MATHEMATICS

A. L. ONISHCHIK

ON COMPACT LIE GROUPS TRANSITIVE ON CERTAIN MANIFOLDS

(Presented by Academician P. S. Aleksandrov, 17 VI 1960)

In the works \((^{1,2})\), Montgomery, Samelson, and Borel determined all connected compact Lie groups acting transitively on spheres. In the present work the problem of finding all connected compact Lie groups transitive on a given manifold is solved for certain other classes of homogeneous manifolds*. At the same time, a general topological method is given for approaching this problem. In particular, a new proof of the results of Montgomery—Samelson—Borel is obtained.

1. We shall denote by \(H(X)\) the cohomology algebra of the space \(X\) with real coefficients, and by \(f^*\) the homomorphism of cohomology algebras associated with the continuous map \(f\). The Poincaré polynomial of a graded space \(X\) or of a topological space \(X\) will be denoted by \(P(X,t)\).

Let \(\mathfrak G\) be a connected compact Lie group, \(\mathfrak U\) its closed subgroup, \(\mathfrak U_0\) the connected component of the identity in the group \(\mathfrak U\), and \(i:\mathfrak U_0\to\mathfrak G\) the embedding. Denote by \(P\) and \(Q\) the spaces of primitive elements of the algebras \(H(\mathfrak G)\) and \(H(\mathfrak U_0)\). Then \(i^*(P)\subset Q\). Hence there exist graded spaces \(P_1, P_2\subset P\) and \(Q_1,Q_2\subset Q\) such that \(P=P_1\oplus P_2\), \(Q=Q_1\oplus Q_2\), \(i^*P_1=0\), and \(i^*\) maps \(P_2\) isomorphically onto \(Q_2\). The basic topological fact used in the present work is that the graded spaces \(P_1\) and \(Q_1\) are topological invariants of the manifold \(X=\mathfrak G/\mathfrak U\), i.e. they do not depend on the choice of a compact group \(\mathfrak G\) transitive on this manifold.

This fact follows from the following theorem, whose proof is based on the results of \((^{3,4})\):

Theorem 1. The graded spaces \(P_1, Q_1\), and also the rational fraction
\[ \frac{P(\mathfrak G,t)}{P(\mathfrak U_0,t)} \]
are topological invariants of the manifold \(X=\mathfrak G/\mathfrak U\). More precisely, if \(r_k\) is the rank of the homotopy group \(\pi_k(X)\), then
\[ P(P_1,t)=\sum_{k=1}^{\infty} r_{2k-1}t^{2k-1}, \qquad P(Q_1,t)=\sum_{k=1}^{\infty} r_{2k}t^{2k-1}, \]
\[ \frac{P(\mathfrak G,t)}{P(\mathfrak U_0,t)} = \prod_{k=1}^{\infty}\left(1+t^{2k-1}\right)^{r_{2k-1}-r_{2k}}. \]

The invariance of the fraction
\[ \frac{P(\mathfrak G,t)}{P(\mathfrak U_0,t)} \]
was already noted in \((^5)\). In the case where \(\mathfrak U\) is a connected subgroup of maximal rank in \(\mathfrak G\), it follows from Hirzebruch’s formula.

Let \(X\) be some topological space. We shall denote by \(r(X)\) and call the rank of the space \(X\) the sum of the ranks of all groups \(\pi_{2k-1}(X)\) \((k=1,2,\ldots)\), if this sum is finite. If \(X=\mathfrak G/\mathfrak U\)—

* All transformation groups are assumed to be effective.

homogeneous space of the compact group \(\mathfrak G\), then from Theorem 1 it follows that \(r(X)=\dim P_1\). In particular, \(r(\mathfrak G)\) is equal to the ordinary rank of the group \(\mathfrak G\).

2. In this section we study homogeneous spaces of compact Lie groups having rank 1, and find all compact Lie groups transitive on these manifolds.

Theorem 2. Let a connected compact Lie group \(\mathfrak G\) act transitively on a manifold \(X\) of rank \(r\). Then there exists a normal divisor \(\mathfrak G' \subset \mathfrak G\), locally isomorphic to a direct product of no more than \(r\) simple groups and transitive on \(X\). In particular, if \(r=1\), then \(\mathfrak G'\) is a simple normal divisor.

Furthermore, if \(\mathfrak G''\) is a normal divisor of the group \(\mathfrak G\), complementary to \(\mathfrak G'\), then \(r(\mathfrak G'') \le r\).

