IC/72A9

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

CONJUGACY AHD STABILITY THEOREMS

IN COHMECTIOB WITH CONTRACTION OF LIE ALGEBRAS

K. Tahir Shah

INTERNATIONAL

ATOMIC ENERGYAGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1972 M.IRAMARE-TR1ESTE

IC/T2A9

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CONJUGACY AND STABILITY THEOREMS

IN CONNECTION WITH CONTRACTION OF LIE ALGEBRAS *

K. Tahir Shah

International Centre for Theoretical Physics, Trieste, Italy,

ABSTRACT

Let a-gCfa). the n-dimensicnal Grassmannian variety, and g be a Lie-algebra

of the dimension m^-n. We study some properties of the set B<£.G(a)-G(a)

where U(a) is the Zariski closure of the orbit of G=GL(g) at the point a,

The group GL( ;) is the group of automcrphiams of ~£. The set B is shown to

be the set of Lie algebras which are contractions of "a. IT is a subalgebra

of f?. The main results are the theorems on the ccnjugacy of contracted

algebras and the stability of Lie -algebra, a . under contraction. The

conjugacy theorem relates the algebras in the set B.

MIRAMARE - TRIESTE

June 1912

* To "be submitted for publication.

O-introduction.

The problem of limit and contraction of Lie -algebras and

their respective groups has been a point of interest to many mathematical

physicists in connection with various problems in physics, especially these

related to the symmetry and dynamical groups- of elementary particle1-1 theory.2)

The operation of contraction was first defined by Segal in the context of

operator algebras of quantum mechanics and later on.it was used by Incnu• 3 )

and Signer to shew the relationship between the relativistic and non-

relativistic theories. There were further studies of this by Saletan, .Levy-.5) l)

lianas and Doebner from the mathematical point of view but using mostly

basis-dependent calculations.

Some of the interesting properties of the contraction

of Lie groups and their representations can be used', to show the relation

among the harmonic functions. For example, one; can obtain the -relation

Limit P ^ (Cos t/i ")' - J (t)

where ?. and J are Legendre and Bessel functions respectiv&Ly,, by usin.g

contraction techniques. There are many more examples where it. j;s possible

to show that limiting techniques are really useful. Although • ;t^ere is a

large amount of work on special cases of Lie--groups and Lie- algebras, there

has been little progress in .obtaining general theorems on the -properties

of contraction, especially those related with the classification of contract-

ed ' algebras and also the stability of algebras under the operation of

contraction.

In this paper, we start from the definition given by1 6)- >

H.Hermann and,u^ing methods of algebraic geometry^prove general theorems

i:ig the existence, stability and classification of Lie-algebra ccn-

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fractions, thanks to the tiethods developed "by D.i mnford. in his book on the.

• geometric invariant theory, it is possible to obtain these results in a

simple Banner. The crucial point in this whole development is the general-

ization & fttHilbert-Muraford theorem to the case of Grassmannian variety.

Let ~Q he a Lie algebra of dimension m and P ( g~ ) "be a Grassmannian variety

where mj^n. Let J3: G—>GL{TI {W)) t>Q a rational representation of algebraic

G and a,eV (g) . If G(a) be the Zariski -closure and the set BCG(aJ-G(a)

is non-empty, then there exists a one-parameter subgroup X,: G —^G such that

as some parainoter t —>0 then ^tjta)-^!)^!;. This is a generalization of Hit

Eil"bert-i-Iumf ord theorem in which case G is (X. linearly reductive algebraic

group which has representations on it-space V and the set 3= {o\, We shall

show in the following that the non-empty set B is a set of all contracted

Lie algebras b of a given algebra *£".

Section 1 deals with some preliminaries and the. notation

to be used. Section 2 describes the connection ffflSe con tract ion problem with

the generalisation dfltle.iiilbert-Mumf ord theorem for Grassmannian variety. In

5ection 3 some theorems on the, existence of contraction are given when G

is reductive (or linearly reductive) and torus, dsfined over an algebraically

closed, field k. The main concern of Sections 4 and 5 is ^ae stability and

classification problems.

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1-J.re^irninaries and i'otatioii*

Definition .1.

