Four-Dimensional Pseudo-Riemannian Homogeneous spaces. Classification of Real Pairs

B. Komrakov seminar

FOUR-DIMENSIONAL

PSEUDO-RIEMANNIAN HOMOGENEOUS SPACES.

CLASSIFICATION OF REAL PAIRS

B. KOMRAKOV, JNR.

International Sophus Lie Centre

Belarus, 220123, Minsk, P.O. Box 70 Current address:

Centre for Advanced Study

at The Norwegian Academy of Science and Letters P.O. Box 7606 Skillebekk

0205 Oslo Norway

e-mail: [email protected]

This publication was typeset using

AMS- T:EX

the American Mathematical Society's 'JEXmacro system.

INTRODUCTION . . . 5

Chapter I. Complex pseudo-Riemannian pairs ... 7

1. CLASSIFICATION OF SUBALGEBRAS IN THE LIE ALGEBRA

so(

4, C) ... 7

2. CLASSIFICATION OF COMPLEX PAIRS . . . 9

Chapter II. Classification of real pairs ... 35

PRELIMINARIES . . . 35

1. REAL FORMS OF PAIRS (g, g) WITH SOLVABLE SUBALGEBRA g ONE-DIMENSIONAL CASE . . . 39

2. REAL FORMS OF PAIRS

(g,

g) WITH SOLVABLE SUBALGEBRA g TWO-DIMENSIONAL CASE . . . 72

3. REAL FORMS OF PAIRS (g, g) WITH SOLVABLE SUBALGEBRA g THREE- AND FOUR-DIMENSIONAL CASE . . . 120

4. REAL FORMS OF PAIRS

(g,

g) WITH NON-SOLVABLE SUBALGEBRA g ... 139

References ... 152

INTRODUCTION

We consider classification of lower-dimensional homogeneous spaces an immedi- ate continuation and global version of classification results obtained by Sophus Lie.

Two-dimensional homogeneous spaces were classified locally by Sophus Lie [L 1]

and globally by G.D. Mostow [M]. (See also preprint [KTD], where the complete classification of two-dimensional homogeneous spaces, both locally and globally, is presented.) S. Lie also obtained some results in classification of three-dimensional homogeneous spaces and described all subalgebras in the Lie algebra so(4, C) (in terms of vector fields). A detailed account of these classifications can be found in [L2]. The local classification of all three-dimensional isotropically-faithful homo- geneous spaces was obtained in [KT], and the classification (local and global) of all two- and three-dimensional pseudo-Riemannian isotropically-faithful homogeneous spaces was given in [DK].

The problem of classification of four-dimensional pseudo-Riemannian homoge- neous spaces is interesting from the point of view of both geometry and physics, and not only in the case of signature (1, 3) (spaces of relativity theory) but also in the case of signature (2, 2) (twistors).

Let

(H, M)

be a homogeneous space,

H

=

Hx

the stabilizer of an arbitrary point x E

M,

and

(b, f))

the pair of Lie algebras corresponding to the pair

(H, H)

of Lie groups.

Lemma. Suppose that the homogeneous space (H, M) admits an invariant pseudo- Riemannian metric. Then the isotropic representation of the pair

(b, f))

p: f)

---7

g((b/fJ),

p(x)(x

+f))= [x,

x]

+f)

(x

E f),

^l

x E f))

^-

is faithful. Moreover, there exists a basis of

b/fJ

such that p(f)) lies in one of the following Lie algebras: so(4), so(3, 1), or so(2, 2), which are the real forms of the complex Lie algebra so(4, C).

In accordance with this, we divide solution of our problem into the following parts:

(1) We find (up to conjugation) all possible forms the subalgebra

(p(fJ))1C = p1C(fJ1C)

can assume. This is equivalent to classifying (up to conjugation) subalgebras .p in the Lie algebra so ( 4, C).

(2) For each subalgebra .p obtained in (1), we find (up to equivalence of pairs) all complex pairs

(g, g)

such that the subalgebra

p1C(f)1C)

is conjugate to .p (here

g = f)1C). ·

(3) For each complex pair (g, g), we find (up to equivalence of pairs) all its real forms

(g

¹⁷

,g

^{17 ),} where() is an anti-involution of the pair

(g,g).

(4) For each real pair obtained in (3), we construct all (up to isomorphism) corresponding homogeneous spaces.

The results of the first part of our work (the classification of the complex pairs) was obtained in [K1] (for a summary of results see [K]). We recall this classification in Chapter I.

This paper presents the results of the second part (the classification of the real pairs) of our work devoted to classification of four-dimensional homogeneous spaces

with an invariant pseudo-Riemannian metric of arbitrary signature. We give this classification in Chapter II. A similar classification for the case of Riemannian metric can be found in [I].

COMPLEX PSEUDO-RIEMANNIAN PAIRS

1. CLASSIFICATION OF SUBALGEBRAS IN THE LIE ALGEBRA so(4, CC) Preliminaries:

1. For the sake of simplicity instead of the standard notation for a subalgebra of so( 4, CC) such as

{ (

X 0 0 0 ) }

0 AX 0 0

g = 0 0 ^-X 0 ^X E CC ' 0 0 0 -Ax

where IAI

<

1, -%

<

argA ~%or IAI = 1, 0 ~ argA ~%we use the following notation:

g= ⁽

X 0 0 0 ) 0 AX 0 0 0 0

-x

0 0 0 0 -Ax

I A I

<

^1, -%

<

arg A ~ % or I A I = 1, 0 ~ arg A ~ %.

Here we imply that variables denoted by Latin letters run through CC and that parameters are denoted by small Greek letters.

2. To refer to subalgebras determined in Theorem 1 we use the following notation:

d.n,

where d is the dimension of the subalgebra; n is the number of the subalgebra in Theorem 1.

