UDC 519.4

MATHEMATICS

L. V. SABININ

ON ISOINVOLUTIVE DECOMPOSITIONS OF LIE ALGEBRAS

(Presented by Academician S. L. Sobolev on 13 IV 1965)

In the present paper, for compact real semisimple Lie algebras, the theory of isoinvolutive decompositions is set forth (without proofs) and its application to compact symmetric spaces of rank 1.

Let \(\Gamma_r\) be a compact semisimple Lie algebra over the field of real numbers; let \(e_1, e_2, \ldots, e_r\) be its basis. The structure is then determined by the commutators of the basis vectors

\[ [e_I e_K]=c^R_{IK}e_R,\qquad c^R_{IK}=-c^R_{KI},\qquad I,K,R=1,\ldots,r, \tag{1} \]

and by the Jacobi identities

\[ [e_I[e_Je_K]]+[e_K[e_Ie_J]]+[e_J[e_Ke_I]]=0. \tag{2} \]

The transformations of the adjoint group \(\Gamma_r^*\) have the form

\[ \bar{\eta}\to e^{\bar c(u)}\bar{\eta},\qquad \bar c(u)=\|c^I_{J K}u^K\|, \tag{3} \]

and the tensor \(c^R_{IK}\) is invariant under such transformations.

In a compact semisimple Lie algebra one can introduce a positive definite metric tensor

\[ a_{IJ}=c^P_{IK}c^K_{PJ},\qquad a_{IJ}=a_{JI} \tag{4} \]

and consider orthonormal bases, in which

\[ a_{IJ}=\delta_{IJ}. \tag{5} \]

Then, as is known,

\[ c^I_{JK}=-c^J_{IK}. \tag{6} \]

An automorphism \(A\) of the algebra \(\Gamma\) is called an involomorphism if

\[ A^2=I,\qquad A\ne I. \tag{7} \]

In some orthobasis any involomorphism \(S\) has a diagonal form with \(\pm 1\) on the main diagonal.

All \(\eta\in\Gamma_r\) such that \(S\eta=\eta\) form a subalgebra (invoalgebra) \(L\subset\Gamma_r\); the pair \(\Gamma_r/L\) will be called an invopair. Obviously,

\[ \Gamma_r=E+L,\qquad E\perp L;\qquad S\bar{\eta}= \begin{cases} \bar{\eta}, & \bar{\eta}\in L,\\ -\bar{\eta}, & \bar{\eta}\in E. \end{cases} \tag{8} \]

The invopair \(\Gamma/L\) generates the symmetric space \(\Gamma^*/L^*\), where \(L^*\) has on \(E\) an irreducible linear representation, if \(\Gamma\) is simple.

Definition 1. We shall say that the algebra \(\Gamma\) is an involutive sum of its subalgebras \(\Gamma=L_1+L_2+L_3\), if \(L_1,L_2,L_3\) are the invoalgebras of commuting involomorphisms \(S_1,S_2,S_3=S_1S_2\), respectively.

Obviously, \(L_1\cap L_2=L_1\cap L_3=L_2\cap L_3=L_0\), and \(L_i/L_0\) \((i=1,2,3)\) are invopairs.

Definition 2. An involutive sum \(\Gamma=L_1+L_2+L_3\) will be called isoinvolutive if \(L_1\) and \(L_2\) are conjugate in \(\Gamma^*\), with the conjugating automorphism of the form \(e^{\bar c(\bar\xi t)}\), where \(\bar\xi\in E_3=L_3-L_0\).

Lemma 1. If \(\Gamma/L\) is a compact semisimple involpara, then there exists \(\bar\xi\in E=\Gamma-L\) such that the subgroup \(e^{c(\bar\xi t)}\) in \(\Gamma^*\) is compact.

Corollary 1. If \(\Gamma\) is a semisimple compact Lie algebra and \(S\) is its involmorphism, then there exist inner automorphisms \(Q_n\) in \(\Gamma^*\) \((n\) a positive integer\()\) such that \((Q_n)^n=1,\; SQ_nS=Q_n^{-1}\).

On the basis of Lemma 1 and its corollary the following theorem is proved:

Theorem 1. Let \(\Gamma\) be a semisimple compact Lie algebra, and let \(L_1\) be any of its invoalgebras; then there exists an isoinvolutive decomposition
\[ \Gamma=L_1+L_2+L_3,\qquad L_1=\varphi L_2,\quad \varphi\in\Gamma^*,\quad \varphi^4=1,\quad S_3=\varphi^2. \]

Corollary 1. The restriction of \(\varphi\) to \(L_3\) acts as an involmorphism or as the identity automorphism.