A Lie algebra"g on a, field, k is a pair (u , V) fcrmed ~oj a

linear vector space V whose coefficients are in k and a "bilinear mappinfa

/A : YXJ-^V which satisfies

x,y,z £ V

where ?(x,y,z} are cyclic permutations cvsr x,y and z.The mappings

bilinear alternating mappings. Let the dimension of g= (yU_,V) "be ni

For each, positive integer n .m, the protective a.lge"bruic

variety of n-dimensional sub-spaces of Lie algebras "g corresponding to the

Lie /jroup G is a Grassmanniaa variety denoted "by P (g) and has a natural

structure of algebraic transformiition space for the algebraic group GL(g)

or GL(V).

Definition 3.

A rational representation of: an algebraic .group G; is a

morphism p: G —^GL V7here characters are defined by % : G—^GL-.» The

orbit of G at'x£V ia denoted as G(x) and is a set: G(x)= | g oc I g.z=

We write g.x for P(g).x where • (°i G -^GL(V) is a rational representation

of G.

Dexinition 4«

A one-parameter sub-group of an algebraic group G is a

morphism ^: G —> G of algebraic groups. .

Let f; G —> X te a morphism of algebraic varieties, then one

Let a and 1> le/fcwo points of f (g), then b is a specialization of a,

Definition

may define f(t)= X i : ) . ^ Y x £K". If f extends to a morphism • /:. Gx/ .. t

then y= f(o) is called the specialization of f(t) as t specializes to aero.

This is written as f(t}—>y as t

Remark 1.

if a —^"b as t —^0 where a = \(t).a. Correspondingly a Lie- algebra b is 1

specialization of Lie- -algebra a if in the limit t —>0 it is approached by

V

Definition 6.

An algebraic group G defined over a field k,which has nc non-

trivial one-parameter subgroup defined over k, is said to be anisctropic

over k. This means G has no ncn-trivial k-split torus.

Definition 7.

An algebraic group G is reductive if its unipotent radical is

trivial and is linearly reductive if all its rational representations are

completely reducible.

Hemark 2.

Tfhon the characteristic of the field k is zero then terms reductive

and linearly reductive are equivalent.

definition 8.

An algebraic set can be defined as follows: let k be an algeb-

raically closed field and A be an affine space. Let k [x...... X J denote

the ring of polynomials in JL , X?,...,X with coefficients in k. Denote

this rin,j as k[ _An J . If B is a proper subset of _An, then the set P^kj^A. J

such that F(x)=oV;:6B is an ideal in k£_An] , denoted as _I (B) . A subset

H of _An is called an algebraic set in A_n if and only if H=V (_I (V)) where

,V(F) is the set of zeroes of F called the locus of F. This means that M'is

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an algebraic set in A? if and only if V is the locus of some ideal in k F A*1],

Also one can write

.. II (F)F

Remark 3.

The set 14 of all Lie--algebra multiplications,with the underlying

vector space V, forms an algebraic set in the space A (V,V) of all bilinear

alternate maps it.: VXV —^V. Let (e ,e , ...,e ) be a base for vector space

V. Then all bilinear mappings B: VxV —^V can be written as

n k •

^ ° k=l 1 0 K

ind B£ ft( V \V ->V ) ,which in the bas i s e. i s C. . <s kn , and

is Ck..£ kn ^n~1^/2 with the condition

ck1

k = ( kxky. .. ,xk } and k ^' = ( kxkjt,. . x)t) are affine spaces. Let2

A .(V,V) be the space of bilinear alternating maps. The Jacobi relation gives

m

whereAsui.i is over all cyclic permutations of i,j,and k. The above -equation

is a polynomial equation. This defines an algebraic variety in k which

is a cone. Denote this variety as M. The group G=GL(V) acts on thj.s variety

as fellows:

Let iffi GL(V) and jUfiii. Then ^,JJ,ell with (f.jut.) (x,y)= .U.( l( x,t y) where

x,y£ V. All JU =^.U-are the Lie algebra multiplications which, give Lie algebra,

( V,U.') isomorphic. to (V,yU.). Looking into this, one can say that the'diagram

VXV

, IfM .

V x V - > Vis commutative.

