Article信州大学理学部紀要 12(1): 45-52(1977)departmental bulletin pape

YOKOTA,  ICHIRO

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JOUR. FAC. SCI,, SHINSHU UNIVERSITY, Vol. 12, No.1 June, 1977Non-compact Sinple Lie Group G2' of Type G2                      By ICHIRO YOKOTA  Department of Mathematics, Faculty of Scjence Shinshu University                  (Received March 15, 1977)   It is known that there exist two simple Lie group of type G2 up to localisomorphism, one of them is compact and the other is non-compact. The compactsimple Lie group G2= G2(-i4) of type G2 is obtained as the automorphism groupAut(G) of the Cayley algebra E and it is a connected, simply connected, simple(in the sense of the center 2(G2) =:1) Lie group. It is also known that the Liealgebra g2' of the non-compact simple Lie group G2(2) of type G2 is obtained asthe derivation Lie algebra of the split Cayley algebra G' and the maximal compactsubgroup of G2(2) has the 'type of AteAi. In this paper, we investigate someglobal properties of the automorphism group G2'= Aut(E') of the split Cayleyalgebra G'. The results are as follows. The group G2' is homeomorphic to SO(4)× R8 and a simple (in the sense of the center z(G2') = 1) Lie group. And hence thecenter x(a2') of the non-compact simply connected simpie Lie group a'2== G2(2) oftype G2 is Z2.   1. Split Cayley algebra G'   Let as' be the split Cayley algebra over the real numbers R. This algebra E'is defined as follows. Let H={a==ai+a2i+a3j'+a4lelaiGR} be the field ofquaternions, In E' == HO He, if we define a multiplication by                    (a + be) (c + de) == (ac + db) + (bc + du)ethen E' becomes an 8-dim. (non-commutative non-associative) algebra over R withthe conjugation a + be = a - be.   The Q-norm Q(x) and the inner product (x, Y)' in as' are definedrespectively by                      Q(a + be) = [a12 - Ib]2                    (a + be, c+ de)' = (a, c) - (b, d)The Q-norm has the following properties.                Q(x) =: xx = xx, Q(xY) == Q(x)Q(Y), x, Y E E'46'･ ' ･ ICHIRo YOKoTA   Remark. In the usual Cayley algebra 6= ff+ ffe, the multiplication is defined                                              tt                      '               (a + be) (c + de) = (ac - db) + (bc + dti)e   2. Definition of group G2'   We denote the automorphism group Aut(e') .of the split Cayley algebra G' byG2', i. e. '               G2' == {cr E IsoR(Q', G')1at(xY) = a(x)a(Y)}tt. ..                        tt     'Obviously a(1)i.=1 holds for aE!G2'. We shall show that the group G2' is contained                                                              '        t ttt     tt   '                                                  'in the Lorentz group O'(E'), where , . , .. ,                                                       ''                                                                  ' ' ' ' O(4, 4)=O'(G')={aEIso.(ng', ts')IQ(ct(x)) -= Q(x)} -                         == {a E IsoR(G', G')l(ev(x), a(y))t = (x, y),}   Lemma 1. For atGG2' anaxGG', we have ' '',,, (1) if xx = -1 then Q(a (x)) =1 and cr (x) = -a(x) .   (2) if xx = 1, x Qt R ,then.Q(a(x)) = -1 and ec(x) = -cr(x) .,   Proof (15 From xx=-1 we have a(x)a(x) = -1, so (Q(a(x)))2=1, 'i.e.' Q(a(x))=±L Assume Q(a(x)) == -1, then .'' .            ev(x) = -Q(ev(x))ev(x) = -(a(x)ev(x))ev(x) = -cr(x) (a(x)cv(x)) ,               = -cr(x)(-1) ･= ev(x)(AIthough 6' is not associative, (x--x)Y=I(xY) is always valid). This means'  ev(x) isa real number a, therefore a(x) ==:a==a(a). Since,a is injective, x=a is real andxx = -1. This is a contradiction. Hence Q(cr(x)) == 1' tind we get iEI(-x) = -cr(x) by acalculation similar tolthe above. (2) is analogously obtained. ' ' '' '                                                           /tt t   Proposition 2. The grouP Ga' is f subgromp of the Loren(.z gromp o'(as').                            '.i..P.r.OtO,f..tgStaiG,gtkfyCthhO.OS,e..adibti.S.iS, .ii tig,{llink'ie,' ie･ 7'e･ lee in Q'･ then these                         tt t. t /. ti.J .11 t. / t .t               i2 =j2 ==: k2 = -1, e2 = (ie)2 == (1'e)2 ::= (ke)2=1                                              t tt/ t                                  ttt/ t tlTherefore, for these elements x we have a(x) = -av(x) = cr(Hx). And since a(1) = 1=ev(1), we get ev(x) =a(I) for ahy element',x of s'. Now ･ ･(. 1･                  Non-compact Simple Lie Group G2t of Type G2 ･47         Q(ev(x)) == cr(x)a(x) = av(I)at(x) = a('Ix) == cr(Q(x)) :-Q(x), x a G'                                                        '                       ･ ･,. 1 .･                                     'This means evEO'(G'). '                                   tt ttt                       '   3. Compact subgroup (G2')fy of G2' .   We shall consider the subgroup (G2')K of G2":                    (G2')K == {a ciE G2'lev(ff) == ff}Since ff and iie are orthogonal with respect to the inner product (x, y)', aG (Gi)Kalso satisfies cr(,Ne) ::ffe. ' / ' ･ '.,1' ' ' '   To determine this group (G2')K, we shall give some remarks on the rotationgroup SO(n) of lower dimensions.                                  '          tt tt'            '                SO(4) = {aGIso.(ff, ll)1[a(x)l == Ixl, deta:=1}                SO(3) - {a ESO(4)la(1) = 1}                     = {a E Iso.(II, ff)lo(xY) == a(x)a(Y)} == Aut(II)      . S3 == {PG"I,IPI=1}                   '                       'ln the topological product of S3 and SO(3), if we give a multiplication by ' (q, T)(P, o) == (qr(P), ra)then S3 × SO(3) becomes a group. We denote this group by S3･SO(3).                                  '   Lemrna 3. The rotation grouP SO(4) is isomorPhic to the grouP S3･SO(3).    Proof Define a mapping f:S3･SO(3)-SO(4) by                           f(P, a) = Lpa                                                  ''                                            'fWhiseraenft.sPoEmoSrOpt4i)smiS. defined bY Lp(x) =Px, xG HL Then it is easy to verify that                                                           '                              ttt                                                                       '    Proposition 4. The grouP (G2')K is isonzorPhic to the rotation grouP SO(4).                                                 '    Proof Define a mapping 9 : S3･SO (3) . (G2')K bY '                    9(P, a) (a + be) =a(a) + (Pa (b))e, a, bG ,[I '                                                                      'Firsit we must show P(P, a) G (G2')K. However it is easy ; in fact, 9(P, a)(R) = ll             (p(P, o) (a + be))(g(P, 'a)(c /l+ de)) 48 ICHIRo YoKoTA               = (o(a) + (Pa(b)>e>(.a(c) + (Po(d))e)               = (a(a)a(c) + Po(d)Pa(b)) + (Pa(b)a(c) + Pa(d)a(a))e               = o(ac + db) + (Pa(bc + da))e               = 9(P, a)((ac + db) + (bc + dtz)e) == 9(P, a)((a + bc)(c + de)) 9 is a homomorphism because             ep(q, T)(P(P, a)(a + be)) =- 9(q, T)(a(a) + (Po(b))e)               = ro(a) + (qr(Pa(b))e = w(a) + (qT(P)ra(b))e               :=:go(qT(P), ra)(a + be) = 9((P, T)(P, a))(a + be) Next we shall show that 9 is onto. For a given a Ei (G2')K, consider the restriction･ a=evIff of a to ff, then ctEAut(ll) =SO(3). Put P= cr9(1, 6)-t, then Plff=1. Set P(e) =:Pe, PG ff, then IPI =1 and          P(a + be) =a+ bP(e) == a+b(Pe) =a+ (Pb)e == op(P, 1)(a + be) i.e. P=P(P, 1). Hence a=P9(1, a) == 9(P, 1)9(1, o) == P(P, a), so that 9 is onto. Obviously 9 is one-to-one. Thus the proof is completed.    