From Theorem 2 it is clear that, in order to find all homogeneous spaces of rank 1, it suffices to consider homogeneous spaces of simple compact Lie groups \(\mathfrak G\). If \(\mathfrak G\) is commutative, then \(X=S^1\), the circle. If \(\mathfrak G\) is simple and noncommutative, then locally isomorphic homogeneous spaces of the group \(\mathfrak G\) have equal ranks. Therefore one must enumerate all pairs \((G,U)\), where \(G\) is a simple noncommutative compact Lie algebra and \(U\) is its subalgebra, that give homogeneous spaces of rank 1.

Theorem 3. All pairs \((G,U)\) that give homogeneous spaces of rank 1 are listed in Table 1, in which the simply connected homogeneous spaces \(X\) corresponding to these pairs are also indicated. Two spaces \(X\) are homeomorphic if and only if they are denoted in the same way.

Table 1

In Table 1, \(i\) denotes the embedding \(U \to G\), and in the corresponding column (for the classical algebras \(G\)) the linear representation of the algebra \(U\) realizing this embedding is indicated; here the notation of [5] is used. If the algebra \(U\) is not simple, then the linear representation of its simple ideal having the greatest rank is indicated. By \(A_1^k\) is denoted a subalgebra of type \(A_1\) and index \(k\) in \(G_2\), and by \(T\) a one-dimensional Lie algebra. In po-

In the last column the following notation is used: \(S^n\) is the \(n\)-dimensional sphere; \(PC^n\), \(PK^n\), \(PO^n\) are, respectively, the complex, quaternionic, and octonionic projective spaces of dimension \(n\); \(V_{n,2}\) is the Stiefel manifold of orthonormal 2-frames in \(n\)-dimensional Euclidean space \(R^n\); \(G_{n,2}\) is the Grassmann manifold of oriented planes in \(R^n\).

Let \(\mathfrak G_1\) and \(\mathfrak G_2\) be two groups of transformations of a manifold \(X\). We say that \(\mathfrak G_1\) and \(\mathfrak G_2\) are similar if there exists a homeomorphism \(A\) of the manifold \(X\) onto itself such that \(A\mathfrak G_1A^{-1}=\mathfrak G_2\).

Corollary of Theorem 3. a) Every connected compact Lie group transitive on \(S^n\) is similar to the group \(SO(n+1)\) or to one of the following subgroups of it: \(SU(m)\), \(U(m)\) \((n=2m-1)\); \(Sp(2m)\), \(Sp(2m)\times U_1\), \(Sp(2m)\times Sp(2)\) \((n=4m-1)\); \(Spin(9)\) \((n=15)\); \(Spin(7)\) \((n=7)\); \(G_2\) \((n=6)\).

b) Every connected compact Lie group transitive on \(PC^n\) is similar to the group \(SU(n+1)\) or (for \(n=2m-1\)) to its subgroup \(Sp(2m)\).

c) Every connected compact Lie group transitive on \(G_{2n+1,2}\), or on the manifold of unoriented planes \(\widetilde G_{2n+1,2}\), is similar to the group \(SO(2n+1)\) or (for \(n=3\)) to its subgroup \(G_2\).

d) If \(n\ne3\), then every connected compact Lie group transitive on \(V_{2n+1,2}\), or on the manifold \(\widetilde V_{2n+1,2}\) obtained from \(V_{2n+1,2}\) by identifying the frames \(e_1,e_2\) and \(-e_1,-e_2\), is similar to the group \(SO(2n+1)\) or \(SO(2n+1)\times SO(2)\). Every connected compact Lie group transitive on \(V_{7,2}\) or \(\widetilde V_{7,2}\) is similar to one of the following groups: \(SO(7)\), \(SO(7)\times SO(2)\), \(G_2\subset SO(7)\), \(G_2\times SU(2)\), \(G_2\times U(1)\).

e) Every connected compact Lie group transitive on \(G_2/A_1^3\) is similar to the group \(G_2\times SU(2)\) or to one of its subgroups \(G_2\), \(G_2\times U(1)\).

f) Every connected compact Lie group transitive on \(PK^n\), \(SU(3)/SO(3)\), \(B_2/A_1^{10}\), \(PO^2\), \(G_2/A_1^3\times T\), \(G_2/A_1^4\), \(G_2/A_1^{28}\), \(PO^1\), is similar to the groups \(Sp(2n+2)\), \(SU(3)\), \(SO(5)\), \(F_4\), \(G_2\), respectively.

Let us also note the following property of manifolds of rank 1.

Theorem 4. Let \(X=\mathfrak G/\mathfrak U\), where \(\mathfrak G\) is a connected compact Lie group and \(\mathfrak U\) is its connected closed subgroup. Then \(r(X)=1\) if and only if \(H(X)\) is an algebra with one generator.

