NEW REPRESENTATION OF THE NON-SYMMETRIC HOMOGENEOUS BOUNDED DOMAINS IN C

Internat. J. Math. & Math. Sci.

VOL. 15 NO. 4 (1992) 741-752

741

NEW REPRESENTATION OF THE NON-SYMMETRIC HOMOGENEOUS BOUNDED DOMAINS IN C

⁴

AND C

^s

GR.TSAGAS and G. DIMOU Divisionof Mathematics

Department

of Mathematics andPhysics

Faculty

of

Technology

UniversityofThessaloniki, Thessaloniki,

GREECE (Received

^December14,1989 and in revised formMarch 1,

1990)

ABSTRACT.

This

paper

deals with thecorrespondingsolvable Liealgebra^toeachofnon-symmetric homogeneousbounded domains in

124

and

125

by special^setof matrices.

Some

interesting propertiesof Kihler manifolds arefound. Thetheoryof s-structure on acompleteRiemann manifoldisalso studied.

KEY WORDS AND PHRASES: J-Algebra,

non-symmetric homogeneousbounded domains, homo- morphismandIhler manifolds.

1991

AMS SUBJECT CLASSIFICATION CODES.

53C55.

1.

INTRODUCTION.

Let D

be a

homogeneous

bounded domain in12".

One

of the

problems

forthe

homogeneous

bounded domains is toclassifythose which arenotsymmetric.

It

is known thatevery homogeneous^bounded domain in

e",

ns3,^issymmetric.

It

has beenprovedthat in each

124,

and

125

there existsonly^one non-symmetrichomogeneousbounded domain.

In

ordertostudythesenon-symmetric homogeneous bounded domains we need a methodtodescribethese manifolds. Thisdescription permitsustounderstand better thegeometryof these manifolds.

Therewere differentdescriptionsof thesenon-symmetric homogeneousbounded domains.

One

of themis

by

meansof

Siegel

domainsof second kind. Thisdescriptioniscomplicated^and

very

difficult toobtainsomegeometrical properties ofthese Kihler manifolds. Anothermethod to describe the non- symmetric homogeneousbounded domain isbymeansofJ-algebraandnormal

J-algebra,

^whichare specialkindsofJ-algebras. ^{This is a}general theory,^{which isnot}alwaysclear in ordertofindthrough somegeometrical propertiesof thenon-symmetric

homogeneous

bounded domains whichcorrespond^to agiven normal

J-algebra.

It

^isknown that to

every

non-symmetric homogeneousbounded domain

corresponds

^a normal

J-algebra.

This normal

J-algebra

^{is a}solvable Lie

algebra.

The aim of thepresent

paper

is to describe thecorrespondingsolvable Lie

algebra

^toeach of non- symmetric

homogeneous

boundeddomains in

124

and

125 by

aspecial setofmatrices. Thisdescription allows to

prove

somevery interesting propertiesof these Kiihler manifolds. Thewhole

paper

^contains four sections.Thesecond section contains the

general theory

about

homogeneous

boundeddomains,and the relation between these manifolds andthe Siegeldomainsof the second kind.

It

alsogivesthe connection between

homogeneous

bounded domains andJ-algebrasandnormal

J-algebra.

Finally,westudythe theoryof s-structure onacompleteRiemannian manifold.

Theunique non-symmetric

homogeneous

^boundeddomain inC is studied in the third section. First of allwegivethe description ofthisnon-symmetrichomogeneousbounded domain bymeans of the Siegeldomains ofsecond kind. Italso containsthecorrespondingnormal

J-algebra,

^whichissolvable byasetof matrices. Thisrepresentationof the normal

J-algebra

allows ustounderstand better this Ktihler manifold.

We

give analyticallythecomplexstructure

J0

^onthisnormal

^J-algebra

s and the innerproduct ons, whichproducesthe Ktihlermetricon thecorrespondingsolvable Lie

group

which isholomorphically isomorphiconto thisnon-symmetrichomogeneousbounded domain in

C4. We

also determine the Lie homomorphismof this solvable Liealgebraandfurthermoreweclassifywhichof themareisometrics.

This classificationpermitsusto

prove

that thisnon-symmetric homogeneousbounded domain admits k-symmetricstructure.