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Definition 9*

Throughout this paper we shall be using Zariski1 tcpolcgy for

algebraic groups. Cne can topologize by taking closed sets A together with

A_ , the affine 1-space and empty set. The topology is a coarse topology.

One can regard F& k[ A 1 as a map* of A_ into A . On £ , the coarsest . tcp-

Ifgy such that all these mappings are continuous,is caller tho Z^riski

topology. If the algebraic set MCA is a closed set in _A_ , then ta inherits

a topology from A1 . Therefore I-TCLM is closed if and only if there exist a

closed subset LCA, such that H = VLClL. This topology on M is also a

Zariski • topology.

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2- Contraction of Lie--algebras.

The operation of contraction of Lie 'algebras can be defined

in many ways wnen basis-dependent calculations are involved. However, there

is one thing common, which is the limiting operation after transforming either

the structure constants or1 the "basis of Lie algebras. In effect, the results

are the sane in all cases. Here we take the definition given by R.Hermann

which can be generalized to algebraic-geometric language easily. In the air • •• •'

cjebraic-geometric approach; the algebraic group G that we shall be using corresponds

.to the transformation of basis or structure constants in basis-dependent cal-

culations.

Definition 10.

Let G be the Lie- group and 'a and b be the Lie -subalgebras of

^corresponding to G. One says that b is a limit of 1~ within g* if there exists

a sequence g ,g ,... of elements of G such that whenever X_,X2,... is a sequence

of elements of "a* such that sequence Adg-fX.), Adg^fX,),.... converges, it

converges to an element of b, that- is

b = Limit AdgfTT) ,n->o©

This definition can be translated into algebraic-geometric language as follows.

Let (U.,V)="a be a subalgebra of g and V (g) be the n- -dim-

enGional Grassmannian variety. Let M be the set of all a defined on V which2

i s an a lgebra ic s e t , as shown in remark 3»in the space A (V,V) . T o dispuss

the act ion of G=GL(g) on P ("g) , we define on ^ ( g ) a s t r uc tu r e of a lgebra ic

transformation space by s e t t i n g

g.a = ( Ad—g).a for a€ P ( g ) and gfiG, where Ad— i s

the adjoint representation of g". Let A (g) be the closed subset of ^

consisting of all n-diraensional subalgebras of g. The closed subset An(#) is

stable under the action of G on P ( g ) . Two n-dimensional subalgebras "a and

"a? are conjugate or G-conjugate if they lie on the same orbit under the action

of G. If t-eT-(g) and V (h) be ,a sub-variety of ^n(g) corresponding to-

subalgebras h" of g* then G(a) C\ V (h) is the union-of H-conjugacy classesg \ n •(

vrhere II is a subgroup of G .

-8 -

Thus the orbits of GL(V) on M are isomorphism classes of Lie -algebras with

tho underlying vector space V. Now we are in a position tc define contraction

Geometrically.

Definition 11.— • /

Let a-=(V,M-) and *b= (V, jtc) theivb is a contraction of a" in

"g, if ^L lies on the'Zariski closure of the orbit of'GL(V) at fL .

Remark 4» ,

SinctM factually a subset of T' {g*), one can also say that if a

and b are points of T1 (g) corresponding respectivly to algebras a and b

and G acts on Y^ (g) in the usual way thenb is a" contraction of a in g if

fc& ofa)' in V[ (g). Here G(a) denotes the closure of the orbit of G at a.

Essentially this is the definition given by Hermann that b is a contract-

ion of a in g" if b is a limit of subalgebras of "g which are G-conjugate

to a. If b is a contraction of "a in g then b is said to be contraction of "a.

More generally if a=(V, AA. ) and a.,=(V, M, ) such that V is isoraorphic to

, then a is a contraction of a if a is isomorphic to some bss. (V,M») which 15

a contraction of a*=(V,iU ),

In the following, we will be using this definition of

contraction and, with the help of . a generalizationotM Hilbert-Mumford

theorem, prove the existence of contraction for various cases 01 Hit algebraic

group G as well as theorems on the conjugacy of contracted algebras b and

stability of a* under the operation of contraction.

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3-Existenoe Theorems for Contraction.

The problem of contraction of Lie -algebras has a deep connection

with the theorem of Hilbert and Mumford in the geometric invariant theory.

