Invariant operators of inhomogeneous unitary group

A NNALES DE L ’I. H. P., SECTION A

M. R ^AFIQUE K. T ^AHIR S ^HAH

Invariant operators of inhomogeneous unitary group

Annales de l’I. H. P., section A, tome 21, n

^o

4 (1974), p. 341-345

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p. 341

Invariant operators

of inhomogeneous unitary group

M. RAFIQUE and K. TAHIR SHAH International Centre for Theoretical Physics, Trieste, Italy

Ann. Henri Poincaré, Vol. XXI, ^n° 4, 1974, )

Section A :

Physique ^’ théorique.

ABSTRACT. ^- Let

u(n)

be the semi-direct sum of an abelian Lie

algebra

with the Lie

algebra

^of

unitary

group

U(n).

^It is shown that ^an invariant

of u(n

⁺

1 )

^{is also an invariant}

u(n)

^{u --+}

u(n

⁺

1 )

is an

expansion.

Sixth-order invariants of semi-direct sum

algebra

^are

calculated

by

this method.

I INTRODUCTION

In this note we discuss the

problem

^of

constructing

invariants of inhomo- geneous

unitary

group

using

^a short-cut method which is

specifically

useful for the calculation of

higher-order

invariants and Casimir operators which are needed for the

representation theory.

There are various methods of

constructing

invariants of the inhomo- geneous

unitary

group. ^{We use a} method first

given by

^Rosen

[1]

^to compute

some of the

invariants

of the

inhomogeneous orthogonal

group and which

was later

generalized by Nagel

^{and Shah}

[2]

^{to include all cases of inhomo-}

geneous

symplectic

^and

orthogonal

groups. We ^start from the Lie

algebra

of the group

T,,(x U(ni, n2)

which is the semidirect

product

of n-dimen- sional translation with

n2)

group. This is

expanded

^to

u(nl,

n2 ⁺

1)

such that the generators ^of

u(nl,

n2 ⁺

1)

^are functions of the generators of

u(ni, n2). Here t"

^and

n2)

^{are the Lie}

algebras corresponding

to the groups

Tn

^and

U(ni, n2).

From the Casimir operator ⁺

1),

which _{is easy} to

calculate,

^{we extract} invariants of

n2) by using

a lemma

given

^below.

In Sec.

II,

^we describe the construction of the generators ^of the

algebra u(nl,

n2 ⁺

1)

^and

give

the commutation relations. Sec. III describes the method of calculation. In Sec. IV we

apply

^this

^method

^to calculate the sixth-order invariants

of tn

Q-

u(n).-

Annales de l’Institut Henri Poincaré - Section A - Vol. XXI, ^{n° 4 - 1974.}

342 M. RAFIQUE ^AND K. TAHIR SHAH

II. EXPANSION OF TO

u(nl, n2 + 1)

In order to use the

expansion

process in the

computation

^of

invariants,

we first have to define it.

Let g

^{be a Lie}

algebra.

Consider the universal

enveloping algebra

^of

Ig

^{in its}

polynomial

^form.

Here, Ig

represents

an

inhomogeneous algebra, being

^a semi direct sum of an abelian part ^I and a

semisimple algebra

g. Now extend the

polynomial algebra

^to

an

algebra EE(Ig)

^{such that not}

only complex

^numbers appear ^as coefficients but also

functions f (C1, C2,

^...,

CN)

^with

Ct, C2,

^...,

CN being

^elements

in

e(Ig) generating

^{its centre. Then an}

injective homomorphism

^{cp of a Lie}

algebra

g ^{into i.} e.,

is called an

Ig

realization

[3]

^{in or an}

expansion.

One should note

that, applying

^{the later}

given mappings

~p ^to the Lie

algebras

of the classical groups, ^one

obtains g

^{of the same type} as g except their dimensions. The method is

equally applicable

^to compact ^{as well as} noncompact groups.

We consider

n2),

^{where n =} n 1 ⁺ n2, is the real Lie

algebra

^of

complex unitary

transformations of an n-dimensional ^vector _space

Vn.

The generators ^of

n2)

^are

Ru

^and

Su,

with the metric

given by

whereas the commutation relations are

Annales de l’Institut Henri Poincare - Section A

INVARIANT OPERATORS ^OF INHOMOGENEOUS UNITARY GROUP 343

Following [4],

^{we construct the}

generators

^{of the Lie}

algebra

^{n2 +}

^{1 )}

as an

Iu(n)

realization ^{via the}

following mapping

^rp

(the generators

^of

n2 ⁺

1 )

^{are denoted as}

etc.) :

where

and

The

braces { , }

^{mean that each term is to be}

symmetrized

ⁱⁿ

Ku and S

with respect ^to

R.

^and

S"

^{but not relative to one}

^another,

^and

divided

by

the number of ^terms needed for

symmetrization.