Thefourthsectiondealswiththesameproblems,whicharestudiedin the third section, for the unique non-symmetric homogeneousbounded domain in 12 Theimportantresultof this section isthat this Ktihlerhomogeneousmanifoldcannotcarry

any

s-structureof korderfor

any

^k

2.

Homogeneous

Bounded Domains

Let C"

be the n dimensional Euclidean complex

space. An open

connected subsetof

D

of

C"

is

c,alled

^domain.

^We

^denote

^{by G(D)}

^the

^group

^ofallholomorphic automorphisms^of

D.

IfDisbounded, then it is called bounded domain in

C". Let D

be abounded domain in

C".

Thereexistson

D

avolume element co which is definedby

co-

(d-i)" gaz ^ ^ az, ^ a, ^ ^

where

z ^,z,,

^arecomplexcoordinates in

C"

and

K

and

Bergman

functionon

D,

which ispositive. The

Bergman

function

K

givesthe Ktihler metric

g

on

D

definedby

g’., ^021gK

If

D

isa bounded domain in

C",

then

(D,g)

isa Ktihler manifold andthe

group G(D)

has a structure of a Lie

group.

The bounded domain

D

in

C"

iscalled

homogeneous,

if the

group G(D)

^actstransitivelyon

D

andtherefore

D,

in thiscase,canbe written

D -G(D)/H (2.1)

where

H

^istheisotropysubgroupof

G(D)

^atthepoint_z0tE

C".

The relation

(:.1)

can alsobe written as follows:

D Go(D)/Ho

where

Go(D)

^istheidentify componentof

G(D)

^and

H0

^{istheisotropy}subgroup of

Go(D)

^atz0 Eachdomain

D

in

C"

isconnected with a

Siegel

domain inthe same Euclidean

complex space C".

Now,

wegivesome basic elements for a

Siegel

domain.

Let V

beaconvex cone in

R . ^We

^{considerthefollowingmap:}

F" e e c F" (w, w’) F(w, w’)

havingtheproperties.

(i)

^{Ifw isfixed,then the}

map

F,"

C C

, F,,,,

^w

F,,,,(w) F(w, w’)

is

complex

linear.

NEW REPRESENTATION OF THE NON-SYMMETRIC DOMAINS 743

(ii) F(w’,w)-F(w,w’).

(iii) F(w,w) _ ^",

^where

^V

^{closure of}

^{V, V--’w}

(iv) F(w, w)

⁰ w O.

The

map F

iscalled

V-hermitian form.

^onC

.

Let

Vbea convexconein t2

. ^{Let F}

^{be a}V-hermitianform on

V. We

considera subset

D(V,F)

^of

C* 13=

2

’’,

where n k + 1,defined asfollows:

D(V,F) {(z, w)

which isa domainin

’.

This iscalled a^Siegeldomainofsecond kind associatedto

(V,F).

If 1-0,then the domain

D(V,F)

iscalled a

Siegel domain

^offirst kind.

Hence

aSiegeldomain of first kind is definedby

O(v)- {z

which is a

special

caseof a

Siegel

domain ofsecond kindforwhich the V-hermitianform

F

is the zero form.

The followingtheorem is true

([3]).

THEOREM

2.1.

Let D’

beaboundeddomain in

" ^C* .

^{Thenthere isa Siegeldomain}

ofsecond kind

D(V,F)

of C which is

biholomorphically

equivalent^onto

D’.

Ifthe

group G(D(V,F))

^{acts on}

D(V,F)

transitively,then

D(V,F)

iscalledlaomogenous

Siegel

domainof

SCcond

kind, whichcanbe written

D(V,F) G(D(V,F))/T,

where

T

istheisotropy

subgroup

^of

G(D(V,F))

^atthe

point (Zo, Wo)

^of

D(V,F).

Thereexists atheorem similar toTheorem2.1.

THEOREM

2.2.

Let D’

be ahomogeneousdomain in

(1 C* .

^{Then thereexistsa}

homogeneous

Siegel

domain ofsecond

kindD(V,F)

in

e 1,

which isbihomographieally equivalent^onto O

To every Siegel

domainof second

ldndD(V,F)

^{in 1. wecanassociatean affine}automorphism

group AF(D(V,F)).

This

group AF(D(V,F)) plays

animportant rolein the

theory

of classificationof

homogeneous

bounded domains in

We

denote

by AF( )

the

group

of allaffineautomorphisms of C

1.