Theorem 1. Any non-zero subalgebra of the Lie algebra so(4, CC) is conjugate (with respect to GL(4, CC)) to one and only one of the following subalgebras:

1.1

0 0

-X 0

dim g

=

¹

_lJ

^1.2

I A I

<

1, -%

<

arg A ~ % or I A I = 1, 0 ~ arg A ~ %

1.3

( ~ ~ ~ ~)

0 0 0 0 0 0 0 0

0 0 -X

-x

dim g

=

²

2.1

(~ ~ ~X jJ

^2.2

^(~ t ^{;~ jJ}

I

A

I <

1 or

I

A

I =

^{1, 0 ~ arg} A ~ ^1r

2.4

(~ ~ ~X D

2.5

G ~ i: ^D

dim g

=

³

3.1

(~ ~ ;~ jJ

^3.2

^(~ t ;: jJ

3.3

( ^{~0 ~ ~z}

_{0 -y}

3.5

4.1

ReA

>

0, or ReA = 0, Im A ~ 0

JJ

y

( ^;~

_{0 -z} 0

0 0

3.4

dim g = 4

(

X y

~ ~X

0 0

( X y

z t

0 0 0 0

0 0 )

~x ~z

-y ^X

~x ~z)

-y - t

dim g

=

5

5.1

(~

^{0 0 y}

^t

^-u

^-x

^{-y 0}

^~z

^{- t}

⁾

dim g

=

6

uv

y 0

~z ⁾

t -u v

-x

0 -y

-t

6.1

Remark. To simplify the computation, instead of so(4, C) we use the linear Lie algebra 6.1, which is conjugate to so(4, C).

2. CLASSIFICATION OF COMPLEX PAIRS

Preliminaries:

1. Let g be one of the subalgebras of the Lie algebra

so(

4, C) determined in Theorem 1. We assume that the Lie algebra g acts naturally on C4 ; then (g, C4 ) is a faithful generalized module. The enumeration of the generalized modules ob- tained in this way coincide with that of the corresponding subalgebras of

so (

4, C) in Theorem 1.

We say that a pair

(g,

g) has type n.m, if the corresponding generalized mod- ule (g,

gjg)

is isomorphic to the generalized module n.m, i.e., to the generalized module (g, C^{4 ),} where g is the subalgebra of so(4, C) supplied with the number n.m in Theorem 1.

2. Let (g, g) be a pair of type n.m. Then without loss of generality we can identify the Lie algebra g with the subalgebra

n. m

of the Lie algebra

so (

4, C).

Let { u1 , u2 , u3 , u4 } be the standard basis of C^{4 :}

3. We define a pair (g, g) by the commutation table of the Lie algebra g only.

Here by {e1, ... , en, u 1, uz, u3, u4} we denote a basis of g (n =dim g). We assume that the Lie algebra g is generated by e1 , . . . , en.

By p, r, s, etc. we denote the parameters appearing in the process of the classi- fication. If there are some complementary conditions on them, it is indicated just after the table. Otherwise we assume that these parameters run through C.

Theorem 2. Any complex isotropically faithful pseudo-Riemannian pair (g, g) of codimension 4 is equivalent to one and only one of the following:

0.1 g = {0}

91 ^UI U2 U3 U4 92 ^{UI U2 U3 U4}

U1 0 U3 U2 0 ^ul 0 U3 0 ul

U2 -U3 0 UI 0 ^{U2 -U3} 0 0 PU2

U3 -u2 -ul 0 0 U3 0 0 0 (p+l)u3

U4 0 0 0 0 U4 -u1 -pu2 -(p+1)u3 0

93 ul U2 U3 U4 94 UI U2 U3 U4

UI 0 0 0 2ul UI 0 0 0 ul

U2 0 0 UI U2 U2 0 0 0 U1 +u2

U3 0 -ul 0 u2+u3 U3 0 0 0 u2+u3

U4 -2u1 -u2 -u2-u3 0 U4 -u1 -u1 -u2 -u2 -u3 0

9s

^{UI U2 U3 U4} 96 ^{UI U2 U3 U4}

UI 0 0 0 UI UI 0 0 0 UI

U2 0 0 0 U1 +u2 U2 0 0 0 PU2

U3 0 0 0 PU3 U3 0 0 0 ru3

U4 -ul -ul-u2 -pu3 0 U4 -u1 -pu2 -ru3 0

97 ^{UI U2 U3 U4} 98 ^{UI U2 U3 U4}

ul 0 0 UI 0 UI 0 0 0 U2

u2 0 0 0 U2 U2 0 0 0 0

U3 -ul 0 0 0 U3 0 0 0 UI

U4 0 -u2 0 0 U4 -U2 0 -ul 0

gg _{UI U2 U3 U4} 910 ul ^{U2 U3 U4}

ul 0 0 0 UI UI 0 0 0 0

U2 0 0 0 0 U2 0 0 UI 0

U3 0 0 0 U2 U3 0 -Ul 0 0

U4 -ul 0 -U2 0 U4 0 0 0 0

911 ul U2 U3 U4 U1 0 0 0 0

U2 0 0 0 0

U3 0 0 0 0

U4 0 0 0 0

1.1

g={ n ^AX

0 ^{0 0 -x 0 0} 0

_jJ

^x

^EC},

A

E

c,

^IAI

<

1,

-

~

<

arg

A :::;;

~ or I

A

I = 1, 0:::;; argA:::;; ~

A=O

1.

[ '] ^{el U1 U2 U3 U4}

el 0 Ul 0 -U3 0

Ul -ul 0 0 U2 0

U2 0 0 0 0 U2 U3 U3 -U2 0 0 U3 U4 0 0 -U2 -u3 0 2. [,] el Ul U2 U3 U4 el 0 Ul 0 -U3 0

Ul - u l 0 0 0 0

U2 0 0 0 0 PU2 U3 U3 0 0 0 U3 U4 0 0 -pu2 -u3 0 3.

