MATHEMATICS

A. I. SIROTA

CENTERS OF NONCOMPACT SIMPLE LIE GROUPS

(Presented by Academician P. S. Aleksandrov, 16 IV 1960)

The centers of the simply connected covering groups of noncompact classical groups were found by A. S. Solodovnikov (¹). In the present paper the centers of all connected simply connected noncompact simple Lie groups are computed. This is done by means of a method used by E. B. Dynkin and A. L. Onishchik (²) for finding the centers of compact (and hence also simple complex) groups, and here extended to arbitrary simple groups. As a result, each element of the center of a group turns out to be represented by a certain vector in the Lie algebra which is carried into it under the canonical mapping.

All groups in what follows are assumed to be connected.

Let \(P\) be a compact simple Lie algebra, \(H\) its Cartan subalgebra, and \([P]\) its complex form. If \(\Sigma\) is the complete system of roots of \(P\), and \(e_\alpha\) is the root vector of \([P]\) corresponding to the root \(\alpha\), then we write the structural formulas in the form

\[ [e_\alpha,e_\beta]=N_{\alpha,\beta}e_{\alpha+\beta}\quad(\alpha+\beta\ne 0);\qquad [e_\alpha,e_{-\alpha}]=2\pi i d; \]

\[ [h,e_\alpha]=2\pi i(h,\alpha)e_\alpha;\qquad h\in H;\quad \alpha,\beta\in\Sigma,\quad \Sigma\subset H. \]

According to F. R. Gantmacher (³), every involutive automorphism of the algebra \(P\) is conjugate to a transformation of the adjoint group by an automorphism \(\tau=\tau_0\exp(\bar h)\), where \(\bar h\) is the matrix of the linear transformation \(x\to[x,h]\), \(x\in P\), \(h\in H\), and \(\tau_0\) is an involutive automorphism of \(P\) carrying into itself some system \(\Pi(P)\subset H\) of simple roots of the algebra \(P\); moreover, if \(\tau_0\) is extended to an automorphism of the whole algebra \([P]\), then \(\tau_0(e_\alpha)=e_{\tau_0(\alpha)}\), \(\alpha\in\Pi(P)\). The automorphism \(\tau_0\) is either an outer automorphism of the algebra \(P\), or the identity.

Denote by \(P_+\) the subalgebra of \(P\) belonging to the characteristic root \(1\) of the automorphism \(\tau\), and put \(H_+=P_+\cap H\). Then \(H_+\) is a Cartan subalgebra of the algebra \(P_+\). If \(P_-\) is the subspace of \(P\) belonging to the characteristic root \(-1\) of the automorphism \(\tau\), then the algebra \(G=P_+ + iP_-\) is a real form of the algebra \([P]\) (⁴). The real forms of simple algebras and the involutive automorphisms defining them are listed, for example, in (³). Denote by \(\mathfrak G\) some simple real group with Lie algebra \(G\), and by \(\mathfrak H_+\) its commutative subgroup generated by the subalgebra \(H_+\). When \(\mathfrak G\) is simply connected, we shall denote it by \(\widetilde{\mathfrak G}\).

Lemma 1. The center \(\mathfrak C(\mathfrak G)\) of the group \(\mathfrak G\) is contained in the commutative subgroup \(\mathfrak H_+\).

Proof. We first carry out the proof for \(\widetilde{\mathfrak G}\). Denote by \(\mathfrak P_+\) the subgroup of \(\widetilde{\mathfrak G}\) corresponding to the subalgebra \(P_+\), and show that \(\mathfrak C(\widetilde{\mathfrak G})\subset \mathfrak P_+\). In the adjoint group \(\mathfrak G^*\) the subgroup \(\mathfrak P_+^*\), corresponding to the subalgebra \(P_+\), is maximal compact and there is a decomposition into a topological product \(\mathfrak G^*=\mathfrak P_+^*\times\mathcal L\), where \(\mathcal L\) is a Euclidean space of the corresponding dimension (⁵). Since \(\widetilde{\mathfrak G}\) is simple

the natural homomorphism \(p:\widetilde{\mathscr G}\to \mathscr G^{*}\) is a covering map. But then the group \(p^{-1}(\mathscr P^{*}_{+})\), being, by virtue of the relation \(\mathscr G^{*}=\mathscr P^{*}_{+}\times \mathscr L\), a connected covering group of \(\mathscr P^{*}_{+}\), is locally isomorphic to \(\mathscr P^{*}_{+}\); whence it follows that \(p^{-1}(\mathscr P^{*}_{+})=\mathscr P_{+}\). \(\mathscr C(\widetilde{\mathscr G})=p^{-1}(e)\), where \(e\) is the identity of \(\mathscr P^{*}_{+}\); consequently,
\(\mathscr C(\widetilde{\mathscr G})\subset \mathscr P_{+}\).