The

group AF(D(V,F))

is defined asfollows:

AF(D(V,F)) { ^F_AF(C )/(D(V,F)) D(I/,F)}

which is aclosed subgroup

ofAF(C

^*/

). ^It

^{isknown that}

AF(c

where

GL((: ’1)

^isthe

group

of invertible linear transformations ofC^x

1, T(C, )

isthe

group

of linear transformationsof

C*

and"."denotesthe semi-direct

product.

Let D(V,F)

^beaSiegeldomainof second kind in

C" C* 1.

Ifthe affineautomorphism

group AF(D(V,F))

^of

D(V,F)

^acts

transitively

onit, then

D(V,F)

iscalled

homogeneous Siegel

domainof

second kind

D(V,F)

in

e " ^1.1,

^whichcan be written

D(V,F) -AF(D(V,F)/H

where

H

istheisotropy

subgroup

of

AF(D(V,F))

atthepoint

(z, w) D(V,F).

We

have the

following

theorem.

THEOREM

2.3.

Let D’

be a homogeneous bounded domain in

C"

12

’

^{1.C x}

^121.

^{Then there}

exists an affinely homogeneous Siegel domain of second kind

D(V,F)

which isbiholomorphically equivalentonto

D’.

From

Theorems 2.2 and2.3weconclude thatevery affinely homogeneous Siegel^{domainof second} kindcan be written

d(V,F) AF(D(V,F))/H (2.1)

which is also a

homogeneous Siegel

domainofsecond kind inthe sense of Theorem2.2,thatmeans

D(V,F) G(D(V,F)) (2.2)

From (2.1)

and

(2.2)

^{we obtain}

D(V,F) G(D(V,F) AF(D(V,F))/H

Bothdescription

(2.1)

and

(2.2)

for aSiegeldomain of second kindare usedtostudy some properties such as

space.

Let D

G/Hbe ahomogeneous boundeddomain in

& a- 1. It

isknown that

D

isacomplex manifold withIrhler metric

g

definedby

0

logf

dz ’ ^ ^dt

On

the manifold

D

there exists acomplexstructure whichgivestheintegrable almost

complex

structure on it denoted

by J.

Let

and b be the Lie

algebras

of

G

and

H

respectively.

Let

^tnbe thetangent

space

olDattheorigin 0of

D. Hence

we obtainthedecomposition:

t-b +m bfm -0,

and since

D

is reductive wehave the relation

[b,m]

^{C m. The Irhlermetric}

g

on

D

induces a Hermitian positivedefinitesymmetricform

B

onm.Thealmostcomplexstructure

J

on

D

defines anendomorphism

Jo

^ontnwiththefollowing properties

Jo

^{m m}

Jo ^{X Jo(X)} J2o

^-id

^(2.3)

We

can extend thisendomorphism

J0

^toanotherendomorphism

J0’

on which is defined asfollows

Jo’

^t,

Jo’ ^{X Jo’(X’)} Jo

^-id

^(2.4)

Jo’(X)-Jo(X),

^if

X m Jo’(X)-O,

if

X

b

(2.5)

Theendomorphism

J0

satisfies thefollowingrelation

IX, Y]

+

Jo([Jo(X), Y])

+

Jo([X,Jo(Y)]) [Jo(X),Jo(Y)] ^{O, X, Y}

^tn

(2.6)

which is obtained from thefactthatthe almost

complex

structure on

D

isintegrable.

Therelations

(2.4), (2.5)

^and

(2.6)

implythat theendomorphismJ0satisfies similar relation as

(2.6),

thatmeans

[X,Y] +Jo’([Jo’(X),Y]) +Jo’([X,Jo’(Y)]-[Jo’(X),Jo’(Y)] O, X, r (2.7) From

the Hermitianpositivedefinite bilinearform

B

onmweobtaina linear form w ontnwhich is definedasfollows

w :m

R,

^w

:X w(X)-B(X,J(X)) (2.8)

This linearform can be extended to another1-form

w’

or

t’

definedby

w’ R, w’ X w’(X)- w(X,) (2.9)

NEW REPRESENTATION OF THE NON-SYMMETRIC DOMAINS 745

This linearform

w’

satisfiesthe relations

’([J’o(X),J’o(r)]) w’([x, rJ), (2. o)

w’([J’0(X),X])

0,

(2.11)

for all

X, Y

tE and

X

b.

