UDC 513.814

MATHEMATICS

D. B. PERSITS

GEOMETRIES OVER DEGENERATE OCTAVES

(Presented by Academician Yu. V. Linnik, June 7, 1966)

1. In the present note the projective plane \(P_2(i,j,\varepsilon)\) over the algebra of degenerate octaves \(R(i,j,\varepsilon)\) is considered, and the connection is indicated between the projective and non-Euclidean geometries of this plane and the limiting special groups of types \(G_2, F_4, E_6\). All considerations are carried out over the field of real numbers.

2. Let \(\mathfrak{G}\) be a connected Lie group, \(G\) its Lie algebra, and

\[ G = K + E \tag{1} \]

a Cartan decomposition of the Lie algebra \(G\), i.e.

\[ [K,K]\subset K;\qquad [E,E]\subset K;\qquad [K,E]\subset E . \]

Following F. A. Berezin and I. M. Gelfand ((\(^{1}\)), p. 349), we define the limiting Lie group \(\mathfrak{G}_0\) for the group \(\mathfrak{G}\) with respect to the Cartan decomposition (1) as follows. Let \(\mathfrak{K}\) be the connected subgroup of the group \(\mathfrak{G}\) corresponding to the Lie subalgebra \(K\). If \(g\in\mathfrak{K}\), then \(A_gE\subset E\), where \(A_g\) is the image of the element \(g\) under the adjoint representation.

The group \(\mathfrak{G}_0\) is constructed on the direct product of the group \(\mathfrak{K}\) and the vector space \(E\). Namely:

\[ (k_1,e_1)(k_2,e_2)\overset{\mathrm{def}}{=}(k_1k_2,\, A_{k_2}^{-1}e_1+e_2), \]

where \(k_i\in\mathfrak{K}\), \(e_i\in E\) \((i=1,2)\).

In addition to the group \(\mathfrak{G}_0\), we define the limiting Lie group with similitudes \(\overline{\mathfrak{G}}_0\) for the group \(\mathfrak{G}\) with respect to the same Cartan decomposition (1). We construct the group \(\overline{\mathfrak{G}}_0\) on the direct product of the group \(\mathfrak{G}_0\) and the set \(R\) of positive real numbers, setting

\[ (k_1,e_1,\lambda_1)(k_2,e_2,\lambda_2)\overset{\mathrm{def}}{=}(k_1,k_2,\,\lambda_2^{-1}A_{k_2}^{-1}e_1+e_2,\,\lambda_1\lambda_2), \]

where \(k_i\in\mathfrak{K}\), \(e_i\in E\), \(\lambda_i\in R\) \((i=1,2)\).

It is easy to verify the correctness of the definition introduced, and also the fact that the naturally embedded space \(E\) forms in \(\mathfrak{G}_0\) (and in \(\overline{\mathfrak{G}}_0\)) a commutative normal divisor.

1. We now consider the algebra of degenerate octaves \(R(i,j,\varepsilon)\). The linear space of the algebra \(R(i,j,\varepsilon)\) is the direct sum of two linear spaces of the quaternion algebra \(R(i,j)\), so that \(\dim R(i,j,\varepsilon)=8\). Multiplication in \(R(i,j,\varepsilon)\) is defined by the formula

\[ (x_1,y_1)(x_2,y_2)=(x_1x_2,\,y_2x_1+y_1\bar{x}_2). \]

The linear mapping

\[ J:R(i,j,\varepsilon)\to R(i,j,\varepsilon),\qquad (x,y)\mapsto \overline{(x,y)}=(\bar{x},-y) \]

is an involution (i.e., an involutive antiautomorphism) in \(R(i,j,\varepsilon)\).

The zero divisors (together with zero) have the form \((0,y)\) and form in \(R(i,j,\varepsilon)\) an ideal \(E\) with trivial multiplication.