The compact algebra \(P_{+}\) admits a decomposition into a direct sum
\(P_{+}=P_{1}+V\), where \(P_{1}\) is compact semisimple, and \(V\) is a commutative algebra (whose dimension is zero or one). Then \(H_{+}=H_{1}+V\), where \(H_{1}\) is a Cartan subalgebra of \(P_{1}\). Correspondingly, in view of the simple connectedness of \(\widetilde{\mathscr G}\), and hence also of \(\mathscr P_{+}\),
\(\mathscr P_{+}=\mathscr P_{1}\cdot\mathscr V\), \(\mathscr H_{+}=\mathscr H_{1}\cdot\mathscr V\), where \(\mathscr P_{1}\) is a semisimple compact subgroup of \(\widetilde{\mathscr G}\) with Lie algebra \(P_{1}\), \(\mathscr H_{1}\) is its maximal torus with Lie algebra \(H_{1}\), and \(\mathscr V\) is a simply connected commutative subgroup of \(\widetilde{\mathscr G}\) with Lie algebra \(V\). From what was proved above it follows that
\(\mathscr C(\widetilde{\mathscr G})\subset \mathscr C(\mathscr P_{+})\); but, since the center of a compact group is contained in its maximal torus,
\(\mathscr C(\mathscr P_{+})=\mathscr C(\mathscr P_{1})\cdot \mathscr V\subset \mathscr H_{1}\cdot\mathscr V=\mathscr H_{+}\).

The extension of the lemma to the case of a non-simply connected \(\mathscr G\) is carried out with the aid of the factorization of the group \(\widetilde{\mathscr G}\).

For an arbitrary semisimple compact algebra \(R\), denote by \(\Gamma_{0}(R)\) the integral lattice in the Cartan subalgebra of \(R\) whose basis consists of the vectors
\(\alpha'=\dfrac{2\alpha}{(\alpha,\alpha)}\), \(\alpha\in \Pi(R)\), and by \(\Gamma_{1}(R)\) the integral lattice whose basis is biorthogonal to the system \(\Pi(R)\). If \(\mathscr R\) is a simply connected group with Lie algebra \(R\), then \(\Gamma_{0}(R)\) is the full inverse image of the identity in the Cartan subalgebra of the algebra \(R\) under the canonical mapping \(R\) into \(\mathscr R\) \((^{2})\).

Introduce \(P_{0}\) into consideration—the subalgebra of \(P\) belonging to the characteristic root \(1\) of the automorphism \(\tau_{0}\). \(P_{0}\) is always semisimple (and, evidently, compact), and \(H_{+}\) is a Cartan subalgebra for it.

Lemma 2. The inverse image of the center of the group \(\mathscr G\) in \(H_{+}\) under the canonical mapping \(c:G\to \mathscr G\) is the lattice
\(\Gamma_{1}(P)\cap H_{+}\), and it coincides with the lattice \(\Gamma_{1}(P_{0})\).

Proof. The canonical mapping induces a homomorphism of the additive vector group \(H_{+}\) onto the group \(\mathscr H_{+}\). An element \(h\in H_{+}\) belongs to \(c^{-1}(\mathscr C(\mathscr G))\) if and only if the linear transformation \(pc(h)\), acting on \(G\), is identical (\(p\) is the adjoint representation of the group \(\mathscr G\)). The transformation \(pc(h)\) extends uniquely to a linear transformation on \([P]\). It is identical if and only if \(h\in \Gamma_{1}(P)\) \((^{2})\). The second assertion of the lemma follows from the fact that the projection of the system \(\Pi(P)\) onto \(H_{+}\) is \(\Pi(P_{0})\).

Theorem. The center of the simply connected group \(\widetilde{\mathscr G}\) is isomorphic to the factor group
\(\Gamma_{1}(P_{0})/\Gamma_{0}(P_{1})\), and the isomorphism is generated by the canonical mapping \(c:G\to \widetilde{\mathscr G}\).