Remark The compact Lie group G2=:Aut(as) also contains a subgroup (G2)K which is isomorphic to SO(4) by a mapping 9 : S3･SO(3) - G2,                     9(P, o)(a + be) =a(a) + (Pa(b))e    4. Polar decornpsition of G2' ･    To give a polar decorriposition of G2' we use the following    Lemma 5 ([1] P. 345). Let G be a real algebraic subgrouP of the general linear grouP GL(n, R) such that the condition A E G imPlies tAe G. Then G is homeomor- Phic to the toPological Product of Gno(n) (which is a maximalcompact subgroup･ of G) and a Euclidean space Rd :                     G= (G nO(n)) × Rd, d = dim (g n lj (n)) where O(n) is the orthogonal subgrouP of GL(n, R), g the Lie algebra of G and b(n) the vector sPace of all reat symmetric matrices of degree n.    To use the above lemma, we define a positive definite inner product (x, Y) in 6' by                     (a + be, c + de) =,(a, c) + (b, d)                  Non-compact Simple Lie Grottp CLY of Type C2 49Two inner products (x, Y), (x, Y)' in G' are combihed with the following relations                    (x, N) :== (x, r(y))', (x, y)' =:= (x, r(y))whcre r=9(-1, 1). We denote by tav the transpose of cr Ei IsoR(g', 6') with respectto (x, Y):(av(x), y)=(x, tcr(y)). . .                                             '   LeTvtma 6. G2' is a real aigebraic sttbgrouP of GL(8, R) ==IsoR(G', E') andsatis.lies the condi.tion av E G2' imPlies ta Ei G2'.   Proof Since (x, r(Y)) == (x, Y)' =(ct(x), ec(Y))' = (ev(x), rcr(y)) = (x, tev7tct(y)) for cr eiG2', we have r= tara. Hence ta == ra-ir E G2'. It is trivial that G2' is real algebraicbecause it is defined by the algebraic relation ev(xy)=ev(x)a(y). ', Let o(Q') be the orthogonal subgroup of IsoR(S', 6'):             O(s) = O(E') = {ev Ei Iso.(G', S')](a(x), ev(Y)) = (x, Y)}                                                                     'Then evEC2'nO(E') induces a linear transformation of ZiL In fact, (a(a), u)=-(a(a), u)t == -(a, ct-1(u))l = -                       (a, ec-i(u)) = -(cr(a), u), for a cl ff, uE ffe,' therefore(a(a), u)=o and av(a)Gff Hence we have . / .                       G2t n･O(St) = (G2')K :il SO(4) '                                             'by Proposition 4. Next we shall determine the Euclidian part g2'nb(G') of G2',where              92' = {9 E HoMR(g', 6')I9(XY) = 9(x)Y + X9(Y)}            b(8) = lj(S') = {g G Hom.(6t, 6')1(g(x, Y) =- (x, g(Y))}Since 92' is a Lie subalgebra of the Lie algebra o(4, 4) =D'(6') == {g ci HomR(6', G')l(g(x), y)' + (x,-g(Y))' = O}, g E g2' nb(G') induces a Iinear transformation 9iff : ,ff --> Ueand gl,ffe =t(glR). Put                    g(i) =Pe, g( 1') = qe, P, qE ffthen g(le) is determined by g(h) = g(i]') = g(i)y' + ig(y')=(Pe) ]' ÷ i(qe) = (-Py' + qi)e andobviously g(1)=O. Conversely two elements P, qGff determine s Qf g2tnb(G').                                        '                                                                     '               . dim (g2' nb(G')) =4+ 4.= 8                 t /t            'Thus we have the following '   Tkeorem 7. The grouP G2' is homeomorPhic to the toPological Product of therotation grouP SO(4) and an s-dim. Eztclidean sPace Rs : '50 ICHIRo YoKoTA                          G2' =t SO(4) × R8in Particular, G2' is a connected (but not simPly connected) Lie grouP.   5. Simgelicity of G,t   Lernma 8. The Lie algebra g2' of G2' is simPle.   