Corollary 2. The restriction of \(\varphi\) to \(L_0\) acts as an involmorphism or the identity automorphism.

Corollary 3. If the restriction of \(\varphi\) to \(L_3\) acts as an involmorphism, then \(L_3\) is an isoinvolutive sum \(L_3=L_0+L'_0+L''_0\), where \(L_0,L'_0,L''_0\) are the invoalgebras of the involmorphisms \(S_1,\;(\varphi^{1/2})^{-1}S_1(\varphi^{1/2}),\;\varphi\), respectively.

Definition 3. An isoinvolutive sum will be called of type I if the restriction of the conjugating automorphism \(\varphi\) to \(L_3\) is the identity automorphism.

We shall consider only simple compact algebras \(\Gamma\).

Let \(\Pi=L_0\cap L'_0\); then
\[ L_0=\Pi+E_0,\qquad L'_0=\Pi+E'_0,\qquad L''_0=\Pi+E''_0, \]
\[ L_1=L_0+E_1,\qquad L_2=L_0+E_2, \]
where \(E,E',E''\) are orthogonal to \(\Pi\), and \(E_1,E_2\) are orthogonal to \(L_0\). From Theorem 1 and its corollaries we have
\[ E_2=\varphi E_1,\qquad E'_0=(\varphi^{1/2})E_0,\qquad (\varphi^{1/2})\Pi=\Pi,\qquad (\varphi^{1/2})E''_0=E''_0. \tag{9} \]

Choosing orthonormal bases in \(\Pi,E_1,E_0,E''_0\), with the help of (9) we construct bases in \(E_2\) and \(E'_0\). The union of all the indicated bases generates an orthobasis in \(\Gamma\), which we shall call isoinvolutive. Let, in this notation,
\[ \begin{aligned} &X_{i_1}\quad (i=1,\ldots,n) &&\text{be a basis in } E_1,\\ &X_{i_2}\quad (i=1,\ldots,n) &&\text{be a basis in } E_2;\\ &Y_{\alpha_2}\quad (\alpha=1,\ldots,\rho) &&\text{be a basis in } E_0,\\ &Y_{\alpha_1}\quad (\alpha=1,\ldots,r) &&\text{be a basis in } \Pi; \end{aligned} \quad\Bigg\}\;Y_\alpha \tag{10} \]
\[ \begin{aligned} &V_{\alpha_2}\quad (\alpha=1,\ldots,\rho) &&\text{be a basis in } E'_0,\\ &V_{\alpha_1}\quad (a=1,\ldots,m) &&\text{be a basis in } E''_0. \end{aligned} \quad\Bigg\}\;V_\alpha \]

By construction
\[ \varphi X_{i_1}=X_{i_2},\qquad \varphi X_{i_2}=-X_{i_1},\qquad \varphi Y_\alpha=\varphi_\alpha^{\beta}Y_\beta \]
\[ \left(\varphi Y_{\alpha_2}=-Y_{\alpha_2},\qquad \varphi Y_{\alpha_1}=Y_{\alpha_1}\right), \tag{11} \]
\[ \varphi V_\alpha=\varphi_\alpha^{b}V_b \qquad \left(\varphi V_{\alpha_2}=-V_{\alpha_2},\qquad \varphi V_{\alpha_1}=V_{\alpha_1}\right). \]

The automorphism \(\theta=\varphi^{1/2}\) acts, as is easily verified, as follows:

\[ \theta X_{i_1}=\frac{1}{\sqrt2}\theta_i^j(X_{j_1}+X_{j_2}),\quad \theta X_{i_2}=\frac{1}{\sqrt2}\theta_i^j(-X_{j_1}+X_{j_2}), \]

\[ \theta_k^p\theta_s^k=\delta_s^p,\qquad \theta_j^i=\theta_i^j, \]

\[ \theta Y_{a_2}=V_{a_2},\quad \theta V_{a_2}=-Y_{a_2},\quad \theta Y_{a_1}=\theta_{a_1}^{\beta_1}Y_{\beta_1},\quad \theta V_{a_1}=\theta_{a_1}^{c_1}V_{c_1}, \]

\[ \theta_{a_1}^{\lambda_1}\theta_{\mu_1}^{a_1}=\delta_{\mu_1}^{\lambda_1},\quad \theta_{a_1}^{c_1}\theta_{b_1}^{a_1}=\delta_{b_1}^{c_1},\quad \theta_{\alpha_1}^{\beta_1}=\theta_{\beta_1}^{\alpha_1},\quad \theta_{a_1}^{c_1}=\theta_{c_1}^{a_1}. \tag{12} \]