Thelinearform w also satisfies the relations

,([SoCX),Jo(]) wC[X, ]), (. t)

wC[JoCX),X])

>0

X

0

(2.13)

for all

X, Y

m.

Thereforefrom the

homogeneous

bounded domainD’-G/H we obtain the set

{t,b,Jo’, w’),

where

J0’

^{is an}endomorphismon havingtheproperties

(2.5)

and

(2.7)

and w a linearformon withthe

properties (2.10)

^and

(2.11).

^Theset

{t’,b,Jo’,w’}

iscalled

J-algebra.

Thereexiststhe following theorem

([3]).

TIIEOREM

:L4.

Let D’

bea

homogeneous

boundeddomain in

.

This manifold

D’

canbewrittenD’-G/H,where

G

isaLiegroupofanalytic automorphismsof

D’

and

H

theisotropy subgroupof

G

at

zo ^D’.

^Wedenoteby and b the Liealgebrasof

G

and

H

respectively. Thecomplex structure

J

on

D’

inducesa linearendomorphism

J0’

^{ont. Then}there exists a linear form

w’

on such

that’the

set

{t,b,Jo’,w’}

^{be comes a}

J-algebra.

In general,

there aremany

J-algebras

which

correspond

^tothe same

homogeneous

bounded domain

D’,

since there are a lot of Lie

subgroups

of the full

group

of analytic automorphismson

D’

which act transitivelyon

D’. Let D’

be ahomogeneousbounded domain int2"

’ ^*.

^{This is}biholomorphically equivalent^ontoahomogeneous Siegeldomainof second kind

D(V,F) G(D(V,F)) (2.14)

andatthe same time isbiholomorphically equivalent^ontoanaffinelyhomogeneous Siegeldomainof second kind

D(V, F) AF(D(V,F))/H’ (2.15)

Let t, bt

^{be the Lie}

^algebras

^of

G(D(V,F))

^and

T respectively.

^Thereare a linear

endomorphism (J)0

and a linearform

w

^on

t

^{such thatthe set}

{t,b,(J)o,w}

^{is a}

^J-algebra.

Similarly,let t2,

b2

^{bethe Lie}algebrasof

AF(D(V,F))

and

H’

respectively. There exist a linear endomorphism

(J2)o

and a linearformw such that the set

{t2, b2, (J)o, w}

^isa

^J-ilgebra.

The three

J-algebras

{t,b,Jo, w}, {t,bt,(J)o,W), ^{tz, b>(Ju)0,wu}

areisomorphic.

From

the above we conclude that the

study

of

homogeneous

boundeddomains in

e"

arereduced to the

study

of

J-algebras

and we have thetheorem

TIIEOREM

^.$.

Let {t,b,Jo’,w’}

be a

J-algebra.

Then there isalwaysaboundedhomogeneous bounded domain

D’

G/H in

e" e

whose

J-algebra

isthegiven. There are alsohomogeneous

Siegel

domain of second kind

D(V,F)

in

e" e* e

andanaffinely

homogeneous Siegel

domain

D(V,F)

in

e*

12 whichhave a

J-algebra

thegiven

{t,b,Jo’,w’}.

Let

D’-G/Hbe a

homogeneous

boundeddomain in

’ ^e .

^{Thereexists a}solvable Lie

subgroup S

of

G(D’)

^{which can}be identifiedwith

D’.

This domain isbiholomorphically equivalentonto ahomogeneous

Siegel

domainofsecond kindD

(V,F)

andsimultaneouslyisbiholomorphically equivalent

ontoanaffinely homogeneous Siegeldomain ofsecondkind

D](V,F)

in

&

C C

t. D(V,F)

and

DI(V,F)

canbe written

D(V,F) G(D(V,F)/H’ D(V,F) -AF(D,(V,F))/T

It

has been

proved

that there are solvable Liesubgroups

S’

and

S

^of

G(D(V,F))andAF(D(V,F)) respectively

suchthat

D(V,F) S’

and

D(V,F) S

Thecorresponding J-algebrasof

S, S’

and

S

^{take theform}

(S,Jo,

^w

(s’,Jo’,W’)

and

(s,(JOo, wl) (2.16)

respectively,where each of the Liealgebras s,

s’

and

s’

^aresolvable.