Before going into this theorem and its generalization, we state two propositions

which will be proved later for variQus different cases of algebraic group G.

Let G be a linear algebraic group which acts on the Grassmannian variety

~p (g) in the way described before. Let ^ : G — ^ G L ^ ) be a rational rep-

resentation defined over the field k. Let a P (g) and G(a) be the Zariski

closure of G^a) and the set C=G(a) - G(a) which contains the set B.

Proposition I..- -

If B C~G(a)-G(a) is a non-empty G-stable set then there exists

a b6B and a one-parameter subgroup ^:G — ^ G defined over k, such that

b= Limit A(t)(a)

where t is some parameter. The Lie,-algebra b is a contraction of a.

The theorem of Hilbert and Mumford is now a special pase of

Proposition"I. It shows that for^linearly reductive algebraic group defined

over k and acting on k-space V, if the set B= Oj then a t= X(t)(a) specializes? •

to zero.

Theorem. [Hilbert-Mumfordj . .

Let /*: G-*GL(V) be a rational representation of a linearly

reductive algebraic group over* algebraically closed field k. If o(i) —G(x;p{bj

then there exists a one-parameter subgroup A : G -*• G such that A(t)(x)the m

specializes to zero. The point x belongs toAlinear vector space over k.

For the case of G=GL(n,C) where C is the complex field of

numbers, the proof was given by Hilbert and. the general case was proved by

D.Mumford. Modifying Mumford!*theorem,. we can say that the contraction of "a

is a null point ^ when G is linearly reductive and acts on T n(g). It i3

to be noted • that when IT is stable . this case differs from that .of , -

a null point contraction of a • In. the first case the orbit G(a) is

closed while in the- other case the set C contains a zero and therefore one

should &eal with them carefully.

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Proposition II.

If G = (g G Q \ ga= a eP(g) f ^ T. where T, is maximal

k-split torus of G, then B=^ and b and a are G-con jugate points. The algebra

~Z is sta"ble under contraction.

Theorem 1.

Let T "be an algebraic torus defined over an algebraically closed

field k, when G=T then proposition I is true.'

Proof.

For the case of G defined over k and acting on the space V such that

^: G —*r GL(V) is a rational representation, the proof is given "by Richardson

9)and Birkes . We use a similar proof for the case of G acting on Grassraannian

variety and then, applying it to the contraction problem^ prove the existence

of a contraction ..."b of a Lie-algebra "a when the algebraic group G is a torus.

Assume the set B to "be a unique closed orbit in the closure

of T(a) where T acts on T (g) in the same way as G, and ag.P (g*). One may

identify the elements of GL(V) as the elements of GL(n,k). Choose the "b sis

°f V (s) s o "that lP(T)OD where D is a subgroup of G and consists of non-

singular diagonal matrices. Now Lemma 3.4 of reference $ shows that when

G=T the orbit T(x) is open for x£V and that set B is a unique closed orbit.

Without loss of generalityf it can be extended to an affine protective or

a Grassmannian variety, if the action of algebraic group G is defined on them.

That is, the set B is a unique closed orbit of G(a) - G(a) and a^= \(i>){a)

specializes to elements of B. By the definition of contraction ( Def, ll) and

the remark following, we have elements of B as those points off7 ('g) • to which

corrssponds the contracted algebra b of "a". Hence whenever G=T, and

W a T - T(a)DB^fe^then we have Limit1 t —>0

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However, one should note that the existence da contraction is bound to the

existence of a one-parameter subgroup X: G --VG. If there ts HO\Xof G then it is

not possible to obtain a contraction of Lie -algebra a.

Hemark.

When torus T of G is defined over an arbitrary field k, it seems that

contraction is possible.

Theorem 2.

When G is a linearly reductive algebraic group G over an algebraically

closed field k then Proposition I is true.

This' statement is rather a generalization/.Nit Hilbert-Mumford

theorem. The case we mentioned before for Ife. Hilbert-Muraford theorem was one

in which the set B=AOr,that is it contains only null element .- and rational

representation defined on space V". Moving from space Vto Grassmann* s variety V (g)

and to the case when B is larger than null set, we have the above theorem.