1. Let \(\mathfrak M_3\) denote the space of square matrices of order 3 with entries from the algebra \(R(i,j,\varepsilon)\). For \(A=\|a_{ij}\|\in\mathfrak M_3\), put
   \[ A^*=\Gamma \widetilde A^{\,t}\Gamma^{-1}, \]
   where
   \[ \widetilde A^{\,t}=\|(\overline{a_{ij}})_{ji}\|,\qquad \Gamma=\operatorname{diag}(\gamma_1,\gamma_2,\gamma_3),\quad \gamma_i=\pm1. \]
   Put
   \[ \mathfrak M_3^+(\Gamma)=\{A:\ A\in\mathfrak M_3;\ A^*=A\}. \]
   On the space \(\mathfrak M_3^+(\Gamma)\) we introduce the structure of a Jordan algebra with identity \(\mathfrak B(R(i,j,\varepsilon),\Gamma)\), putting
   \[ AB=\frac12(A\cdot B+B\cdot A), \]
   where \(A,B\in\mathfrak M_3^+(\Gamma)\); \(A\cdot B\) is the ordinary product of the matrices \(A\) and \(B\). In what follows we shall consider only the cases
   \[ \Gamma=\Gamma_+=\operatorname{diag}(1,1,1),\qquad \Gamma=\Gamma_-=\operatorname{diag}(1,1,-1). \]

We also define
\[ \mathfrak B(E,\Gamma)=\{A=\|a_{ij}\|:\ A\in\mathfrak B(R(i,j,\varepsilon),\Gamma);\ a_{ij}\in E\}. \]
Clearly, \(\mathfrak B(E,\Gamma)\) is an ideal in \(\mathfrak B(R(i,j,\varepsilon),\Gamma)\) with trivial multiplication.

1. To construct the projective plane, we introduce into consideration the spaces
   \[ \Pi_\Gamma=\{X:\ X\in\mathfrak B(R(i,j,\varepsilon),\Gamma)\setminus\mathfrak B(E,\Gamma);\ X^2=\operatorname{Sp}X\cdot X\}, \]
   where \(\operatorname{Sp}X\) is the trace of the matrix \(X\), and
   \[ \Pi_\Gamma^*=\{\rho X\}, \]
   where \(X\in\Pi_\Gamma\), \(\rho\ne0\) is real; that is, \(\Pi_\Gamma^*\) is the set (or manifold) of lines passing through \(0\) and lying in \(\Pi_\Gamma\).

We now define the projective plane \(P_2(i,j,\varepsilon)\) over the algebra \(R(i,j,\varepsilon)\) as the union of two sets—the set of points \(P\) and the set of lines \(L\)—with an incidence relation defined between the elements of these sets: \(x\circ a\), where \(x\in P\), \(a\in L\), satisfying the following properties: there exist two one-to-one mappings
\[ \mu_1:\ P\xrightarrow{\text{onto}}\Pi_{\Gamma_+}^*,\qquad \mu_2:\ L\xrightarrow{\text{onto}}\Pi_{\Gamma_+}^* \]
such that, if \(\rho X=\mu_1x\), \(\rho A=\mu_2a\), then
\[ x\circ a\Longleftrightarrow XA=0. \]

1. We now define the elliptic polar transformation (elliptic polarity) \(\pi_+\):
   \[ \pi_+:\ P\to L,\qquad x\mapsto a=\mu_2^{-1}\mu_1x. \]
   Clearly,
   \[ x\circ a\Longleftrightarrow \pi_+^{-1}a\circ\pi_+x. \]
   The projective plane \(P_2(i,j,\varepsilon)\) with the given polarity \(\pi_+\) will be called the elliptic unitary plane \(\overline S_2(i,j,\varepsilon)\) over the algebra \(R(i,j,\varepsilon)\) (cf. (2)).