1. Using the fact that the Poincaré polynomials of simple compact Lie groups are known, from Theorem 1 one can obtain some general results about simple transitive transformation groups.

Theorem 5. Let \(X=\mathfrak G/\mathfrak U\), where \(\mathfrak G\) is a simple connected compact Lie group and \(\mathfrak U\) is its closed subgroup, and let \(\mathfrak G'\) be a simple compact Lie group transitive on \(X\). Denote by \(G,U,G'\) the Lie algebras of the groups \(\mathfrak G,\mathfrak U,\mathfrak G'\). If \(G\) is the algebra \(A_n\) or an exceptional algebra, then, as a rule, \(G'\cong G\). If \(G\) is one of the algebras \(B_n,C_n,D_{n+1}\), then, as a rule, \(G'\) is also one of these algebras. The only exceptions to this rule are the following cases: the pair \((G,U)\) defines one of the rank-1 manifolds listed in parts a)—d) of the corollary to Theorem 3; \(G=A_{2n}\), \(U=C_{n-1}\); \(G=A_{2n-1}\), \(U=C_n\). In the last two cases we have \(G'\cong G\) or \(G'=A_{2n+1}\), \(G'=A_{2n-2}\), respectively.

Corollary. Let \(\mathfrak G\) be a simple compact noncommutative Lie group, and let \(\mathfrak T\) be its maximal torus. Then every simple connected compact Lie group transitive on \(\mathfrak G\) or on \(\mathfrak G/\mathfrak T\) is similar to the group \(\mathfrak G\) (we assume that \(\mathfrak G\) acts on itself by left translations).

* The assertion concerning spheres is a result of Montgomery—Samelson—Borel. The question of groups transitive on a simply connected manifold having the same integral cohomology as the nonhomogeneous sphere was considered in paper \((^6)\), where the corresponding special case of Theorem 2 was proved.

1. Let \(X\) be some compact manifold. Theorem 5 shows that, for us, the question of when all compact groups transitive on \(X\) are simple is an important one. Consider the case when \(X\) is a simply connected manifold with positive Euler characteristic. Then \(X\) has the property indicated above if and only if it is not a direct product of two homogeneous spaces. We shall say that the manifold \(X\) is indecomposable if it is not a direct product of two manifolds of positive dimension. It can be proved, for example, that a simply connected orientable compact manifold of rank 1 is always indecomposable. We now give one more sufficient condition for the indecomposability of a manifold.

A manifold \(X\) of dimension \(2km\) is called \(k\)-symplectic if there exists an element \(\omega \in H^{2k}(X)\) such that \(\omega^m \ne 0\).

Lemma. Let \(X\) be a \(k\)-symplectic manifold and let \(H^i(X)=0\) \((0<i<2k)\), \(\dim H^{2k}(X)=1\). Then \(X\) is indecomposable.

1-symplectic manifolds are called symplectic. In [7] it is proved that a homogeneous space of a semisimple compact Lie group is symplectic if and only if it is Kähler. Therefore, from the lemma and from Theorem 5 the following theorem follows.

Theorem 6. Let \(X=\mathfrak{G}/\mathfrak{U}\), where \(\mathfrak{G}\) is a simple compact Lie group. Denote by \(G\) and \(U\) the Lie algebras of the groups \(\mathfrak{G}\) and \(\mathfrak{U}\). If \(G\) is an algebra of type \(A_n\) or an exceptional algebra, and \(U\) is the centralizer of its center in \(G\), and if the center of the algebra \(U\) is one-dimensional, then every connected compact Lie group transitive on \(X\) is simple and, as a rule, is similar to the group \(\mathfrak{G}\). The only exceptions to this rule are the manifolds \(X=PC^{2m-1}\), \(G_{7,2}\), \(\widetilde{G}_{7,2}\).

In particular, if \(X=C_{n,k}\) is the Grassmann manifold of \(k\)-dimensional subspaces of an \(n\)-dimensional complex space, then for \(k>1\) every connected compact Lie group transitive on \(X\) is similar to the group \(SU(n)\).

Moscow State University
named after M. V. Lomonosov

Received
17 VI 1960

CITED LITERATURE

1. D. Montgomery, H. Samelson, Ann. Math., 44, 454 (1943).
2. A. Borel, C. R., 230, 1378 (1950).
3. H. Cartan, J. P. Serre, C. R., 234, 393 (1952).
4. A. L. Onishchik, Matem. sborn., 44, 3 (1958).
5. A. L. Onishchik, DAN, 129, 261 (1959).
6. Y. Matsushima, Nagoya Math. J., 2, 1 (1951).
7. A. Borel, Proc. Nat. Acad. Sci., 40, 1147 (1954).