[ '] ^{el ul U2 U3 U4}

e1 0 U! 0 -U3 0

Ul - u l 0 0 e1+u2 0

U2 0 0 0 0 0

U3 U3 -el-u2 0 0 0

U4 0 0 0 0 0

4. [

,]

el Ul U2 U3 U4 el 0 Ul 0 -U3 0

Ul -Ul 0 0 U2 0

U2 0 0 0 0 0

U3 U3 -u2 0 0 0

U4 0 0 0 0 0 5. [

,]

e1 U1 U2 U3 U4

el 0 Ul 0 -U3 0

ul -ul 0 0 el 0

U2 0 0 0 0 U2 U3 U3 -el 0 0 0

U4 0 0 -U2 0 0

6.

[ ' ] ^{e1 U1 U2 U3 U4}

e1 0 U1 0 -U3 0

U1 -u1 0 0 0 0

u2 0 0 0 0 U2 U3 U3 0 0 0 0

U4 0 0 -u2 0 0 7. [,] e1 u1 U2 U3 U4 e1 0 U1 0 -U3 0

U1 -U1 0 0 e1 0

U2 0 0 0 0 0

U3 U3 -e1 0 0 0

U4 0 0 0 0 0

A - l _{- 2}

8. [,] e1 U1 U2 U3 U4 e1 0 U1 _2U2 ¹ -U3 -2U4 ¹

U1 -u1 0 0 -2e1 u2

U2 -~U2 ^{0 0 U4 0}

U3 U3 2e1 -U4 0 0

U4 _2U4 ¹ -U2 0 0 0 9. [,] e1 U1 U2 U3 U4

e1 0 U1 2U2 ¹ -u3 -2U4 ¹

U1 -u1 0 0 0 U2

U2 -2U2 ¹ 0 0 0 0

U3 U3 0 0 0 0

U4 _2U4 ¹ -U2 0 0 0

I

A

I <

1, - ~

<

arg A ~ ~ or

I

A

I

= 1, 0 ~ argA ~ ~

10. [,] e1 U1 U2 U3 U4 e1 0 U1 AU2 -U3 -AU4

u1 -u1 0 0 0 0

U2 -AU2 0 0 0 0

U3 U3 0 0 0 0

U4 AU4 0 0 0 0

1.2

1.

[

' ]

e1 U1 U2 U3 U4

1.3

1.

[

' ]

e1 u1 U2 U3 U4

2.

[

,]

e1 Ut U2 U3 U4

3.

[

'

]

e1 U1 U2 U3 U4

e1

0 0

-x

-x

U1 U2 U3 U4

0 u1 u1 +u2 -u3 -u4 -u4

-u1 0 0 0 0

-u1-u2 0 0 0 0

u3+u4 0 0 0 0

U4 0 0 0 0

e1 u1 U2 U3 U4

0 e1 0 U1 U2 -e1 0 -2U2 U3 ¹ 2U4 ¹

0 2U2 ¹ 0 ~U4 ⁰

-u1 -U3 -2U4 ¹ 0 0

-u2 -2u4 1 0 0 0

e1 U1 U2

0 0 0

U3 U1

U4 U2

0 0 0 ->.e1 +(>.+1)u1 +.Au2 0

0 0 0 0 U2

-u1 >.e1- ( >. + 1 )u1->.u2 0 0 0

-u2 0 ^-u2 0 0,

1>-1

<

1 or 1>-1 = 1, 0 :=:;; arg ^>. :=:;; 1r

e1 U1 U2 U3 U4

0 0 0 u1 U2

0 0 0 U1 0 0 0 0 0 U2 -u1 -u1 0 0 e1 -U2 0 ^-u2 -e1 0

4. [

,]

el

U!

U2

ua U4

el 0 0 0

U!

U2

U!

^{0 0 0 X} y U2 0 0 0 y z

ua

-u1

-x

-y 0 0

U4

^-u2 -y -z 0 0, where

1

A

1

x =

¹

^+A

^{el + 1}

^+A

^{ul - 1}

^+A

^u2'

1 1 1

Y

= - _1+A

^{e1 +}

_1+A

^{u1 +}

_1+A

^{u 2,}

A A

1+2A

z

=

- - - e 1 + - - u l + - - u 2

1+A 1+A 1+A '

A#

-1 5.

[ 'l

^el

^U!

^U2

^{ua U4}

el 0 0 0

U!

U2

U!

^{0 0 0 0 0}

U2 0 0 0

U!

U2

ua

^{-ul 0 -ul 0}

-ua U4

-u2 0 -u2

ua

⁰

6.

[,]

el

U!

U2

ua U4

el 0 0 0

U!

U2

U!

0 0 0 0 0

U2 0 0 0

AU!

-Ael +(A+1)u2

ua

-ul 0

-AU!

0

-Aua

U4

-u2 0 Aei-(A+1)u2

AU

a 0 7. [

,]

el U1 U2

ua U4

el 0 0 0

U!

U2

ul 0 0 0 0 0

U2 0 0 0 0 U2

ua

-ul 0 0 0 el

U4

-u2 0 -u2 -el 0 8.

[ ' l

^el

^U!

^U2

^{ua U4}

el 0 0 0

Ul

U2

U!

0 0 0 0 0

U2 0 0 0 -ul el

ua

^{-ul 0}

Ul

0 e1

+ua

U4

-U2 0 -e1 -e1-u3 0

9.