When the characteristic of the'field k is zero then we have the same case as

that of torus /P. However, if the characteristic ?of k is not zero, then we

9)follow Richardson-Birkes to prove the theorem.

Proof. *

Let T be a maximal torus of G such that if Bf\T{a)^= 6 then from

T.heorem 1 there is an element b<= Bf\T(a) and a one-parameter subgroup of T

such that for ±->0 we have \ ( t ) (a) -^-b.Suppose BfiT(a) = fy V" TCG. Choose

one T and for nny a '= g(a}£ G(a), we have B O ^ a " ) = Bf\T(g.a) which i s

equal to Bf\g( g~ T( g .a ) ) = g( Bf ls" T(g .a ) ) = <fi because g~ T g i s maximal

torua.Since B and T(a") are two disjoint T-stable closed subsets of [U

therefore there is a T-invariant regular function fnJ, such that f /(B)-0

and f ,( T(a*)) = 1 . Let 0 ,= i Y € ^ ( g ) • f -( T )f 0 1 be an open seta a i n • I ti »j

and l e t G=MTM where M i s a compact subgroup of G. Since M(a) i s compact,

one has K(a)<£ 0 , j j 0 , 1 1 . . . (J 0 , for some a* a* . . . a ' <£M(a). Definea 1 a 2 v an x ^ n

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n

f(Y) =

H t h e H a u s d o r f f - c o n t i n u o u s T - i n v a r i a n t f u n c t i o n f ( X ) w h e r e

. ( K )n

." Then f attains a positive mini-

mum value on the compact set M(a) and hence on TM(a) and TM(a) , where*

represents the Hauadorff-closure. But f(B)=O, so Bf\ TM(a) s= (j>. Since B is

G-stable therefore 3/1 TlUnT = ^ . Since we have G(a)= MTM(a)OM. TM(a)*

which contains in H.G(a) = b"fa!) and M.TM(a) = G(a) = G"fa")", which is a

contradiction because BCG(a). Now the set BOiT'a) and therefore BflG(a)

is not empty. Theitfore we have a set of contracted algebras B~ correspending

tc points b £ B C P (g).

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Stability Theorems for a under Contraction.

Under what conditicn does a Lie algebra survive contraction or is

it stable under contraction? This question is similar to the case of deformation

of Lie algebras where one has a condition which-says that whenever second

cohomology group of that algebra with coefficients in itself vanishes,that is

H ("a.,*a)=O, then the algebra a is "rigid" under deformation. Here we'consider

the case when "a is either stable,meaning that its isotropy group G.. for

a £ *\g). is equal to G, or all of its contracted Lie algebras b are iso-n

raorphic to "a. In both cases the result is the same. In terms of the properties

of set 33, the stability condition require that B = ^ which, in other words,

"roughly"means that the orbit G(a) is closed. Since we considered B C C = G(a)-G(a),

therefore, when B is contained in G(a) - G(a) we leave out some points.in the

boundary of the orbit. It follows that their algebras, which are also contracted

algebras of "a, are net included in the discussion of theorems on existence.

However,it makes little difference when orbits are closed. ^n he os.se when

G = G, it is trivial to notice that 'a is contraction-stable or C-stable.a '

The more complicated case is when G s=G and the proposition II is importanta

to show that B=fi . The proposition II is true for the following cases-

LGT'.m"-. 1.

Let G be the connected component of G, If "a is C-stable for G theno • o

it is also C-stable for G.

Proof.

From proposition 1.2 of reference 10, one can prove fjfc above lemma when

the group G acts on linear space V. Generalizing this to the case of P (g),

oi:e obtains G(a) = G(a) for a e P ( i ) . Hence G(a) is closed and there do not

c::ist any points which are limit points of algebras conjugate to "a. The Lie

•Q'-OCT-'.. a is C-stable.

-Ill-

Lecma 2.

Let P be a parabolic subgroup of G. if "a is C-stable for P then it

is also C-stable for G.

Proof.

