UDC 519.46

MATHEMATICS

L. V. SABININ

ON THE CLASSIFICATION OF TRISYMMETRIC SPACES

(Presented by Academician I. G. Petrovskii, 12 III 1970)

In the present note all nontrivial trisymmetric spaces ((^1)) with simple compact Lie groups of motions will be indicated.

As is known ((^1)), a homogeneous Riemannian space (V = Q/H) is called trisymmetric if through each of its points (x) there pass three pairwise orthogonal mirrors (W_x^{(1)}, W_x^{(2)}, W_x^{(3)}), and locally (in some neighborhood (\Omega_x) of the point (x))
[
W_x^{(1)} \cap W_x^{(2)} \cap W_x^{(3)} = {x}, \quad
\dim V =
\sum_{\alpha=1}^{3} \dim W_x^{(\alpha)}.
]
If, moreover, there exists (p \in \operatorname{Aut}(Q)), (p^3 = I), such that
[
pW_x^{(1)} = W_x^{(2)}, \quad
pW_x^{(2)} = W_x^{(3)}, \quad
pW_x^{(3)} = W_x^{(1)},
]
then we shall say that (V = Q/H) is supertrisymmetric. A mirror ((^2)) at a point (x \in V = Q/H) is the maximal set of points (y \in V) fixed with respect to an isometric subsymmetry ((^2)) (S_x) (((S_x)^2 = I, S_x(x)=x)). A mirror is a totally geodesic surface in the Riemannian (V = Q/H) ((^1,^2)). If (\Gamma) is the Lie algebra of the group (Q), then we introduce the notation
[
\Gamma \stackrel{\mathrm{def}}{=} \ln(Q);
]
if (L \subseteq \Gamma) is a subalgebra, and (H \subseteq Q) is the subgroup corresponding to it, then we shall use the notation
[
L \stackrel{\mathrm{def}}{=} \ln_Q(H).
]

A notation of the form (Q/H) means that this is the homogeneous space generated by the group (Q) and the subgroup (H), but such that (H) may contain a nontrivial normal divisor of the group (Q). The equality
[
Q'/H' = Q''/H''
]
means an isomorphism (Q' \cong Q'') inducing an isomorphism (H' \cong H''). A notation of the form (Q/H) means that this is the homogeneous space generated by the group (Q) and the subgroup (H), where (H) contains only the trivial normal divisor of the group (Q). Finally, the notation
[
Q'/H' \underset{\varphi}{=} Q''/H''
]
means the existence of a homomorphism (\varphi) such that (Q' \xrightarrow{\varphi} Q''), (H' \xrightarrow{\varphi} H''), and (\ker \varphi) is the maximal normal divisor of the group (Q') belonging to (H').

Definition 1. We shall say that a Lie group (Q) is an invoproduct of the groups (Q_1, Q_2, Q_3), and write
[
Q = Q_1 \boxtimes Q_2 \boxtimes Q_3
]
(or, more briefly,
[
Q = \boxtimes_{\alpha=1}^{3} Q_\alpha
]),
if (Q_\alpha) is the maximal subgroup of fixed elements of an involutive automorphism (S_\alpha) (((S_\alpha)^2=I)) of the group (Q), with
[
S_\lambda S_\mu = S_\nu \quad (\lambda \ne \mu,\ \mu \ne \nu,\ \nu \ne \lambda).
]
If, moreover, there exists (p \in \operatorname{Aut}(Q)) such that
[
pQ_1 = Q_2, \quad pQ_2 = Q_3, \quad pQ_3 = Q_1,
]
then we shall call the invoproduct (\boxtimes_{\alpha=1}^{3} Q_\alpha) a superinvoproduct.

Definition 2. We shall say that a homogeneous space (Q/H) is an invoproduct of homogeneous spaces
[
Q_\alpha^/H_\alpha \quad (\alpha = 1,2,3)
]
and write
[
Q/H = Q_1^/H_1 \boxtimes Q_2^/H_2 \boxtimes Q_3^/H_3
]
(or, briefly, ...).