Alltheproperties,whichare validforthesolvable

J-algebra (s,J, w)

are validforthe othertwo

(s’,J’, w’) (s,Ja, w).

Since

(s,J,w)

is a

J-algebra

we havethat

J0

^isanendomorphismon s with theproperties.

(i) Jo:

^{s s,}

Jo ^{:X--,J(X), Jo--1, (2.17)}

(ii) [X, Y]

+

Yo([Jo(X), Y])

+

Jo([X,So(Y)]) [Jo(X),So(V)]

⁰

(2.18)

wisa linear form on s with theproperties

(iii)

^{w s}

R,

^w

:X (X), w([Jo(X),Jo(Y)] w([X,Y]) (2.19)

(iv) w([Jo(X),X])

^>0

(2.20)

andtheoperator

(v)

^{ad, s} s,

ad’go ad:00;) [0, ;]

hasonlyreal characteristic roots,

’%

⁶s, that is,

ado,

as amatrix isR-triangular.

The set

(s,J,w)

^iscalled normalJ-algebra. Thethree Lie

algebras,

definedby

(2.16),

arenormal

J-algebras,

whichareisomorphic.

Now,

weconsider ahomogeneousbounded domain

D’

in

C’. Let G

be the maximalsolvableLie subgroup^of

G(D’)

^{which acts}transitivelyon

D’

andsplitsoverthereal numbers.

We

knowthat such a

subgroup

is defineduniquely

up

toconjugacy.

The

J-algebras

whichcorrespondto such

subgroups

arenormal.

Let (s,J, w)

bea normal

J-algebra.

From

this we obtain an inner

product

on s asfollows

(X, r 3 ^-,([S0(X),

We

have the followingtheorem

TIIEOREM 2.6.

The correspondence betweennormal

J-algebras

and

homogeneous

bounded domain in

e"

is one-to-one.

Let M

bean n-dimensional connected Riemannianmanifold,and

I(M)

the

group

of isometrics of

M.

Ifthere is a

map

s

:M I(M)

suchthatfor

every

x

tEM

the

image s(x) s,,

^{is an}isometry of

M

having xas anisolated fixed

point,

thenthe

isometry s,

^{iscalledRiemannian}

symmetry

atxorsimply symmetryatx.TheRiemannian manifold

M

withthispropertyiscalled Riemannians-manifold. Ifthere isa

positive integer

ksuch that

s ^{-/d., x E M,}

^then

^M

^iscalledaRiemannian s-manifoldof order kor

simply

k-symmetric Riemannian

space.

The usual Riemannian symmetric

spaces

^are Riemannian s-manifolds of order2.

We

refer three resultsconcerningRiemannian s-manifolds

[9].

NEW REPRESENTATION OF THE NON-SYMMETRIC DOMAINS 747

I.

Thegroupof all isometrics

I(M)

on aRiemannian s-manifold

M

actstransitivelyonit.

II. Let M

be a Riemannian s-manifold. Then

M

G/H,where

H

is theisotropy subgroupof

G

atany pointof

M.

III. Let M

beaconnected Riemannian s-manifold. There exists another

s’-structure {sx’

^x

M

on

M

suchthat with

{sx’

^x

M}

becomes ak-symmetricRiemannian manifold.

3. Non-symmetric BoundedDomain inC

In

12 there existsonlyone non-symmetric homogeneousbounded domain

D’.

^{First we}givethe affinely

homogeneous

Siegel domain

D(V,F)

which isbiholomorphically equivalent ^onto

D’. We

describebelow

V

and

F.

Let Q -H(2,R)

bethe vectorspace^over

R

ofallrealsymmetricmatrices2 2. This is athree dimensionalreal vector

space,

that isisomorphiconto

R 3.

Thereforek 3and hence 1. Theconvex cone

V

is

V H’(2, R),

thatmeans allpositive symmetricmatrices3x3. The mapping

F

definedby

Then

D(H/(2,R),F)

^isanaffinely

homogeneous Siegel

domain of second kindbiholomorphically equivalentonto

D’.

Now,

ourmethod consistsof describing

D’

orequivalently

D(H’(2,R),F) by

anormal

J-algebra (s,&,w).