Next, we define the hyperbolic unitary plane \({}^{1}\overline S_2(i,j,\varepsilon)\). For this purpose consider the one-to-one mappings
\[ \psi_1:\ \Pi_{\Gamma_+}^*\to\Pi_{\Gamma_-}^*,\qquad \rho \begin{vmatrix} \lambda_1 & x_3 & \bar x_2\\ \bar x_3 & \lambda_2 & x_1\\ x_2 & \bar x_1 & \lambda_3 \end{vmatrix} \mapsto \rho \begin{vmatrix} \lambda_1 & x_3 & -\bar x_2\\ \bar x_3 & \lambda_2 & -x_1\\ x_2 & \bar x_1 & -\lambda_3 \end{vmatrix}; \]
\[ \psi_2:\ \Pi_{\Gamma_+}^*\to\Pi_{\Gamma_-}^*,\qquad \rho \begin{vmatrix} \lambda_1 & x_3 & \bar x_2\\ \bar x_3 & \lambda_2 & x_1\\ x_2 & \bar x_1 & \lambda_3 \end{vmatrix} \mapsto \rho \begin{vmatrix} \lambda_1 & x_3 & \bar x_2\\ \bar x_3 & \lambda_2 & x_1\\ -x_2 & -\bar x_1 & -\lambda_3 \end{vmatrix}. \]

The hyperbolic polar transformation \(\pi_-\) is defined as follows:
\[ \pi_-:\ P\to L,\qquad x\mapsto a=\mu_2^{-1}\psi_2^{-1}\psi_1\mu_1x. \]

It turns out that

\[ x \circ a \Longleftrightarrow \pi_-^{-1}a \circ \pi_-x . \]

By the plane \({}^{1}\bar S_2(i,j,\varepsilon)\) we shall mean the plane \(P_2(i,j,\varepsilon)\) with the polarity \(\pi_-\) defined above.

In the spaces \(\bar S_2(i,j,\varepsilon)\) and \({}^{1}\bar S_2(i,j,\varepsilon)\) one can introduce the structures of semi-Riemannian spaces (see, for example, \((^2)\)), defining in them a metric as follows. Consider the mappings

\[ p_1:\bar S_2(i,j,\varepsilon)\xrightarrow{R_8}\bar S_2(i,j), \qquad p_2:{}^{1}\bar S_2(i,j,\varepsilon)\xrightarrow{{}^{4}R_8}{}^{1}\bar S_2(i,j) \]

(see \((^2,^3)\)).

Here by unitary non-Euclidean planes are meant the sets of interior points, \(R_8\) is eight-dimensional Euclidean space; \({}^{4}R_8\) is a pseudo-Euclidean space of index 4. The mappings \(p_i\) \((i=1,2)\) are fibrations. We shall denote the distance between points \(x\) and \(y\) by \(d(x,y)\). Then

\[ d(x,y)\overset{\mathrm{def}}{=} d(p_i x,p_i y), \qquad \text{if } \quad p_i x\ne p_i y, \]

\[ d(x,y)\overset{\mathrm{def}}{=} d_R(x,y), \qquad \text{if } \quad p_i x=p_i y, \]

where \(d_R(x,y)\) is the distance in the metric of the fiber.

1. A collineation \(\varphi\) of the plane \(P_2(i,j,\varepsilon)\) will mean a pair of one-to-one mappings \(\varphi=(\varphi_1,\varphi_2)\)

\[ \varphi_1:P\overset{\text{onto}}{\longrightarrow}P, \qquad \varphi_2:L\overset{\text{onto}}{\longrightarrow}L \]

such that

\[ x\circ a \Longleftrightarrow \varphi_1x\circ \varphi_2a . \]

In what follows we shall write \(\varphi x=\varphi_1x,\ \varphi a=\varphi_2a\).

A similarity of the plane \(\bar S_2(i,j,\varepsilon)\) (or \({}^{1}\bar S_2(i,j,\varepsilon)\)) is a collineation \(\varphi\) such that \(\varphi\pi_+=\pi_+\varphi\) (respectively, \(\varphi\pi_-=\pi_-\varphi\)). A motion of the planes \(\bar S_2(i,j,\varepsilon)\), \({}^{1}\bar S_2(i,j,\varepsilon)\) is a similarity preserving the structure of semi-Riemannian spaces, i.e. the metric \(d(x,y)\).