[

,]

e1 U1 U2 us U4

e1 0 0 0 U1 U2

U1 0 0 0 0 U1

U2 0 0 0 f.LU1 -AJ.Le1 +(:X.+J.L)u2

us -u1 0 -J.LU1 0 (1-J.L )us

U4 -u2 -u1 AJ.Le1-(A+J.L)u2 (J.L-1)us 0 10.

[

,]

e1 U1 U2 us U4

e1 0 0 0 U1 u2

U1 0 0 0 0 U1

U2 0 0 0 1

2u1 -~e1 +(:X.+~)u2

us -u1 0 1

-2u1 0 1

e1 +2us U4 -u2 -u1 ~e1-(:X.+~)u2 -e1-~us ⁰ 11.

[,] e1 u1 U2 us U4

e1 0 0 0 U1 U2

U1 0 0 0 0 U1

U2 0 0 0 (1->..)u1 >..(>..-1)e1 +u2 us -u1 0 (:X.-1)u1 0 e1 +>..us U4 -u2 -u1 >..(1-:X.)e1 -u2 -e1 ->..us 0

:X.#~

12.

[

,]

e1 U1 U2 us U4

e1 0 0 0 U1 U2

U1 0 0 0 -e1 +2u1 U2

U2 0 0 0 U2 -e1 +u1

us -u1 e1-2u1 -U2 0 0

U4 -u2 -u2 e1-u1 0 0

13.

[ 'l

e1 U1 U2 us U4 e1 0 0 0 U1 U2 U1 0 0 0 0 0 U2 0 0 0 0 U1 us -u1 0 0 0 e1 U4 -U2 0 -ul -el 0

14.

[' J el UI U2 U3 U4

el 0 0 0 U1 U2 UI 0 0 0 0 0 U2 0 0 0 0 UI

U3 -ul 0 0 0 0

U4 -u2 0 -ul 0 0 15.

[ ' J el UI U2 U3 U4

e1 0 0 0 UI U2

UI 0 0 0 0 UI

U2 0 0 0 UI -e1 +u1 +2u2

U3 -ui 0 -ul 0 0

U4 -u2 -u1 e1 -u1 -2u2 0 0 16.

[ ' J el UI U2 U3 U4

el 0 0 0 UI U2

UI 0 0 0 0 0

U2 0 0 0 u 1 U2-Ul

U3 -UI 0 -ul 0 -U3 U4 -U2 0 UI-U2 U3 0 17.

[' J el UI U2 U3 U4

el 0 0 0 UI U2

UI 0 0 0 0 UI

U2 0 0 0 AU! -Aei +(1-A)ui +(1+A)u2

U3 -ul 0 -AUI 0 (1- A)u3

U4 -u2 -u1 Aei+(A-1)ui-(1+A)u2 (A-1)u3 0

A#1 18.

[

,]

el ui U2 U3 U4

e1 0 0 0 UI U2

UI 0 0 0 0 UI

U2 0 0 0 _2u1 ¹ -2e1 +2u1 +2u2 ^{1 1 3}

U3 -ul 0 -2u1 ¹ 0 _{e1 +} 2u3 ¹

U4 -u2 -u1 2e1- 2u1- 2u2 -e1-1 1 3 2u3 1 0

['] ^{el ul Uz us U4}

el 0 0 0 ul Uz

ul 0 0 0 2ul 2uz

Uz 0 0 0 uz e1-2u1 ¹ us -ul -2ul -uz 0 U4 U4 -u2 -2u2 ~u1 -e1 -u4 0 23.

[

,]

el Ul Uz Us U4 el 0 0 0 ul Uz ul 0 0 0 ^X y Uz 0 0 0 y z us -ul -x -y 0 0 U4 -uz -y -z 0 0,

where

Two pairs corresponding to parameters ( ^)q, J-£1 ) and ( .\2 , J-£2 ) are equivalent if and only if the points (>.1^,~-£1^{), (>.}2^,~-£2^{) E} C* x C* lie in the same orbit of the action of the symmetric group 6 3 on C* x C* generated by the transformations

(>.,

J-t)

^--+ (J-£,

>.); c>-,

1-£) --+ (

~,

^-

x).

24.

[ ' l

e1 u1 U2 ua U4 e1 0 0 0 u1 U2 U1 0 0 0 0 0

U2 0 0 0 0 0

ua -u1 0 0 0 e1 U4 -u2 0 0 -e1 0 25.

[ ' l

e1 U1 U2 U3 U4

e1 0 0 0 u1 U2 U1 0 0 0 0 0

U2 0 0 0 0 0

ua -u1 0 0 0 0

U4 -U2 0 0 0 0 1.4

g={ G

^{X 0 0 0 0 0 X 0}

D

^xE

^C}

1. [,] e1 U1 U2 ua U4 e1 0 0 u1 U2 e1 U1 0 0 U1 U2 U1 U2 -u1 -u1 0 ua 0

ua -u2 -u2 -u3 0 -ua U4 -e1 -u1 0 ua 0

2.

[

'

] ^{el UI Uz us U4}

el 0 0 ul uz el

UI 0 0 0 0 PU1

uz -u1 ^{0 0 0} (p-l)uz

us -uz ^{0 0 0} (p-2)us

U4 -e1 -pu1 (1-p)uz (2-p)us 0 3.

[,] el UI uz us U4 el 0 0 UI Uz el ul 0 0 0 0 2ul

Uz -ul 0 0 el Uz us -uz ^{0 -el 0 0}

U4 -e1 -2u1 -uz 0 0 4.

[

,]

el UI Uz us U4 el 0 0 UI Uz ⁰ UI ^{0 0} UI Uz ⁰ Uz ^{-ul -ul} 0 us ⁰ us -uz -uz -us ^{0 0}

U4 0 0 0 0 0

5.