We

have

proved

thatthe solvable Lie

algebra

s can be describedbythesetof matrices

0 x

y

c

s-{A-

^{0 0 /x,}

ct,,r,R V,13,/5,.R’} ^(3.2)

0 0 6

0 0 0

From

this construction of sweconcludethat theendomorphism

Jo

^{has theform}

Jo ^{(fS, u) fS,t R,}

^{1 kl<5}

^(3.3)

whichmustsatisfythe relations

(2.17)

^and

(2.18). ^From

these conditions andafter a lot of estimateswe obtain

i 0 0 0

t

^{0 0 0}

0 x 0 0 0 v 0 0

o o o o o p o

0 0 0

:

0 0 0 o

1+:

Jo- ---

⁰⁰0

_----

⁰0^{0 0}0⁰

-

⁰

^/----]

⁰

^o

^-c0

^o

^{0 -x0}

^o

⁰

^-,

^{000 00}

^o

wherela,

v, p,

o

R

Thelinearform w on this Liealgebra^sdefinedby

,(x) (xo,x)

(3.4)

(3.5)

where the usual innerproducton sand

X0 ^(Kt, K2,Ka, K4,Ks, IQ, Kv, Ks)

^afixedvector.

In

ordermat wsatisfies the conditions

(2.19)

^and

(2.20)

^{we musthave}

Kl.t>0, K2v>0, K3p>0, Ko>0 (3.6) Now,

wehaveprovedthefollowingtheorem.

THEOREM

3.1.

Let D’

be the unique non-symmetric homogeneous bounded domain in

124 t23

x 12. Thecorrespondingnormal

J-algebra

^is

(S,Jo, w),

where s,

J0

^{and waregiven by}

(3.2), (3.3)

and

(3.4)

respectively.

Now,

wedeterminethe solvable Lie

group S

whichcorresponds^tothe solvable Liealgebra^s.

Wedenoteby

GL(S)

the

group

of allnonsingular endomorphismsof s. The Lie

algebra gl(s)

of

GL(S)

consists ofall endomorphismsof swiththe standard bracketoperation

IX, Y]-XY-YX.

^The mapping

ad s

gl(s

ad

B

adB

whereadB s s, adB

T adB(T) [T,B

isahomomorphismof s onto asubalgebra

ad(s)

of

gl(s).

Let Int(s)

be the analyticsubgroupof

GL(s)

whose Liealgebrais

ad(s), lnt(s)

^iscalled theadjoint group ofs.

The

groupAut(s)

ofallautomorphismsofsis aclosedsubgroupof

GL(s). ThusAut(s)

has aunique analyticstructure underwhich it becomes atopologicalLiesubgroupof

GL(s). We

denoteby

d(s)

the Liealgebra^of

Aut(s). Now,

^the

group Int(s)

^isconnected so it isgenerated bythe elements e

ax, X

s.

Therefore

Int(s)

^{is a}normal

subgroup

of

Aut(s).

From

the above we conclude that the solvable Lie

group S

of s is definedby

’(e 1) ice

^x

¹⁾

1

Ce’-l) Ce ^-1)

^g

e^y 0 0 0

S-

_{0 e 0 0}

0 0 e 0

0 0 0 e

’

(3.7)

The innerproductonthe solvable Liealgebra^sis definedby

<x, Y> ^{w([Sox, q) (3.8)}

where w isgiven by

(3.4).

This innerproductdeterminesthe Khler metric on

S

which isessentiallythe

Bergmann

metricon it.

Nowwecanstatethefollowingtheorem.

THEOREM

3.2. The homogeneous non-symmetricbounded domain in

12

isbiholomorphically isomorphiconto the solvable Lie

group S

definedby

(3.7).

^{The Ihlermetric}

^g

^on

^S

^{is definedbythe}

relation

(3.8).

Let F

be a Lieautomorphismon s. This

F

can berepresented bythe matrix

’art

^{0 0 0}

as

^{0 0}

^0’

0 a-_a 0 0 0 a2 0 0

0 0 a33 0 0 0 a37 0

0 0 0

a

^{0 0 0}

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

El

NEW REPRESENTATIONS ON THE NON-SYMMETRIC DOMAINS 749

whichbecomes an

isometry

withrespect^tothe inner

product

when

fr, r3 ^{(Xo, [jx, ,,,(ix, q)}

a

n-+_l, a22-+.1, a-+/-l, a.-+/-l,

%6-1+x ’a22 a37"1+2 ^a

Thereforeif

F

isanisometry,then it hastheform

x(1-art)

a

^{0 0 0}

0/.. 0

,,t-"-a)

0

az

^{0 0 0}

0 0 a3 0 0 0

+ all }

0"

a

_+z

0 0

0 0

t-ajs

/ ^ass 0

0 0 0

an

^{0 0 0} -a

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

where

an

^-,-1,

azz

^+/-1,

aaa

^+/-1,

^a,

^+/-1.