1. Theorem 1. The group of automorphisms of the algebra \(R(i,j,\varepsilon)\) is isomorphic to the group \((\bar G_2^+)_{0}\), where \(G_2^+\) is the compact group of type \(G_2\).

Theorem 2. The group of collineations of the plane \(P_2(i,j,\varepsilon)\) is isomorphic to the group \((\bar E_6^-)_{0}\), where \(E_6^-\) is the noncompact group of type \(E_6\), which is the group of collineations of the octavian projective plane \(P_2(i,j,k)\) (see, for example, \((^3)\)).

Theorem 3. The groups of similarities and motions of the plane \(\bar S_2(i,j,\varepsilon)\) are respectively isomorphic to the groups \((\bar F_4^+)_{0}\) and \((F_4^+)_{0}\), where \(F_4^+\) is the compact group of type \(F_4\).

Theorem 4. The groups of similarities and motions of the plane \({}^{1}\bar S_2(i,j,\varepsilon)\) are respectively isomorphic to the groups \((\bar F_4^-)_{0}\) and \((F_4^-)_{0}\), where \(F_4^-\) is the noncompact group of type \(F_4\), which is the group of motions of the octavian unitary hyperbolic plane \({}^{1}S_2(i,j,k)\).

1. A line \(P_1(i,j,\varepsilon)\) of the plane \(P_2(i,j,\varepsilon)\), considered as the set of points incident with one and the same line, is a cylinder \(S^4\times R^4\) with a 4-dimensional sphere \(S^4\) as base and a 4-dimensional plane \(R^4\) as generator. The line \(P_1(i,j,\varepsilon)\) may be regarded as the absolute in the semi-Euclidean space \({}^{1}S_9^5\) (see \((^4)\)). The group of motions of the space \({}^{1}S_9^5\) induces on the absolute the group \(C\) of conformal transformations, and the group of similarities induces the group \(\bar C\) of conformal transformations with similarity.

On the other hand, the line \(P_1(i,j,\varepsilon)\), considered as a submanifold of the semi-Riemannian space \(\bar S_2(i,j,\varepsilon)\) (or \({}^{1}\bar S_2(i,j,\varepsilon)\)), is isometric to a connected component of the sphere of the semi-Euclidean space \(R_9^5\) (respect-

respectively, \(^{1,0}R_9^5\)). Therefore, after identifying diametrically opposite points, the lines of the planes \(\bar S_2(i,j,\varepsilon)\), \(^{1}\bar S_2(i,j,\varepsilon)\) will be isometric respectively to the spaces \(S_8^4\) and \(^{1,0}S_8^4\) (see (4)). Moreover, the following theorems hold:

Theorem 5. The group of collineations \((\bar E_6^{-})_0\) of the plane \(P_2(i,j,\varepsilon)\) induces on the line \(P_1(i,j,\varepsilon)\) the group \(\bar C\), while the subgroup \((E_6^{-})_0\) induces the group \(C\) (at least locally).

Theorem 6. The group of similarities of the plane \(\bar S_2(i,j,\varepsilon)\) (respectively, \(^{1}S_2(i,j,\varepsilon)\)) induces on the line, as on the space \(S_8^4\) (respectively, on the space \(^{1,0}S_8^4\)), the group of similarities, and the group of motions—the group of motions (at least locally).

The author expresses his deep gratitude to I. M. Yaglom, under whose supervision this work was carried out.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
30 V 1966

CITED LITERATURE

¹ F. A. Berezin, I. M. Gelfand, Tr. Mosk. matem. obshch., 5, 311 (1956).
² B. A. Rosenfeld, L. M. Karpova, Uch. zap. Kolomensk. ped. inst., 8, 8 (1964).
³ T. A. Springer, F. D. Veldkamp, Indagations Math., 25, 413 (1963).
⁴ I. M. Yaglom, B. A. Rosenfeld, E. U. Yasinskaya, UMN, 19, No. 5, 51 (1964).