[,]

el UI Uz us U4

el 0 0 UI uz ⁰

UI 0 0 0 0 UI

Uz -ul 0 0 0 Uz

us -uz 0 0 0 u1 +us

U4 0 -Ul -Uz -UI-US 0 6.

[ '] ^{e1 U1 uz us U4}

el 0 0 UI Uz ⁰

ul 0 0 0 0 UI uz -ul 0 0 0 Uz us -uz ^{0 0 0} us

U4 0 -u1 -uz -us 0 7.

[,] el Ul Uz us U4

el 0 0 ul Uz ⁰

UI ^{0 0 0} U1 0

Uz -ul 0 0 re1 +uz+u4 ⁰

us -uz -u1 -re1 -uz -u4 0 PU4

U4 0 0 0 -pu4 0

8.

[ ' l

^{el Ul U2 U3 U4}

el 0 0 Ul U2 0

Ul 0 0 0 ul 0

U2 - u l 0 0 re1 +u2 0

U3 -u2 -u1 -re1 -u2 0 PU4

U4 0 0 0 ^-pu4 0

9.

[,] el Ul U2 U3 U4

el 0 0 Ul U2 0

Ul 0 0 0 ul 0

U2 - u l 0 0 re1 +u2+u4 0

U3 -U2 -Ul - r e1 - U2- U4 0 U!-U4

U4 0 0 0 ^U4-Ul 0

10.

[ ' l

^{el Ul U2 U3 U4}

el 0 0 UI U2 0

U1 0 0 0 Ul 0

U2 -ul 0 0 re1 +u2 0

U3 -u2 - u1 - r e1 -u2 0 ^U!-U4

U4 0 0 0 U4-Ul 0

11.

[ ' l

^{el Ul U2 U3 U4}

el 0 0 Ul u2 0

Ul 0 0 0 0 0

U2 -ul 0 0 re1 +u4 0

U3 -U2 0 - r e 1 -u4 0 U4

U4 0 0 0 ^-u4 0

12.

[ ' l

^{el UI U2 U3 U4}

el 0 0 Ul U2 0

Ul 0 0 0 0 0

U2 - u l 0 0 re 1 0

U3 -u2 0 - r e 1 0 U4 U4 0 0 0 -U4 0 13.

[ ' l

^{el Ul U2 U3 U4}

el 0 0 ul U2 0

Ul 0 0 0 0 0

U2 - u l 0 0 e1 +u4 0

U3 -u2 0 -el-u4 0 Ul

U4 0 0 0 -ul 0

14.

[

,]

el U! U2 U3 U4

el 0 0 U! U2 0

U! 0 0 0 0 0

U2 -U! 0 0 U4 0

U3 -u2 0 -U4 0 U!

U4 0 0 0 -U! 0 15.

[ ' l

^{el U! U2 U3 U4}

el 0 0 U! U2 0

U! 0 0 0 0 0

U2 -U! 0 0 e1 ^+u4 0

U3 -u2 0 -e1-u4 0 0

U4 0 0 0 0 0

16.

[ ' l

^el U! U2 U3 U4

el 0 0 U! U2 0

U! 0 0 0 0 0

U2 -ul 0 0 U4 0

U3 -u2 0 -U4 0 0

U4 0 0 0 0 0 17.

[ 'l

^{el U! U2 U3 U4}

el 0 0 U! U2 0

U! 0 0 0 0 0

U2 -U! 0 0 el 0

U3 -u2 0 -el 0 U!

U4 0 0 0 -u1 0 18.

[ '] ^el U! U2 U3 U4

el 0 0 U! U2 0

U! 0 0 0 0 0

U2 -U! 0 0 0 0

U3 -u2 0 0 0 U!

U4 0 0 0 ^-U! 0 19.

[,] el U! U2 U3 U4

el 0 0 U! U2 0

U! 0 0 0 0 0

U2 -ul 0 0 el 0

U3 -u2 0 -el 0 0

U4 0 0 0 0 0

20. [,] el UI U2 U3 U4 el 0 0 U1 U2 0

UI 0 0 0 0 0

U2 -ul 0 0 0 0

U3 ^-u2 0 0 0 0

U4 0 0 0 0 0

2.1

g= {

(~

^{0 y 0 0 -X 0 0 0}

jJ

^{x,y E}

^C}

1.

[ ' l

^{el e2 UI U2 U3 U4}

el 0 0 UI 0 -U3 0

e2 0 0 0 U2 0 -U4 ul -ul 0 0 0 el 0

U2 0 -U2 0 0 0 e2 U3 U3 0 -el 0 0 0

U4 0 U4 0 ^-e2 0 0 2.

[ ' l

^{el e2 UI U2 U3 U4}

el 0 0 UI 0 -U3 0

e2 0 0 0 U2 0 -U4 U1 -UI 0 0 0 e1 0

U2 0 ^-u2 0 0 0 0

U3 U3 0 -el 0 0 0

U4 0 U4 0 0 0 0 3.

[ ' l

^{el e2 UI u2 U3 U4}

el 0 0 UI 0 -U3 0

e2 0 0 0 U2 0 -u4 UI -UI 0 0 0 0 0

U2 0 ^-U2 0 0 0 0

U3 U3 0 0 0 0 0

U4 0 U4 0 0 0 0

2.2

g={ G ^AX

^{0 0 y}

^-x

^{-y 0 0}

jJ ^x,yEi+

.:\ E C, /.A/

<

1 or /.A/= 1, 0 :=:; arg

>.