Taking into account the action of the automorphisms \(S_1,S_2,S_3\), we write the structure of the algebra \(\Gamma\) in the form

\[ [X_{i_1}X_{j_1}]=-b_{i_1j_1}^{a}Y_a,\quad [X_{i_2}X_{j_2}]=-b_{i_2j_2}^{a}Y_a,\quad [X_{i_1}X_{j_2}]=q_{i_1j_2}^{a}V_a, \]

\[ [X_{i_1}V_a]=t_{i_1a}^{j_2}X_{j_2},\quad [X_{p_2}V_a]=t_{p_2a}^{j_1}X_{j_1},\quad [V_aV_c]=-b_{ac}^{\alpha}Y_\alpha,\quad [X_{i_1}Y_\alpha]=a_{i_1\alpha}^{p_1}X_{p_1}, \tag{13} \]

\[ [X_{i_2}Y_\alpha]=a_{i_2\alpha}^{p_2}X_{p_2},\quad [V_cY_\alpha]=a_{c\alpha}^{b}V_b,\quad [Y_\alpha Y_\beta]=c_{\alpha\beta}^{\gamma}Y_\gamma. \]

Introduce the notation

\[ a_{k_1\alpha}^{p_1}=a_{k\alpha}^{p},\quad b_{k_1j_1}^{a}=b_{kj}^{a},\quad q_{k_2j_1}^{a}=q_{kj}^{a},\quad t_{i_1a}^{p_2}=t_{ia}^{p}. \tag{14} \]

Then the automorphism \(\varphi\) from (11), for the structure (13), gives

\[ b_{i_2j_2}^{\alpha}=\varphi_\beta^\alpha b_{ij}^{\beta},\quad q_{ij}^{b}=\varphi_a^b q_{ji}^{a},\quad t_{i_2b}^{j_1}=-\varphi_b^a t_{ia}^{j},\quad a_{i_2\alpha}^{p_2}=\varphi_\alpha^\beta a_{i\beta}^{p}, \]

\[ b_{ac}^{\alpha}=\varphi_a^f\varphi_c^d\varphi_\beta^\alpha b_{fd}^{\beta},\quad a_{c\alpha}^{b}=\varphi_c^d\varphi_f^b\varphi_\alpha^\beta a_{d\beta}^{f},\quad c_{\alpha\beta}^{\gamma}=\varphi_\alpha^\lambda\varphi_\beta^\mu\varphi_\nu^\gamma c_{\lambda\mu}^{\nu}. \tag{15} \]

From the orthonormality of the isoinvobasis it also follows that

\[ a_{j\alpha}^{i}=b_{ji}^{\alpha},\quad q_{kj}^{a}=t_{ja}^{k},\quad t_{i_2b}^{j_1}=-t_{jb}^{i},\quad a_{c\alpha}^{f}=b_{cf}^{\alpha},\quad c_{\beta\gamma}^{\alpha}=-c_{\beta\gamma}^{\alpha}. \tag{16} \]

The automorphism \(\theta\) from (12) for the structure (13) leads to the relations

\[ t_{i\alpha_2}^{p}=\theta_i^p\theta_i^k a_{k\alpha_2}^{s},\quad \theta_i^k\theta_s^b t_{ia}^{p}=\theta_a^b t_{kb}^{s},\quad \theta_i^k\theta_s^p a_{k\beta_1}^{s}=\theta_{\beta_1}^{\alpha_1}a_{i\alpha_1}^{p}, \]

\[ c_{\alpha_2\beta_2}^{\lambda_1}=-b_{\alpha_2\beta_2}^{\sigma_2}\theta_{\sigma_1}^{\lambda_1},\quad b_{a_1\beta_2}^{\sigma_2}=\theta_{a_1}^{c_1}a_{c_1\beta_2}^{\sigma_2},\quad a_{\alpha_2c_2}^{e_1}=\theta_{b_1}^{e_1}a_{c_2\alpha_2}^{b_1},\quad a_{e_2\alpha_1}^{b_2}=\theta_{\alpha_1}^{\beta_1}c_{e_2\beta_1}^{b_2}, \tag{17} \]

\[ a_{c_1\alpha_1}^{b_1}\theta_{b_1}^{e_1}=\theta_{c_1}^{d_1}\theta_{\alpha_1}^{\beta_1}a_{d_1\beta_1}^{e_1},\quad c_{\alpha_1\beta_1}^{\lambda_1}\theta_{\lambda_1}^{\mu_1} =\theta_{\alpha_1}^{\sigma_1}\theta_{\beta_1}^{\nu_1}c_{\sigma_1\nu_1}^{\mu_1}. \]

