For the ase of the graded Lie algebra, the following results are investigated

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SUMMARY OF Ph.D. DISSERTATION

Shool StudentIdentiationNumber SURNAME,First name

Miyashita Toshikazu

Title

Realizations of xed pointsubgroups by the automorphismsof nite order

in exeptional simple Liegroups and itsappliations

Abstrat

It is well known that the involutiveautomorphisms of theompat Lie groupsplay anim-

portantrole in thetheory of symmetrispaes (.f. Berger[1℄)

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In[42℄, [43℄and [44℄ Yokota

showed that the exeptionalsymmetri spaes G=H are realized denitely by alulatingthe

xed pontsubgroupsoftheinvolutiveautomorphisms

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J. A.Wolf andA. Graylassiedthe

automorphismsoforder3intheonnetedompatLie groupsofenterfreeandthestruture

ofthexedpointsubgroupsbyitsautomorphisms([56℄). Yokotarealizedtheautomorphsmsof

order 3in exeptionalompatLiegroupsG

2

;F

4 andE

6

,anddeterminedthestrutureofthe

xed pointsubgroupsbyitsautomorphisms([41℄).

For the ase of the graded Lie algebra, the following results are investigated. Kaneyuki

lassiedtheseondkindgraded deompositionsofsimpleLiealgebras:g=g

2 g

1 g

0

g

1 g

2

andthesubalgebrasg

ev

=g

2 g

0 g

2

;g

0

([18℄). Haralassiedthethirdkindgraded

deompositionsofsimpleLiealgebras:g=g

3 g

2 g

1 g

0 g

1 g

2 g

3

andthesubalgebras

g

ev

;g

0 andg

ed

=g

3 g

0 g

3 ([10℄).

InthispaperweshallrealizethoseresultsbyLiegroupwhentheLiealgebragisexeptional

type. For the purpose we onsider the problem realizing xed point subgroups by automor-

phismsofnite orderand alsotheintersetionsofthosexedpointsubgroups. Fortheaseof

exeptionalLiegroupG,theinvolutiveautomorphisms; 0

2F

4

; ; 0

2G

2

playanimportant

role. Here wedetermine thegroupstrutures of theintersetion of thexed point subgroups

G

\G

0

;G

\G

0

;G

\G

;G

\G

0

\G

\G

0

. FortheexeptionalompatLiegroupE

7 ,

wedeterminetheautomorphismsoforder3and thestrutureofthexed pointsubgroupsby

itsautomorphism. ThereexiststhespinorgroupssequeneofexeptionalompatLiegroups:

Spin(1) Spin(8) Spin(14) Spin(15) Ss(16) E

8

. Then we an prove

that thesequeneisdeeplyrelatedto thexed pointsubgroupofeah spinorgroupby 0

. As

theappliationsrelatedtotheautomorphismofniteorder,weonsiderthegrouprealizations

G

ev

;G

0

;G

ed

whih orrespond to the Lie algebras g

ev

;g

0

;g

ed

investigated by Kaneyuki and

Hara mentionedasabove. ThenweonstrutdenitelythesubgroupSpin(14;C)oftheom-

plex exeptionalLie groupE

8 C

and thestruture ofthe group(E

8 C

)

0

in ase ofseond kind

graded deomposition. We also onstrut Spin(12;C) of the omplex exeptional Lie group

E

7 C

andthestrutureofthegroup(E

7 C

)

ev

in aseofthird kindgradeddeomposition.

Remark

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[1℄,[10℄,[18℄,[41℄,[42℄,[43℄,[44℄,[56℄arethenumbersofthereferenesofthedesser-