:=:; 1r

.:\=0

1. [

,]

e1 e2 UI U2 us U4 ei 0 e2 UI 0 -us 0 e2 -e2 0 0 UI -U4 -2e2 UI -ul 0 0 0 U2 -ui

U2 0 -ul 0 0 0 U2

us us U4 -u2 0 0 2us U4 0 2e2 UI -u2 -2us 0

2. [,] ei e2 UI U2 us U4

ei 0 e2 UI 0 -us 0

e2 -e2 0 0 UI -U4 0

UI -ul 0 0 e2 U4 0

u2 0 -ul -e2 0 (p-1)us pu4 us us U4 -u4 (1-p)us 0 0

U4 0 0 0 -pu4 0 0

3. [

,]

el e2 U! u2 us U4 el 0 e2 UI 0 -us 0 e2 -e2 0 0 UI -U4 0 UI -ul 0 0 0 0 0 U2 0 -ui 0 0 us U4 us us U4 0 -us 0 0 U4 0 0 0 -U4 0 0 .:\=1

4. [

,]

el e2 UI U2 us U4 el 0 0 U! U2 -us -u4 e2 0 0 0 UI -U4 0 U! -ui 0 0 0 e2 0 U2 -U2 -UI 0 0 el e2 us us U4 -e2 -ei 0 0 U4 U4 0 0 -e2 0 0

5.

[ ' l

^{e1 e2 Ul U2 U3 U4}

e1 0 0 U1 U2 -u3 -u4 e2 0 0 0 Ul -u4 0

u1 -u1 0 0 0 0 0

U2 -u2 -u1 0 0 e2 0

U3 U3 U4 0 ^-e2 0 0

U4 U4 0 0 0 0 0 ,\ = -~

6. [,] e1 e2 u1 U2 U3 U4 e1

e2

0 ³ U1 1 1

ze2 - 2^{u2 -u3 zU4} -~e2 0 0 u1 -U4 0

u1 U2

- u l 0 0 U4 0 0

1 - u l -u4 0 0 0 2^u2

U3 U4

U3 U4 0 0 0 0

1 0 0 0 0 0

- 2^u4

l-\1 <

1 or

l-\1 =

^{1, 0:::;} arg ,\:::; ^1r 7.

[ ' l

e1 0 (1--\)e2u1

e2 ( ,\ -1) e2 0 0

u1 -u1 0 0

U2 -AU2 -Ul

0

U3 U3 U4 0

U4 AU4 0 0

2.3

AU2 -U3 -AU4 U1 -U4 0

0 0 0 0 0 0 0 0 0 0 0 0

g= {

(~ ^-x

^{y 0}

0

0 .

X) }

-x 0

-x ₀ x,yEC -y X

1.

e1 0 .

e2 -2e2 u1 -u1

U2 U2

u3 u2+u3 U4 -U1-U4

2e2 u1

0 0

0 0

0 0 0

U3

-u2 -u2-u3 u1 +u4

U1 -U4 0

0 0 0

0 0 0

0 0 0

0 0 0

2.4

g={ G

^{y 0 0 0 -x 0 0 y}

D

^{x,y E}

^C}

1.

[

,]

el e2 U! U2 us U4

ei 0 e2 U! 0 -us 0 e2 -e2 0 0 U! U2 0

U! -ui 0 0 U! U2 0

U2 0 -U! -U! 0 us 0

us us -u2 -u2 -us 0 0

U4 0 0 0 0 0 0 2.

[ 'l

^{ei e2 U! U2 us U4}

ei 0 e2 U! 0 -us 0 e2 -e2 0 0 U! U2 0

U! -ui 0 0 0 0 U!

U2 0 -U! 0 0 0 u2 us us -U2 0 0 0 us U4 0 0 -u1 -u2 -us 0 3.

[ 'l

^{el e2} U! U2 us U4

e1 0 e2 U! 0 -us 0 e2 -e2 0 0 U! U2 0

U! -U! 0 0 0 0 0

U2 0 -U! 0 0 0 0

us us -u2 0 0 0 0

U4 0 0 0 0 0 0 2.5

g= { G _~

_-x ^{-y 0 0}

1.

[

,]

ei e2 U! U2 us U4

ei 0 0 0 U! -U4 -2e1

e2 0 0 0 -2e2 -u2 U!

U! 0 0 0 2e2-ul u2+u4 2ei-ui

U2 -U! 2e2 u 1 -2e2 0 -2us U2-U4 us U4 U2 -u2-u4 2us 0 2us

U4 2el -u1 u 1 -2el ^U4-U2 -2us 0

2.

[

' ]

^{el ez Ul Uz U3 U4}

el 0 0 0 Ul -u4 0 ez 0 0 0 -2ez -uz Ul Ul 0 0 0 -Ul U4 0 uz -ul 2ez Ul 0 -2u3 -u4 U3 U4 Uz -U4 2u3 0 0

U4 0 -ul 0 U4 0 0

3.

[

'

] ^{el e2 Ul Uz U3 U4}

el 0 0 0 Ul -U4 0

ez 0 0 0 0 -uz Ul

Ul 0 0 0 0 Ul 0

Uz -Ul 0 0 0 e1 +gez+(l-h)uz hu1

U3 U4 u2 -u1 -el-gez+(h-l)uz 0 -(g+h)el +kez-(1 +h)u4

U4 0 -ul 0 -hu1 (g+h)el-kez+(l+h)u4 0

Re h

>

⁰ or Re h

=

^{0, Im h ~ 0} (if k =/= 0), hE C (if k

=

⁰⁾

4.

[ ' ] ^{el ez Ul Uz U3 U4}

el 0 0 0 Ul -U4 0

ez 0 0 0 0 -uz Ul

Ul 0 0 0 0 Ul 0

Uz -Ul 0 0 0 gez+(l-h)uz hu1

U3 U4 Uz - u1 -gez+(h-l)uz 0 -(g+h)el-(l+h)u4

U4 0 -ul 0 -hu1 (g+h)e1 +(l+h)u4 0

Re h

>

0 or Re h

=

0, Im h ~ 0 5.

