

A(qiqk

⁺

pipk)

^form

together

^{with the}

Mjk

another Lie

algebra

in

AW2

^with

Under the action of R this Lie

algebra

^is

isomorphic

^to

u(n, R, E) :

with L a real

antisymmetric

^{and K a real}

symmetric n

^{x n matrix}

(see § 1-1)~

and M resp. H ^a linear combination of the

Mik

resp. the

Hik.

^{The corres-}

ponding

matrix group

U(n, R, E)

^is

given by

^{the matrices}

(4).

§ 11-9:

Realization of some Derivations and

Automorphisms.

We want to

identify derivations,

_resp.

automorphisms,

with elements of the form

ad(Z)

{2n), resp.

exp(ad(Z)

for suitable Z ^E

weyl(E, ~).

Besides the inner derivations of

heis(2n),

^{which are}

given by

^ad

(E) |heis

(2n) we have the derivations ad

(AW2)

(2n) of

heis(2n) :

(note (66)).

^{In the} notation of

(32)

From

(64)

^{now follows that no elements of ad}

(weyl(E, u)) Iheis

(2n) ^can

be identified with derivations

given

^{0 in}

(32).

^The

corresponding

facts hold for

automorphisms given by

^{1 in}

(21)

^{and the}

special

^auto-

morphism

0,

1,

^-

1).

From

(64)

follows that there is no

representation by

elements of

ad

(weyl(E,

^of those derivations of which

are

given by nonvanishing

p

and J3

ⁱⁿ

(40).

^The

remaining

derivations

A CLASS OF SOLVABLE LIE GROUPS 123

in

(40)

^are

given by

the elements of ad

(E)

^and

by

those elements Z E

A W 2

^which

obey

the conditions

The

corresponding automorphisms

^{of the Lie}

algebras

^{RR (B E fl3}

^R,

see

(38),

^{are now}

generated by

^{the matrix}

exponential mapping

^{of these}

derivations.

§

^{II-I0: The}

Special

^Case

of the

(Nonrelativistic )

^{Free Particle.}

We discuss the above results for the

free particle

^{whose Hamilton} operator is

given by

Ri

^{= 0.} Because of this

nilpotency

^theorem

(38)

^{is not}

applicable.

We show that for n 2 theorem

(38)

^{holds in the same}

^form,

^{i. e. that}

the

automorphism

group of

IRRF +) ^heis(2n)

^is

^{given by}

the matrices

(38)

with

0 #= ð #=

^{1 in}

general.

^It follows that the elements of

are

given by

the matrices

(40)

^{with v # 0 in}

general.

^{First note that}

from

(b)

^{in the}

proof

^{of theorem}

(38)

^{we have for}

^RZ

^{= 0}

(multiplying by

^{R from the}

left)

^{RGR = 0. So from}

(e)

^{we have for x = Rz with the}

help

^of

(a) a

^{= 0. This is true for all R with}

^RZ

^{= 0.}

What remains ^to show is d ⁼ 0. From

RFGRF

^{= 0 and G invertible}

follows that G has the form

O

^{with A} invertible. It follows from

(e)

^{for x = d}

Writing yT

⁼

y2T)

^{and the same} for d where now yl, y2,

di , d~

^are

n column vectors, ^from

(a) follows di =

^0.

Denoting

the scalar

product

in n dimensions

by , ),

^{we have from}

(e’)

^{since A} is invertible for all n vec- tors y~ the

condition d~, ~2 = ~ d~,

^{from which} for n &#x3E; 2 follows

d2

^{= 0.} To prove this we used two

peculiarities

^of

Rp,

^{which will}

not hold in

general

^{for all R with}

^{R 2}

⁼

0, namely ~

^{= 0 and the}

special

form of G above.

The set of elements of

spl (2n, IR, E) obeying

^the

equation (36)

^for

RF

is the set of matrices

( B0 ~

^{with B =}

^B~

^{= v}

For

A - 1 2

^v

^idn

^{= : L we get L = -}

^LT.

^{’ The}

ⁿ²

⁺ 1-dimensional Lie

algebra zent(RR)

^is

given by

^{the matrices}

() + 1 2v().

Contrary

^to the harmonic oscillator

(R

⁼

3),

^{where v = 0 from}

(41),

^for

the free

particle

^{v ~ 0. It is}

straightforward

^to show that because of theorem

(38)

^the

automorphism

group of

tRRp +) ^heis(2n)

^is

^{given by}

^the

set of matrices

where the 2n x 2n matrix G has the form

and the n n matrices

A, B, C,

^{D are}

subject

^to the conditions ⁼

C~C

⁼ A~B and

C~D symmetric.

The one-dimensional Lie is the

image

^of

1 2 R03A3^qlpl

^{under the}

isomorphism R ; the 1 2 ^{n(n -} 1)-dimensional

^Lie

/=i

algebra

of the matrices

(~ ~) ⁾

^{is the}

^R-image

^{of the}

^IR)

^Lie

^alge-

bra

(78),

^{and the}

commutative 1 2 ⁿ⁽ⁿ

⁺

1)-dimensional

^Lie

algebra

^{of the}

matrices

(~ ~)

^{is the}

^R-image

^{of the}

Note that the kernel of the

homomorphism