From

theform

F,,,

we obtainthat it has theeigenvalue 1withmultiplicityatleast4times. Therefore we haveprovedthefollowingtheorem.

THEOREM

3.3.Thehomogeneous non-symmetricbounded domain inC withthe standardK/ihler metric does not admit

any

k-symmetricstructure.

4. Non-symmetricBounded Domain inC

In

^C there isonlyonenon-symmetric

homogeneous

bounded domain

D’. It

is well known that there exists theaffinely

homogeneous Siegel

domain

D(V,F)which

^isbiholomorphically equivalentonto

D’.

Thereforeweneed

V

and

F

whichare describedbelow.

The convex cone

V,

in thiscase,isthesame as in

paragraph

3,that is

V-H/(2,R).

The mapping

F

isdefined

by

F" C2C2-’HC2,R),F

u- v-

--FCu,v)-(u’+’u)-(

v2

"i-

^{/; u2k’}

Hence

wehave,in thiscase, k 3and1 2.

From

aboveweconclude that

D(H/(2,R),F)

^isan affinely

homogeneous Siegel

domainof second kindbiholomorphieallyonto

D’.

Now,

^ourmethod consists ofdescribing

D’

orequivalently

D(H/(2,R),F)

by^anormal,/-algebra

(S,Jo, w).

We

have

proved

that the solvable Lie

algebra

s canbedescribed

by

the set of matrices.

s-- 0 _1/2 0

/

0 0 0

(4.1) (4.4)

From

thisconstructionof s we conclude thattheendomorphism

Jo

has the form

J0-(13k,), IER

^{1-:k, 5}

which mustsatisfythe relations

(2.17)

^and

(2.18).

From

these conditions and after a lotof calculations we have

pt 0 0 0 0 ql 0 0 0 0

0

_P2 0 0 0 0 q2 0 0 0

0 0 P3 ^{0 0 0 0} q3 0 0

0 0 0 P4 ^{0 0} 0 0

q,

0

0 0 0 0

Ps

^{0 0 0 0}

q

+p

0 0 0 0 pl 0 0 0 0

qt

0

l/d

0 0 0 0 P2 ⁰ 0 0

qz

0 0 0 0 0 0 p 0 0

0 0 0

+

0 0 0 0

p,

0

0 0 0 0 0 0 0 0

Ps

(4.2)

where PitE

R, 5,

ql

E R"

5.

Thelinearform w on this Liealgebra^{s is}defined asfollows

w(X) (Xo, X (4.3)

where isthe usual innerproducton s

andXo(K1,

K2,K3,K,,,K5, KI,

KT,Ks,Kg,Km)

afixed vector.

In

order w tosatisfythe conditions

(2.19)

and

(2.20)

^{we musthave}

ql

21

^>0,

q_

₂₃>0,

q :

^>0,

^qs rs

^>0

^(4.4)

Now,

wehaveprovedthefollowingtheorem

THEOREM

4.1.

Let D

be theunique non-symmetric homogeneousbounded domain in 12

123 12".

Thecorresponding normal

J-algebra

is

(s,J0, w),

where

S,Jo

and waregiven

by (4.2)

^and

(4.3)

respectively.

Now,

we determine thesolvable Lie

group S

which

corresponds

tothe solvable Lie

algebra

^s.

We

denoteby

GL(s)

the

group

of nonsingular endomorphismsof s. TheLie

algebra gl(s)

of

GL(s)

consists of allendomorphismsof s with thestandardbracketoperations.

[x,r]-xY- YX.

Themapping

ad s

gl(s)

ad

B

adB where

adB s s, adB

T

adB

T T, B

is ahomomorphismof sontoa

subalgebra ad(s)

^of

gl(s). Let int(s)

be theanalyticsubgroup^of

GL(s)

whose Lie

algebra

^is

ad(s). ^It

^isknown that

Int(s)

^iscalled theadjoint

group

of s.