[,] e1 e2 Ul Uz U3 U4

el 0 0 0 U! -u4 0

ez 0 0 0 0 -uz Ul

Ul 0 0 0 0 0 0

Uz -ul 0 0 0 e1 +gez-Uz Ul

U3 U4 Uz 0 -e1-ge2+uz 0 -ge1 +kez-u4

U4 0 -Ul 0 -Ul ge1-kez+u4 0

6.

[,] el e2 Ul Uz U3 U4

el 0 0 0 Ul -U4 0

ez 0 0 0 0 -uz Ul

Ul 0 0 0 0 0 0

Uz -Ul 0 0 0 gez-Uz Ul

U3 U4 Uz 0 -gez+uz 0 -ge1-u4

u4 0 -Ul 0 -Ul ge1 +u4 0

7.

[ ' J ^{el e2 UI U2 U3 U4}

el 0 0 0 UI -u4 0

e2 0 0 0 0 -u2 ul

ul 0 0 0 0 0 0

u2 -ul 0 0 0 e1 +e2 0

U3 U4 U2 0 -el-e2 0 -e1 +ke2 U4 0 -ul 0 0 e1 -ke2 0 8. [,] el e2 Ul u2 U3 U4

el 0 0 0 UI -u4 0 e2 0 0 0 0 -U2 UI ul 0 0 0 0 0 0

U2 -ul 0 0 0 e2 0

U3 U4 U2 0 -e2 0 -el U4 0 -ul 0 0 el 0 9.

[ ' J ^{el e2 UI U2 U3 U4} el 0 0 0 UI -u4 0 e2 0 0 0 0 -u2 u1 ul 0 0 0 0 0 0

u2 -ul 0 0 0 el 0

U3 U4 U2 0 -el 0 e2 U4 0 -ul 0 0 -e2 0 10.

[ ' J ^{el e2 UI U2 U3 U4} el 0 0 0 ul -u4 0 e2 0 0 0 0 -u2 u1 ul 0 0 0

0

^{0 0}

U2 -ul 0 0 0 e1 0

U3 U4 U2 0 -el 0 0

U4 0 -ul 0 0 0 0 11.

[ ' J ^{el e2} UI U2 U3 U4 el 0 0 0 u1 -u4 0 e2 0 0 0 0 -U2 U1 Ul 0 0 0 0 0 0

U2 -ul 0 0 0 0 0

U3 U4 U2 0 0 0 0

U4 0 -ul 0 0 0 0

3.1

g={ 0

^{z y 0} ₀ ^-x _-z ^{0 0}

_jJ ^x,y,zEC}

1. [,] _el ez es u 1 Uz Us U4 el 0 0 es u1 0 -us 0 ez 0 0 -es 0 Uz 0 -u4 es -es es 0 0 u1 -u4 0 U1 -ul 0 0 0 0 0 0 Uz 0 -Uz -Ul 0 0 0 0 us us 0 U4 0 0 0 0 U4 0 U4 0 0 0 0 0 3.2

g= { 0 ^AX

^{0 0 y -x -y -z 0}

jJ ^x,y,zEC},

A. EC, ReA.> 0 or ReA.= 0, ImA.?: 0 A.=O

1.

[ ' l

^{e1 ez es Ul Uz us U4}

e1 0 ez es U1 0 -us 0

ez -ez 0 0 0 U1 -U4 -2ez

es -es 0 0 0 -2es -uz Ul

ul -ul 0 0 0 2es-ul uz+u4 2ez-UI Uz 0 -ul 2es u1-2es 0 -2us Uz-U4 us us U4 uz -Uz-U4 2us 0 2us U4 0 2ez -u1 u1 - 2ez U4-U2 -2us 0 2.

[ ' l

^{el ez es Ul uz us U4}

el 0 ez es Ul 0 -us 0 ez -ez 0 0 0 Ul -U4 -2ez es -es 0 0 0 0 -uz ul Ul -ul 0 0 0 0 Uz -ul Uz 0 -Ul 0 0 0 0 Uz us us U4 Uz -uz 0 0 2us U4 0 2ez -u1 Ul -uz -2us 0

.A=l 3.

[,] el e2 e3 Ul U2 U3 U4

el 0 0 2e3 Ul U2 -U3 -U4 e2 0 0 0 0 Ul ^-U4 0

e3 -2e3 0 0 0 0 -U2 U1 ul -ul 0 0 0 0 0 0

u2 -U2 -ul 0 0 0 e2 0

U3 U3 u4 U2 0 -e2 0 0

U4 U4 0 -ul 0 0 0 0 Re.A>OorRe.A=O, Im.A~O

4.

[,] el ^{e2 e3} Ul ^{U2 U3 U4}

el 0 (l-.A)e2 (l+.A)e3 u1 AU2 -u3 -AU4 e2 (.A -l)e2 0 0 0 U1 ^-U4 0

e3 -(l+.A)e3 0 0 0 0 -u2 ul

ul -ul 0 0 0 0 0 0

U2 -.Au2 -ul 0 0 0 0 0

U3 U3 U4 U2 0 0 0 0

U4 AU4 0 -Ul 0 0 0 0

3.3

g={ 0

^{X y 0 0 -z -y 0 0}

JJ ^x,y,zEC}

1.

[ ' ] ^{el e2 e3 ul U2 U3 U4}

el 0 -e2 e3 0 U2 0 ^-U4

e2 e2 0 0 0 Ul -U4 0

e3 -e3 0 0 0 0 -U2 U1

Ul 0 0 0 0 0 ul 0

U2 -u2 -u1 0 0 0 pe3+u2 0

U3 0 U4 U2 -u1 ^-pe3-u2 0 ^-pe2-u4

U4 U4 0 -Ul 0 0 pe2+u4 0

2.