We assume as before that *f: G—»GL( .P(g)) is a rational representation

of algebraic group G, Let P act on p ("g) as (g.a*){p)=s: (gp,a*) where g<SG,

p £ P and a<5r*(e}. There exists a quotient variety G XP(g)/p = G/p *"P(g)

and let O< : G*P(g) ^-yG/pXp(g) be an open quotient morphism. Let the inverse

image of P(a) be mapping [p(a)]"1 » j(s,a')£ Gxrj(g) | a*€g?(a)j

sending (g,a') to g"1(a'). The set [ P ^ " 1 ~ fe^itl"1 since K"1( <* [

to [":'(a)] and it follows that c* [ P(a)] ~ is closed in

because oC is open. Since G/p is complete, the image G(a) of c< •.O*(a)l~-

under projection map G/p xP(g) — ^ P (g) is closed in P {g). Therefore, the

orbit G(a) « G(a) and "a is C-stable. "a is the .Lie algebra corresponding to

the point a. •

Lenrna 3.

Let G be a k-anisotropic algebraic group defined over k. If proposition I

is true the a is C-sta"ble.

Prcof.

For a k-anisotropic algebraic group,a one-parameter subgroup does not

exist. (See the definition , Bef. 10.) "Let .'G -*GL(TMg)) be a rational

representation. If B ffl then proposition I would imply the existence cf ,which

is a contradiction. Therefore B is empty.

Lemma 4.

Let U be a unipotent group. If G=U the a is C-stable.

Proof.•

Uefoe Ivostant-Rosenlicht theorem9^ which says that for U=G acting morph-

ically on

C-£3t:i'.le.

V(c) the crbit G(a) is closed. It follows that B= <j> and hence "a is

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Conjugacy Theorems for Contracted Lie algebras.

• The theorems on the conjugacy of Lie -algebras .relates the

algebras to equivalence classes. The problem of classification of. contraction

is related to the problem of finding equivalence relation among the contrac-

tions of a''given Lie-algebra a. Let b, and b~ be the contractions of a. If

there exists some element g£ G such that g~ big = b ? then "b. and b~ belong

to the same class. In other words, if the points b, and b ? of the set B belong

to the same orbit under some group then they are equivalent. In discussing

existence theorems, we assumed B to be a unique closed orbit because for G

reductive and torus such is the case. It is obvious then that all.b£B are

conjugate to each other and one gets an equivalence class of contracted algebras

of a' when the group G is reductive or torus. The group G is a transformation

group which acts on the basis of'Lie • algebra or on .the structure constant

space.

Definition 12.

Let G be a linearly reductive algebraic group and X be a one-

parameter subgroup, of G, then we define P{ X ) to be the set of all "K <£ G

such that X(t) Q X(t) has a specialization in G as t —^0. P{ X) is called

parabolic group.

In the set of one-parameter -rsub groups of G, we define an

equivalence relation denoted by ,AJ as follows:

A W a if there exist positive integers p and q and an element

Q£?( X) such that

Conjugacy Theorem.

Let \Q: G — > Gh{V (g)) be a rational representation and. G

be a linearly reductive algebraic group. Let a . H (g) and O.X^ e * w o one~.

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subgroups of G. Suppose that ^ A » I f X("t)(a)-^b as t-*0 then

<f (t)(a)->g(b) as t-»G for some g £ G.

Corollary,

If D=s(yU.,V) is a contraction of a=(M-,V) corresponding to points

b and a of T^g) respectrwly and there is an equivalence relation ^r^o then

there exists a contraction K of "a which corresponds to the point ,'g(b) for g£ G.

Remark 5.

It is to be noted that equivalence relation A r^o exists only if

there is parabolic subgroup .of G. This ueans tint whenever a para-

bolic subgroup of G exists, If is always possible to obtain an' equivalent

contraction to b. One nay classify all contractions of a given Lie -algebra

thein this way. In physics, for example, one contracts^DeSitter group and looks

for groups which are equivalent to (lie. Poincare''group. ' All one has to do

is to find some suitable transformations G which acton the Lie algebra struoture

constant space or "basis.

Proof. [ConjugECy Theorem]

From the definition of P(A) > there exists some /3€. G such that

A(t) tf M t " 1 ) - * ^ as t —»Q. Since p > 0 then also \ ( t P ) ^ >(t~P}-»/3as t -*0.