^{An invariant}

and a Casimir

operator

^{are defined as follows :}

Invariant: let

S , R )

^{be a}

homogeneous polynomial

ⁱⁿ

L v,

^{S and}

R".

Then I is said to be ^an invariant ^if it commutes with

S

^and

R,..

^{The order of an invariant is the} order of the

poly-

nomial.

Casimir operator: ^a Casimir

operator

^{is an invariant which cannot be}

expressed

^{as a linear} combination ^of lower-order invariants. ^In

general,

an mth-order Casimir

operator CJG)

^{for a group G is}

where are the

generators

of the Lie

algebra

^{of G.}

III .

COMPUTATION

^OF INVARIANTS ^OF

Let

u(nl, n2)

^{which is}

expanded

^to

u(nl,

^{n2 +}

1)

^{and let}

+

1)

^such

that ha

⁼ ~2...

gN)

^{where N = n(n +}

1)/2.

The Casimir operators ^{of n2 +}

1)

^are functions of gl, ^..., ~ since

where

Cm

^{is the} mth-order Casimir operator ^of

u(n 1, n2

⁺

1 )

^{and M is the}

Vol. XXI, n° 4 - 1974.

344 M. RAFIQUE ^AND K. TAHtR SHAH

number of generators n2 ⁺

1 ).

^Let

X 1

^{be an invariant}

of tn~u(n1, n2),

then

Let

X2

^{be an} invariant of ₊

1),

^{then we have}

LEMMA.

If u(nl,

n2 ⁺

1)

^{is reatized in}

u(nl, n2)) through

^the

expansion

cp, then _any operator ^X

commuting

^with

ha

^E

u(nI,

^{n2 +}

1)

^atso

commutes with _gi E tn C

u(nl, n2).

Proof

^The generators

L’ ,n+

^{1 and}

Q’ ,n+

^{1 are,}

^by construction

&#x3E;

functions of

S~

^{and One can}

^verify

^{that any operator X which}

satisfies the relation satisfied

by X2

also satisfies the relation satisfied

by X 1.

We show that

[Rp., ^X]

^and

[S,", ^X]

^{are zero as} follows. Consider the commu-

tator

[Q~+

l,n+ ^{1 ~}

X]

and substitute for _{1,n + 1} to get

which reduces to an

expression involving

commutators of the type

X], [R"RB X]

^and

[S"SB X]

^because

X]

^and

X]

vanish due to

@.

Now

[Ln+ 1 v’ X]

^{= 0}

implies

^that

or

Notice that we have divided

by

/!, the last three

equations

^of

@.

^{Now we}

may remark that if X ⁼

X(x) =

^{i =}

^0,

^{1 ... with parameter a,}

and if I is another operator

independent

^{of x, then I commutes with}

Xi

if and

only

^{if it commutes with}

X(x).

^{This means that}

Using

further the commutation relations of and

Q

^with

^X,

^{we have}

also

[S~ X]

^{= 0.}

^So,

^{if X is an} invariant of _n2 +

1 )

^then it is also an

invariant of

n2). Q.

^{E. D.}

IV . SIXTH-ORDER INVARIANTS OF

Using

^{the method}

developed

^{in Sec.}

III, starting

^from the second-order Casimir operator ^{of +}

1 )

^we determine invariants of

second,

Annales de l’Institut Henri Poincaré - Section A

345

INVARIANT OPERATORS OF INHOMOGENEOUS UNITARY GROUP

fourth and sixth order for the Lie

algebra n2).

The second-order Casimir operator ^{of +}

1 )

^is

In

polynomial

^{form one can have}

where

^{is some} parameter ^{related to 03BB.}

Substituting

^from

equations @

and

using

commutation relations we have the

following

invariants

(all

the indices are written as

subscripts

because for

simplicity

^{we take}

ACKNOWLEDGMENTS

The authors wish to thank Prof. H. D. Doebner for

correcting

many errors, for discussion and critical

reading

^{of the}

manuscript.

^{Thanks are}

due to Prof. Abdus

Salam,

the International Atomic

Energy Agency

^and

UNESCO for

hospitality

^at the International Centre for Theoretical

Physics,

^Trieste.

REFERENCES

[1] ^J. ROSEN, ^{J. Math.} Phys., ^t. 9, 1968, p. 1305 ; Nuovo Cimento, ^t. 46B, 1966, p. 1.

[2] ^J. G. NAGEL and K. T. SHAH, ^{J. Math.} Phys., ^t. 11, 1970, p. 1483.

[3] ^H. D. DOEBNER and T. D. PALEV, ICTP, Trieste, preprint IC/71/104.

[4] ^{J. G.} NAGEL, Ann. Inst. Henri Poincaré, ^t. 13, 1970, p. ^1.

(Manuscrit reçu le 17 décembre ^, 1973)

Vol. XXI, n° 4 - 1974.