[

' ]

^{el e2 e3 U1 U2 U3 U4} el 0 -e2 e3 0 U2 0 -U4 e2 e2 0 0 0 Ul ^-U4 0

e3 -e3 0 0 0 0 -u2 Ul Ul 0 0 0 0 0 0 0

U2 -u2 -u1 0 0 0 e3 0

U3 0 U4 U2 0 -e3 0 -e2 U4 U4 0 -ul 0 0 e2 0

3. [,] el e2 e3 Ul U2 U3 U4 el 0 -e2 e3 0 ^u2 0 -u4 e2 e2 0 0 0 ul -u4 0

e3 ^-e3 0 0 0 0 -U2 Ul Ul 0 0 0 0 0 0 0

U2 -u2 - u l 0 0 0 0 0

U3 0 U4 U2 0 0 0 0

U4 U4 0 ^-Ul 0 0 0 0 3.4

g={ 0 ^-x

^{y 0} ₀

^-x

_-y ^{0 0}

_~z) ^x,y,zEC}

1.

[ ' l

^{e1 e2 e3 U1 U2 U3 U4}

el 0 2e2 -2e3 u1 -U2 -U3 U4 e2 -2e2 0 el 0 ul -U4 0

e3 2e3 -e1 0 U2 0 0 -U3 Ul - u l 0 ^-U2 0 0 0 0

u2 U2 - u l 0 0 0 0 0

U3 U3 U4 0 0 0 0 0

U4 -U4 0 U3 0 0 0 0 3.5

g={ ei ^-z

^{y 0} ₀ ^{-2y -2x 0} ₀

_D ^{x,y, z}

^E

^C}

1.

[ ' l

^{el e2 e3 Ul U2 U3 U4}

el 0 2e2 -2e3 2ul 0 ^-2u3 0

e2 -2e2 0 el 0 Ul -2u2 0

e3 2e3 - e l 0 ^{2u2 -U3} 0 0

Ul -2u1 0 ^-2u2 0 0 0 Ul U2 0 ^{- u l} U3 0 0 0 U2 U3 2u3 2u2 0 0 0 0 U3 U4 0 0 0 - u l -u2 -U3 0

2.

[

' ]

^{e1 e2 e3 ul U2 U3 U4}

el 0 2e2 -2e3 2ul 0 ^-2u3 0 e2 -2e2 0 el 0 U! -2u2 0 e3 2e3 -el 0 2u2 -U3 0 0

U! -2ul 0 -2u2 0 e2 el 0 U2 0 -ul U3 -e2 0 e3 0

U3 2u3 2u2 0 -el -e3 0 0

U4 0 0 0 0 0 0 0

3.

[ '] ^{e1 e2 e3 U! U2 U3 U4}

el 0 2e2 -2e3 2ul 0 -2u3 0 e2 -2e2 0 el 0 Ul -2u2 0 e3 2e3 -el 0 2u2 -U3 0 0 ul -2ul 0 -2u2 0 0 0 0

U2 0 -ul U3 0 0 0 0

U3 2u3 2u2 0 0 0 0 0

U4 0 0 0 0 0 0 0

4.1

g={ G

^{z y 0 -x - t 0}

_jJ ^x,

^y,

^z,

^{t E}

^IC}

0 -z

1. [

,]

el e2 e3 e4 ul u2 U3 U4

e1 0 0 e3 e4 U! 0 ^-U3 0 e2 0 0 -e3 e4 0 U2 0 -u4

e3 -e3 e3 0 0 0 u 1 - u4 0 e4 -e4 -e4 0 0 0 0 -u2 ul ul -ul 0 0 0 0 0 0 0 U2 0 -u2 -ul 0 0 0 0 0

U3 U3 0 U4 U2 0 0 0 0

U4 0 U4 0 -ul 0 0 0 0

4.2

g= {

(~

^{y 0}

^{t -x}

^{0 0}

~z) ^x,y,z,t

^E

^<C}

0 -y

-t

1.

[' l

^{el e2 es e4 U! U2 us U4}

el 0 0 2es -2e4 U! -u2 -us U4

e2 0 0 0 0 U! U2 -us -U4

es -2es 0 0 el 0 U! -U4 0

e4 2e4 0 -el 0 U2 0 0 -us

U! -U! -U! 0 -u2 0 0 e1

+

3e2 2es

U2 U2 -U2 -U! 0 0 0 2e4 -e1

+

3e2

us us us U4 0 -e1- 3e2 -2e4 0 0

U4 -u4 U4 0 us -2es e1- 3e2 0 0

2.

[

,]

el e2 es e4 U! U2 us U4

el 0 0 2es -2e4 u1 -u2 -us U4

e2 0 0 0 0 U! U2 -Us -U4

es -2es 0 0 el 0 U! -U4 0 e4 2e4 0 -el 0 U2 0 0 -us U! -U! -U! 0 -u2 0 0 0 0

U2 U2 - u2 -u1 0 0 0 0 0

us us us U4 0 0 0 0 0

U4 -U4 U4 0 us 0 0 0 0 4.3

g= {

(~ ^{-x -t}

^{y 0} 0

^-x

-y ⁰

~z) ^{x,y,z, t}

^E

^<C}

1.

[

,]

el e2 es e4 U! U2 us U4

el 0 2e2 -2es 0 U! -U2 -Us U4

e2 -2e2

0

el 0 0 U! -U4 0 es 2es -el 0 0 U2 0 0 -us

e4 0 0 0 0 0 0 -u2 U!

U! -ul 0 -U2 0 0 0 0 0

U2 U2 -U! 0 0 0 0 0 0

us us U4 0 U2 0 0 0 e4

U4 -U4 0 us -U! 0 0 -e4 0