Proof. Let us find the inverse image of the identity \(e\) in \(H_{+}\) under the canonical mapping, i.e. \(c^{-1}(e)\cap H_{+}\). In view of the simple connectedness of \(\widetilde{\mathscr G}\), and consequently also of \(\mathscr V\), the homomorphism \(V\to\mathscr V\) generated by the canonical mapping is an isomorphism. Therefore \(c^{-1}(e)\cap H_{+}\subset H_{1}\), i.e. \(c^{-1}(e)\cap H_{+}\) coincides with the inverse image of the identity in \(H_{1}\) under the canonical mapping of the algebra \(P_{1}\) into the group \(\mathscr P_{1}\). Since \(H_{1}\) is a Cartan subalgebra of \(P_{1}\), and \(\mathscr P_{1}\) is semisimple compact and simply connected (in view of the simple connectedness of \(\widetilde{\mathscr G}\)), we have
\(c^{-1}(e)\cap H_{+}=\Gamma_{0}(P_{1})\). The assertion of the theorem now follows from Lemma 2.

Below are given the centers \(\mathscr C(\widetilde{\mathscr G})\) of simply connected noncompact simple real Lie groups, computed as complete sets of representatives of the cosets \(\Gamma_{1}(P_{0})\) modulo \(\Gamma_{0}(P_{1})\) for each Lie algebra \(G\). \(Z_{m}(z)\) denotes the additive cyclic group of order \(m\) with generator

\(z\), \(Z(z)\)—the infinite cyclic group. The numbering of the simple roots is the same as in (2). The real forms of exceptional algebras are specified by the signatures \(\delta\) of their Cartan metric.

Real forms \(A_n\) \((n>1)\)

1. \(G=A_n^l\)—the algebra of matrices of order \(n+1\) with trace \(0\), preserving the invariant Hermitian form

\[ -\sum_1^l x_k\bar y_k+\sum_{l+1}^{n+1}x_k\bar y_k,\qquad l=1,\ldots,\left[\frac{n+1}{2}\right]. \]

Introduce the notation:

\[ u_1=\frac{1}{n+1}\sum_1^n k\alpha_k,\qquad u_2=\frac{1}{n+1}\sum_1^n (n-k+1)\alpha_k, \]

\(d=\gcd(l,n-l+1)\), \(z_1=\dfrac{l}{d}u_2-\dfrac{n-l+1}{d}u_1\), \(z_2=M_1u_1+M_2u_2\), where \(M_1,M_2\) are integers such that \(M_1l+M_2(n-l+1)=d\). Then the center has the form \(Z_d(z_1)+Z(z_2)\).

1. \(G=I_n\)—the algebra of real matrices of order \(n+1\) with trace \(0\). Introduce the notation

\[ z=\frac12\sum_0^{[n/2]}\alpha_{2k+1},\qquad z_1=\alpha_{\frac{n+1}{2}},\qquad z_2=\alpha_{\frac n2}+\alpha_{\frac n2+1}. \]

Then the center has the form \(Z_4(z)\), if \(n\) and \((n+1)/2\) are odd; \(Z_2(z)+Z_2(z_1)\), if \(n\) is odd and \((n+1)/2\) is even; \(Z_2(z_2)\), if \(n\) is even.

1. \(G=J_n\) (\(n\) odd)—the algebra of quaternionic matrices of order \((n+1)/2\), considered up to real positive factors. The center has the form \(Z_2(z)\), where

\[ z=\frac12\sum_0^{[n/2]}\alpha_{2k+1}. \]

Real forms \(B_n\)

1. \(G=B_n^{2l}\)—the algebra of real matrices of order \(2n+1\), preserving the invariant quadratic form

\[ -\sum_1^{2l}x_k^2+\sum_{2l+1}^{2n+1}x_k^2,\qquad l=1,\ldots,n. \]

The center has the form \(Z(z_1)+Z_2(z_2)\) for \(l=1\), and \(Z_2(z_1)+Z_2(z_2)\) for \(l>1\), where \(z_1=\alpha_l'\), \(z_2=\tfrac12\alpha_n'\).

Real forms \(C_n\)

Denote

\[ z=\frac12\sum_0^{[(n-1)/2]}\alpha_{2k+1}',\qquad z_1=\alpha_n'. \]

1. \(G=C_n^{2l}\)—the algebra of matrices of order \(2n\), preserving the invariant skew-symmetric bilinear form

\[ \sum_1^n(x_{2k-1}y_{2k}-x_{2k}y_{2k-1}) \]

and the Hermitian form

\[ -\sum_1^{2l}x_k\bar y_k+\sum_{2l+1}^{2n}x_k\bar y_k. \]

The center has the form \(Z_2(z)\).

1. \(G=IC_n\)—the algebra of real matrices of order \(2n\), preserving the invariant skew-symmetric bilinear form

\[ \sum_1^n(x_{2k-1}y_{2k}-x_{2k}y_{2k-1}). \]

The center has the form \(Z(z)\), if \(n\) is odd, and \(Z_2(z)+Z(z_1)\), if \(n\) is even.