Finally, consideration of all Jacobi identities leads to the relations

\[ a_{k\alpha}^{p}b_{ij}^{\alpha} =2a_{k\alpha_2}^{p}b_{ij}^{\alpha_2} +t_{j\alpha_2}^{p}q_{ki}^{\alpha_2} -t_{i\alpha_2}^{p}q_{kj}^{\alpha_2} +t_{j\alpha_1}^{p}q_{ki}^{\alpha_1} -t_{i\alpha_1}^{p}q_{kj}^{\alpha_1}, \]

\[ -b_{\alpha_1c_1}^{b_1}a_{i\alpha_1}^{p} =t_{ja}^{p}t_{ic_1}^{j}-t_{jc_1}^{p}t_{ia_1}^{j}, \]

\[ -a_{c_2\alpha_2}^{b_1}t_{ib_1}^{k} =t_{pc_2}^{k}a_{i\alpha_2}^{p}+a_{p\alpha_2}^{k}t_{ic_2}^{p}, \]

\[ a_{c_1\alpha_2}^{b_2}t_{ib_2}^{k} =t_{pc_1}^{k}a_{i\alpha_2}^{p}+a_{p\alpha_2}^{k}t_{ic_1}^{p}, \]

\[ a_{c_1\alpha_1}^{b_1}t_{ib_1}^{k} =a_{p\alpha_1}^{k}t_{ic_1}^{p}-t_{pc_1}^{k}a_{i\alpha_1}^{p}, \tag{18} \]

\[ c_{\alpha_2\beta_2}^{\sigma_1}a_{j\sigma_1}^{l} =a_{i\beta_2}^{l}a_{j\alpha_2}^{i}-a_{i\alpha_2}^{l}a_{j\beta_2}^{i}, \]

\[ c_{\beta_2\alpha_1}^{\sigma_2}a_{j\sigma_2}^{l} =a_{i\beta_2}^{l}a_{j\alpha_1}^{i}-a_{i\alpha_1}^{l}a_{j\beta_2}^{i}, \]

\[ c_{\alpha_1\beta_1}^{\sigma_1}a_{j\sigma_1}^{l} =a_{i\beta_1}^{l}a_{j\alpha_1}^{i}-a_{i\alpha_1}^{l}a_{j\beta_1}^{i}. \]

Definition 4. An isoinvolutive sum, if it is not of type I, shall be called type II if \(\theta_s^p=+\delta_s^p\), and type III otherwise.

The use of the relations obtained above leads to the theorem:

Theorem 2. If a compact simple Lie algebra \(\Gamma\) admits an isoinvolutive decomposition

\[ \Gamma=L_1+L_2+L_3 \]

of type I or II, then the maximal subalgebra of elements fixed under the action of the conjugating auto-

morphism \(\varphi\), has center \(\xi \in L_3 - L_0\). Hence it follows that, if the involara \(\Gamma/L_1\) has rank \(1\), then \(L_3 - L_0\) is one-dimensional for type I, and \(L''_0 - \Pi\) is one-dimensional for type II.

This permits one to obtain the following theorems:

Theorem 3. If the isoinvolutive sum of type I
\[ \Gamma = L_1 + L_2 + L_3 \]
is simple and compact, and \(\Gamma/L_1\) is an involara I, then \(\Gamma/L_1\) is of the form
\[ SO(n+1)/SO(n), \]
where \(L_1/L_0\) is of the form
\[ SO(n)/SO(n-1) \]
(with the natural embedding).

Theorem 4. If \(\Gamma/L_1\) is a simple compact involara of rank \(1\), and
\[ \Gamma = L_1 + L_2 + L_3 \]
is an isoinvolutive decomposition of type II and \(E_0\) is one-dimensional, then \(\Gamma/L_1\) is of the form
\[ SU(n+1)/S(U(n)\times \tilde U(1)), \]
and \(L_1/\tilde L_0\) is of the form
\[ SU(n)/S(U(n-1)\times U(1)) \]
(with the natural embeddings)
\[ (L_0-\tilde L_0 = Z \]
is a one-dimensional center in \(L_1\)).

Theorem 5. If \(\Gamma/L_1\) is a simple compact involara of rank \(1\),
\[ \Gamma = L_1 + L_2 + L_3 \]
is an isoinvolutive decomposition of type II and \(E_0\) is three-dimensional, then \(\Gamma/L_1\) is of the form
\[ Sp(n+1)/Sp(n)\times Sp(1), \]
and \(L_1/\tilde L_0\) is of the form
\[ Sp(n)/Sp(n-1)\times Sp(1) \]
(with the natural embeddings)
\[ (L_0-\tilde L_0 = Z \]
is a three-dimensional ideal in \(L_1\)).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
13 IV 1965