In short, (Q/H=\prod_{\alpha=1}^{3} Q_\alpha/{}^{}H_\alpha), if (Q=\prod_{\alpha=1}^{3} Q_\alpha), (H=\prod_{\alpha=1}^{3} H_\alpha). In this case (Q_\alpha/{}^{}H_\alpha) will be called the mirrors in (Q/H). We note that (Q_\alpha/{}^{*}H_\alpha) are mirrors in (Q/H) in the sense of ((^{2})).

We now reformulate the definition of a trisymmetric homogeneous space, using the notion of an involutive product.

Definition 3. A homogeneous space (Q/H=\prod_{\alpha=1}^{3} Q_\alpha/{}^{}H_\alpha) is a trisymmetric space if (locally) (\bigcap_{\alpha=1}^{3} Q_\alpha=Q_0\subseteq H), moreover trivial if (Q_0=H), semitrivial if (Q_0=H_\alpha) (for some (\alpha)), and nontrivial in the remaining cases. If, in addition, (\prod_{\alpha=1}^{3} Q_\alpha) and (\prod_{\alpha=1}^{3} H_\alpha) are superinvolutive products with respect to (p\in \operatorname{Aut}(Q)), then we shall say that (Q/H) is supertrisymmetric*. We note that the mirrors in a trisymmetric space are symmetric spaces.

We formulate two auxiliary theorems.

Theorem 1. Let (Q) be simple, and let (Q/H=\prod_{\alpha=1}^{3} Q_\alpha/{}^{}H_\alpha) be a nontrivial trisymmetric space with compact group (H); then
[
Q_\alpha/{}^{}H_\alpha=(\widetilde Q_\alpha\times \widetilde H_\alpha)/(\widetilde Q_\alpha\cap \widetilde H_\alpha)\times \widetilde H_\alpha
=\widetilde Q_\alpha/(\widetilde Q_\alpha\cap H_\alpha),
]
where (\widetilde Q_\alpha\cap H_\alpha\subseteq \bigcap_{\alpha=1}^{3} Q_\alpha=Q_0), and (\widetilde H_\alpha) is a normal divisor in (Q_\alpha).

Theorem 2. Let (Q/H=\prod_{\alpha=1}^{3} Q_\alpha/{}^{}H_\alpha) be a trisymmetric space with compact group (H), and suppose the symmetric space (Q_\alpha^{}/Q_0) (\bigl(Q_0=\bigcap_{\alpha=1}^{3} Q_\alpha\bigr)) is irreducible for some (\alpha); then (Q/H) is trivial or semitrivial.

The problem of classifying trisymmetric spaces
[
Q/H=\prod_{\alpha=1}^{3} Q_\alpha/{}^{}H_\alpha
]
is naturally solved by passing from groups to the corresponding Lie algebras. Then the involutive products (Q=\prod_{\alpha=1}^{3} Q_\alpha) and (H=\prod_{\alpha=1}^{3} H_\alpha) generate involutive sums ((^{3,5})) of Lie algebras
[
\ln_Q(Q)=\ln_Q(Q_1)+\ln_Q(Q_2)+\ln_Q(Q_3),
\qquad
\ln_Q(H)=\ln_Q(H_1)+\ln_Q(H_2)+\ln_Q(H_3),
]
where
[
L_0=\ln_Q\left(\bigcap_{\alpha=1}^{3} Q_\alpha\right)\subseteq \ln_Q(H).
]
Taking Theorems 1 and 2 into account, we also obtain, for a nontrivial trisymmetric space (Q/H), that
[
\ln_Q(Q_\alpha)/{}^{}L_0
]
((\alpha=1,2,3)) are reducible involutive pairs ((^{3,4})). Thus, first of all, one should seek involutive decompositions of simple compact Lie algebras satisfying the condition stated above. Next one should seek nontrivial ideals (M_\alpha) in
[
L_\alpha=\ln_Q(Q_\alpha)\qquad (\alpha=1,2,3)
]
so that
[
K=M_1+M_2+M_3+L_0
]
is a subalgebra in (\Gamma=\ln Q). Such a subalgebra also determines (K=\ln_Q(H)). Finally, from the pair of algebras (\Gamma/K) we uniquely (locally) reconstruct the trisymmetric space (Q/H), and from the involutive pairs ((^{3,4}))
[
L_\alpha/{}^{}L_0
]
we reconstruct its mirrors (Q_\alpha/{}^{}H_\alpha). In the cases of the classical simple algebras (so(n)), (su(n)), (sp(n)), the problem is solved using known matrix models. The case of the exceptional algebras (g_2, f_4, e_6, e_7, e_8), owing to the absence of good matrix models, is considerably more difficult. With the aid of the apparatus of involutive sums ((^{3-6})) and of a certain procedure for reconstructing involutive sums, all the data needed for the classification of trisymmetri-