Real forms \(D_n\)

Let

\[ z=\frac12(\alpha_{n-1}+\alpha_n),\qquad z_1=\frac12\sum_1^{[(n-1)/2]}\alpha_{2k-1}+\frac n4(\alpha_{n-1}+\alpha_n)-\frac12\alpha_n. \]

\[ z_2=\alpha_l+\frac12(\alpha_{n-1}+\alpha_n),\qquad z_3=\frac12\sum_1^{[(n-1)/2]}\alpha_{2k-1}+\frac n4(\alpha_{n-1}+\alpha_n)-\frac12\alpha_{n-1}. \]

\[ z_4=\alpha_l. \]

1. \(G=D_n^{2l}\)—the algebra of real matrices of order \(2n\) leaving invariant the quadratic form
   \[ -\sum_1^{2l} x_k^2+\sum_{2l+1}^{2n} x_k^2,\qquad l=1,\ldots,\left[\frac n2\right]. \]
   The center has the form \(Z_2(z)+Z(z_1)\), if \(l=1\); \(Z_4(z_1)+Z_2(z_4)\), if \(l>1\), \(n\) is odd; \(Z_4(z_1)+Z_2(z_2)\), if \(l>1\), \(n\) is even, \(l\) is odd; \(Z_2(z_1)+Z_2(z_2)+Z_2(z_3)\), if \(l>1\), \(n\) and \(l\) are even.

2. \(G=JD_n\)—the algebra of matrices of order \(2n\) leaving invariant the quadratic form
   \[ \sum_1^n x_{2k-1}x_{2k} \]
   and the Hermitian form
   \[ \sum (x_{2k-1}\overline{x}_{2k-1}-x_{2k}\overline{x}_{2k}). \]
   The center has the form \(Z(z_1-sz)\), if \(n=2s+1\); \(Z_2(z_1-sz)+Z(z_3-sz)\), if \(n=4s+2\); \(Z(z_1-2sz)+Z_2(z_3-2sz)\), if \(n=4s^*\).

3. \(G=D_n^{2l+1}\)—the algebra of real matrices of order \(2n\) leaving invariant the quadratic form
   \[ -\sum_1^{2l+1} x_k^2+\sum_{2l+1}^{2n} x_k^2,\qquad l=0,1,\ldots,\left[\frac n2\right]. \]
   The center has the form \(Z_2(z)+Z_2(z_4)\).

Real form \(G_2\)

1. \(\delta=2\). The center has the form \(Z_2(\alpha'_2)\).

Real forms \(F_4\)

1. \(\delta=-52\). The center is trivial.
2. \(\delta=4\). The center has the form \(Z_2(\alpha'_1)\).

Real forms \(E_6\)

Put \(z=\frac13(\alpha_1-\alpha_2+\alpha_4-\alpha_5)\), \(z_1=\alpha_6\).

1. \(\delta=-14\). The center has the form \(Z(z)\).
2. \(\delta=2\). The center has the form \(Z_6(z)\).
3. \(\delta=-26\). The center is trivial.
4. \(\delta=6\). The center has the form \(Z(z_1)\).

Real forms \(E_7\)

Put \(z=\frac12(\alpha_4+\alpha_6+\alpha_7)\), \(z_1=\alpha_5\).

1. \(\delta=-5\). The center has the form \(Z_2(z)+Z_2(z_1)\).
2. \(\delta=7\). The center has the form \(Z_4(z)\).
3. \(\delta=-25\). The center has the form \(Z(z)\).

Real forms \(E_8\)

1. \(\delta=-24\). The center has the form \(Z_2(\alpha_1)\).
2. \(\delta=8\). The center has the form \(Z_2(\alpha_7)\).

The author expresses gratitude to P. K. Rashevsky for his attention to the present work.

Received
15 IV 1960

CITED LITERATURE

\({}^{1}\) A. S. Solodovnikov, DAN, 129, No. 2 (1959).
\({}^{2}\) E. B. Dynkin, A. L. Onishchik, UMN, 10, No. 4, 3 (1955).
\({}^{3}\) F. R. Gantmakher, Matem. sborn., 5 (47), 101, 217 (1939).
\({}^{4}\) É. Cartan, The Geometry of Lie Groups and Symmetric Spaces, Moscow, 1949, p. 175.
\({}^{5}\) E. Cartan, La topologie des groupes de Lie, Paris, 1936.

* What is stated in the corresponding item in (1) is valid only for odd \(n\).