Since jTCt^Ca) = ( f W ) Y C») - <f( A(tP) $ >(t"P) > Ctp)<a)-^"-I

t —> 0. Let us put J A a g. Since q > 0 so we have

?T(t)(a)-» g(b) as t-VO.

Proof. [Corollary]By the definition of contraction b of a correspond to the

points of P (g). Since from the theorem of conjugacy there exists a g£G

such that b,=*=g(b) therefore we have correspondingly an algebra b_ which is

. equivalent to "b. Since all b. lie on the orbit of G at b, they art fee isomorphism

class of contracted Lie-algebras.

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Conclusion and. ou few more Remarks.

It is well known that there is a definite relationship "between

the problem of contraction and deformation of Lie -algebras and hence Lie

Croups.In the case of deformation theory , whether it is for Lie-algebras

or'Lie'-groups, there are some definite mathematical results with which one

can compare the results of the contraction theory because they seem to be,-

rather intuitively, inverse operations, Gerstenhaber ^ has shown that if

H ('a,'a)=O then/ a is rigid with respect to deformation. Also there is a

12)result of Richardson ; which, is more relevant in this connection,that if

the orbit of G at a is open in (1) then "a is a rigid subalgebra of *g"

with respect to deformation. As a consequence one has 1-cchomology space

H (H'jS/a) ™sO. However, when the orbit G(a). is closed, we have shown that "a"

is C-stable. One may say that in such a case H ("a, "g/a") sfc 0 which seems to

be consistent with the deformation theory. At present we have no proof of this

conjecture. Prom the properties of variety LldP (g) one can say that:

1) * The contraction of a solvable Lie,-algebra is a solvable

Lie algebra.

2) The contraction of a nilpotent Lie- algebra is a nilpotent Lie

algebra.

3) The contraction of a unimodular Lie • algebra is a uniwadular

Lie'algebra.

4) Let cb Ca) "be the 1-derivative algebra of a" then

dimension of D (b)

(Tb)

dimension of §J (a)

where £0 (Tb) i s the 1-der ivat ive algebra of b .

5) M.Levy-Hahas-5'' has shorn tha t i f Z (a) , Z (b) and B...(a)f

o ^ • ^ -

B (b) are the 2-cocycle •. and 2-coboundajry operators of algebras "a and b, then

one has

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(a) dim Z {tO > dim Z (i.)

(V dim £2fb) ^ dim B2(a"). •

Finally , we nay remark that there are many aspects ox'

contraction tbeory which, are not known and much has to lie done in this direction.

Por instance, stability theorem in terms of condition on cohomology groups,

generalization to Lie groups of this approach and G-conjugacy structure of

.of the set B for different clasaical groups are some unsolved problems.

ACKHOWLEDGMEHTS

The author vishes to thank Prof. H.D. Doebner for discussions and

for reading the manuscript. He also vishes to thank Prof. Afcdus Salam,

the International Atomic Energy Agency and UNESCO for hospitality at the

International Centre for Theoretical Physics, Trieste.

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REFERENCES

1) Doebner, H.D.. and Melsheimer, 0 7 Kuovo Cimento A^» 306 (1967);

J. Math. Phys. £, 1638 (1968) and 11., 1^36 (1970).

2) Segal, I.E., Duke Mat. J. l8_, 2\ (1951).

3) Wjgner, E.P. and Inonii, E., Proc. Natl. Acad. Sci. (USA) 39_, 510 (1953).

k) Saletan, E., J. Math. Phys. 2_, 1 (1961).

5) Levy-Nahas, M.; Thesis 1969, University of Paris.

6) Hermann, R -; Lie Groups for Physicists {Ben j amin, New York 1966).

7) Mumford, P., Geometric Invariant Theory (Springer-Verlag, Kev York I965);

Fogarty, J., Invariant Theory (Benjamin, New York 1969).

8) Richardson, R.,Ann. Math. 86_, 1 (1967).

9) Birkes, D.; Ann. Math. 9_3_, h59 (1971).

10) Borel, A., Linear Algebraic Groups (Benjamin, New York 1969);

Borel, A vAnn. Math. 6U, 20 (1956).

11) Gerstenhaber, U.t Ann. Math. 72., 59 (1961*).

12) Richardson, R.; J. Diff. Geom. 3., 289 (1969).

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