...spaces the involutive decompositions of exceptional algebras can be found. Thus, if (\Gamma=L_1+L_2+L_3,\ L_0=L_\alpha\cap L_\beta) ((\alpha\ne\beta)), then for the involutive pairs (\Gamma/L_\alpha) and the reducible pairs (L_\alpha/L_0), respectively, we have:

1) (g_2/so(4),\ so(4)/so(2)\oplus so(2);)
2) (f_4/su(2)\oplus sp(3),\ su(2)\oplus sp(3)/u(1)\oplus u(3);)
3) (e_6/su(2)\oplus su(6),\ su(2)\oplus su(6)/u(1)\oplus s(u(3)\oplus u(3));)
4) (e_7/su(2)\oplus so(12),\ su(2)\oplus so(12)/u(1)\oplus u(6);)
5) (e_7/u(1)\oplus e_6,\ u(1)\oplus e_6/f_4;)
6) (e_8/su(2)\oplus e_7,\ su(2)\oplus e_7/u(1)\oplus u(1)\oplus e_6.)

We note that in 1), 2), 3), 4), 6) (\Gamma/L_\alpha) are principal unitary involutive pairs ((^4)), while in 5) (\Gamma/L_\alpha) is a central involutive pair ((^4)); moreover, all the decompositions obtained are superinvolutive ((^4)). Using the decompositions obtained for exceptional algebras and matrix models of classical algebras and their involutive automorphisms, we arrive at a complete classification of nontrivial trisymmetric and nonsymmetric spaces

[
Q/H=\mathop{\boxtimes}{\alpha=1}^{3} Q\alpha^{*}/H_\alpha
]

with simple compact groups (Q) of motions and maximal rotation groups

with natural embeddings.

All the spaces obtained are supertrisymmetric with principal unitary or central mirrors (i.e. with mirrors generated by principal unitary or central automorphisms ((^{4,6})) of the group (Q)); moreover, the conjugating automorphism (p) belongs to (SU(3))—the normalizer of the rotation group (\widetilde H) in the case of a principal mirror—and (p) belongs to (SO(3))—the normalizer of the rotation group (\widetilde H) in the case of a central mirror.

Peoples’ Friendship University
named after Patrice Lumumba
Moscow

Received
23 II 1970

REFERENCES

(^1) L. V. Sabinin, Siberian Mathematical Journal, No. 2, 2 (1961).
(^2) L. V. Sabinin, Scientific Reports of Higher Schools, Series of Physical and Mathematical Sciences, No. 3 (1958).
(^3) L. V. Sabinin, Doklady Akademii Nauk SSSR, 165, No. 5 (1965).
(^4) L. V. Sabinin, Doklady Akademii Nauk SSSR, 175, No. 1 (1967).
(^5) L. V. Sabinin, Proceedings of the Seminar on Vector and Tensor Analysis, Moscow State University, 14 (1968).
(^6) L. V. Sabinin, Proceedings of the Geometry Seminar, Institute of Scientific Information, Academy of Sciences of the USSR, No. 2 (1969).