NEW REPRESENTATION OF THE NON-SYMMETRIC DOMAINS 751 Thegroup

Aut(s

of allautomorphismsof s is aclosed subgroupof

GL(s ). ThusAut(s)

has aunique analyticstructureunder which it becomes atopological subgroupof

GL(s). We

denoteby

d(s)

the Lie

algebra

of

aut(s). ^Now,

^the

group Int(s)

isconnected,so it isgenerated bytheelementse

’c’), X

s.

Therefore

Int(s)

isanormal

subgroup ofAut(s).

From

theabove we conclude that the solvable Lie

group S

of s is definedby

0

S.L.

⁰

0 0 0

. ^{,, z} (eV 1) (e

^v’

1) x. ,, ^x _{(eV, 1)}

e*’

0 0 0 0

0 e

"

^{0 0 0}

0 0 e

’’

^{0 0}

0 0 0 e

’’

⁰

0 0 0 0 e

’

xjR

^{i-1 5}

/

^pi

^R"

^{i-1 5}

Theinnerproductonthe solvable Lie

algebra

^sisgiven

by (4.3).

This inner

product

determinesthe Kiihler

onS

which isessentiallymetricthe

Bergman

metric on it.

Now,

wecan statethefollowingtheorem.

THEOREM

4.2. Thehomogeneous non-symmetricbounded domain in

s

^isbiholomorphically

is,

^omorphicontothe solvable Lie

group S

defined

by (4.5).

^{The Khlermetric}

g

^on

S

defined bythe relation

(4.3).

Let F

be a Lieautomorphismon s. This

F

canbe

represented

by^{the matrix}

’all

0 0 0

F-

₀ 0 0 0 0

0 0 0 0

a

^{0 0 0 0}

az

^{0 0 0 0}

az

^{0 0 0}

0 a3 0 0 0 0

a

^{0 0}

0 0

a

^{0 0 0 0 a,9 0}

0 0 0

ass

^{0 0 0 0}

as

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

which becomes an isometry withrespectto theinnerproduct when

a_n +/-1,

az ^-,-1, a.

+1, a44 +1,

ass ^+1,

P:’q2 1

, az _as-

P3 1 a3s

a2

l+p

^a, a3

If

F

isanisometry,thenithas the form

PsPs

¹

ass

l+ps ass

Plql

as1 0 0 0 0 0 0 0 0

+pl all

P2q2 1-a22

0 a22 0 O 0 0 0 0 0

/p22 a22

P3q3 -a

0 0 a33 0 0 0 0 0 0

+p32 a33

O

0 0

a

^{0 0 0 0 v,,}_p2

^-o.

^O

0 0 0 0 _a55 0 0 0 0 Psq5

p2 a

0 0 0 0 0 1 0 0 0 0

0

0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1

where

an

^_+1, a22 _+1, _a33 _+1,

au

_+1, a55 ^_+1.

From

theform of

Fz,

weobtainthat ithas theeigenvalue1 withmultiplicityat least 4. Therefore wehaveprovedthefollowingtheorem.

THEOREM

4.3. The homogeneous non-symmetricbounded domain inC with thestandard Kiihler metric does not admit

any

k-symmetricstructure.

REFERENCES

CARTAN, E. Sur

les domains homeshomogeneousde

l’espace

de n variablescomplexes, Abh.

Math.

Sere., Hamburg

Univ.11

(1936),

^116-162.

HANO, J. On

Kihlerian

homogeneous

ofunimodular,

Am. J.

Math. 78

(1957),

^885-900.

KANEYUKI, S. Homogeneous

BoundedDomainsandSiegerDomains,Springer-Verlag,

New

York

(1971).

KOSZUL, J. Sur

laforme herrnitiennecanoniquedes

espaces

homogenes complexes,

Can. J.

Math.

_7 (1955),

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LEDGER, A.

and

OBATA, M.

Affineand Riemannians-manifolds,

J.

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451-459.

MOTSUSHIMA, Y. Sur

les

espaces homogenes

Kihleriensd un

groupe

de Liereducti, Math.

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PYATETSKIJ-SHAPIRO, J. On

aproblem

proposed

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E. Cartan, Do.kl.

Akad. Nauk,

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PYATETSKIJ-SHAPIRO, J. Automolvhic

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Journal of Applied Mathematics and Decision Sciences

Special Issue on

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Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

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Guest Editors

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[email protected]

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