Proof The complexification g2'C of the Lie algebra g2' is isomorphic to thecornplexification g2C of the derivation Lie algebra g2 of the Cayley algebra as, becausethe complexification 6'C of 6' is isomorphic to the one asC of 6. As is well knowng2a is simple, so that g2'C is so, hence asi is also simple.   Since G2' is a connected group from Theorern 7 and a sirnple group as Liegroup from Lemma 8, any normal subgroup of G2' is contained in the center z(G2')of G2'. We shall show z(G2') =: 1.   Let aEz(G2'). First we show that a induces a linear transformation of .if:aE(G2')K. In fact, put a(a)=c+de for aGif, then from the commutativitycondition ep (-1, 1)a =:= ev9(-1, 1), we have     c- de =9(-1, 1)(c + de) =9(-1, 1)a(a) = cr9(-1, 1)(a) = ev(a) =c+ deTherefore de =O and a(a) = cE ,ffL Hence there exists an element (P, ff)(iiS3･SO(3)such that a==9(P, ff) by Proposition 4. Furthermore from the commutativity conditioncv9o(q, 1) :: 90(q, 1)ev, crSO(1, a) :=; 90(1, T)ev, we have                   Pq=qP for all qes3                   ar == ra for allTEi SO(3)so that P=:±1, a=L Hence a=9(1, 1) or a=9(-1, 1). We shalt show thatep(-1, 1) is not an element of the center z(G2') using the following P. Let P be alinear transformation of G' satisfying               P(1) - 1, fi (i) - ie, P( 1') = je, P(le) - le,               P(e) ::: e, P(ie) - i, P(de) -f', P(lee) ==: hethen PE G2' and P(-1, 1) does not commute with P because 9(-1, 1)P(i) == 9(-1,1)(ie) -- -ie and P9(-1, 1)(i) =:P(i) == ie. Thus we have the following   Theorem 9. The grouP G2' is simPle (in the aigebraic sense) Lie group.   Since the fundamental group of G2' is Z2 from Theorem 7 and G2' is a simplegroup, we have the following   Theorem 10. The center z(&2') of the non-comPact simply connected Lie group                 Non-compact Simple Lie Group G2t of Type G2 51G2' = G2(2) of tyPe G2 is Z2.   6. Some subgroups of G2' ･･ I. We shall investigate the structures of subgroups of G2' under which someelememts of the basis 1, i, 1' , k, e, ie, ]'e, lee of as' are invariant. ･    (1) The group G2'(i) == {aEG2'la(i) =i} is isomorphic to the group               SU(1, 2) -= {AGM(3, C)]AJA'= f, detA=1}where C is the complex numbers, A' the conjugate transposed matrix of A and1==(g -gt -i) And G2'(i) is homeomorphic to                          G2'(i) = U(2) × R4where U(2) is the unitary group.   (2) The group G2'(e) ==: {aEG2'[a(e) == e} is isomorphic to the group               SU(3, a) == {A EM(3, a)IAA" =E, detA =- 1}where a is the algebra a == {a + bcla, b E R, e2 = 1} with the conjugation a + bt =a - be and A" is the conjugate transposedmatrix of A. And G2'(e) is homeomorphicto                          G2'(e) =i SO(3) × R5   (3) The group Gi(i, 7', k) = {aGG2']cr(i) =i, at(1')= j, a(le) =le} is isomorphicto the group S3.   (4) The group G2'(i, e, ie) = {aEG2'la(i) = i, ct(e) =e, ev(ie) =ie} is isomorphicto the general linear group GL(2, R), hence G2'(i, e, ie) is homeomorphic to Si ×   II. Let Mm,n be the manifold defined by                                           ?Tl fjll-n     Min,n=={(Xt, ''', Xm, Xm+1, ''', Xm+n) ERM+n[ = Xi2- Z] Xj2==1}                                          t=1 j==7n+1Then we have the following homogeneous spaces.               G27G2t(i) ] M3,4, G2'(i)IG2'(i, 1', le) :M2,4,               G2'IG2e(e) [r M4, 3, G2'(e)IG2'(i, e, ie) r M3,352 IC}IIRo YOKOTA[1] s.1962.                         ReferencesHELGAsoN ; DiLffbrential Geometry and Symmetric Spaces, Pure and Applied Math. ,

Non-compact Simple Lie Group G₂' of Type G₂

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Non-compact Simple Lie Group G₂' of Type G₂

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