Octonions, Albert vectors and the group E ₆ ( <i>F</i> ) (Q6561442)

The paper under review presents a uniform and self-contained approach to the construction of the groups of type \( E_{6} \) over arbitrary fields without using Lie theory. In brief, the main points of the construction can be exposed as follows.\N\NLet \(V\) be a vector space of finite dimension over a field \(F\). After giving concise historical remarks concerning the development of the subject considered, the authors begin with a discussion of some properties of the group of nonsingular linear transformations of \(V\) that preserve a non-singular quadratic form defined on \(V\). Further, let \( \mathbb{O} \) be an octonion algebra over \(F\) and let \( ^{-}\colon \mathbb{O} \rightarrow \mathbb{O} \) be the conjugation (standard involution) in \( \mathbb{O} \). Then, the Albert space \( \mathbb{J} \) is the \(27\)-dimensional vector space over \(F\) spanned by the elements of the form\N\[\N( a, b, c \mid A, B, C) = \begin{bmatrix} a & C & \bar{B} \\\N\bar{C} & b & A \\\NB & \bar{A} & c \end{bmatrix},\N\]\Nwhere \( a, b, c, A, B, C\in \mathbb{O} \) and \( a, b, c \) belong to the subspace of \( \mathbb{O} \) generated by the identity element \( 1_{\mathbb{O}} \) of the octonion algebra \( \mathbb{O} \). The elements \( ( a, b, c \mid A, B, C) \) of \( \mathbb{J} \) are called Albert vectors. If \(T\) denotes the trace function on \( \mathbb{O} \), then the Dickson-Freudenthal determinant \( \Delta ( X ) \) of the element \( X = ( a, b, c \mid A, B, C) \) is defined as\N\[\N\Delta ( X ) = abc - aA\bar{A} - bB\bar{B} - cC\bar{C} + T ( ABC),\N\]\Nwhere the last term has sense because \( T ( (AB)C) = T ( A(BC) ) \). This \( \Delta ( X ) \) is a cubic form on \( \mathbb{J} \). The group \( SE_{6} ( F ) \) is defined to be the group of all \(F\)-linear maps on \( \mathbb{J} \) preserving the Dickson-Freudenthal determinant. The group \( E_{6} ( F ) \) is the quotient of \( SE_{6} ( F ) \) by its center.\N\NThe mixed form is the map \( M\colon \mathbb{J}\times\mathbb{J}\rightarrow F \) such that if \( X = ( a, b, c \mid A, B, C) \) and \( Y = ( d, e, h \mid D, E, H) \), then\N\begin{align*}\NM ( Y, X ) = bcd + ace + abh - dA\bar{A} - eB\bar{B} - hC\bar{C} - a ( D\bar{A} + A\bar{D} ) \\\N- b ( E\bar{B} + B\bar{E} ) - C ( H\bar{C} + C\bar{H} ) + T ( DBC + ECA + HAB).\N\end{align*}\NIf the field \(F\) contains more than two elements, then the mixed form can be obtained from the Dickson-Freudenthal determinant in view of the equality\N\[\NM ( X, Y) = \frac{1}{\alpha ( \alpha - 1 )} \Delta ( X + \alpha Y ) - \frac{1}{\alpha - 1 } \Delta ( X + Y ) + \frac{1}{\alpha} \Delta ( X ) - ( \alpha + 1 ) \Delta ( Y ),\N\]\Nwhere \( \alpha \) is an element of \(F\) different from 0 and 1.\N\NThe following notions are borrowed from [\textit{A. Cohen} and \textit{B. Cooperstein}, Geom. Dedicata 25, 467--480 (1988; Zbl 0643.20025)] to distinguish non-zero elements of \( \mathbb{J} \). Namely, a non-zero Albert vector \(X\in \mathbb{J}\) is called white if \( M ( Y, X ) = 0 \) for all \( Y\in \mathbb{J} \); grey if \( \Delta ( X ) = 0 \) and there exists \( Y\in \mathbb{J} \) such that \( M ( Y, X ) \not= 0 \); black if \( \Delta ( X ) \not= 0 \) and \(X\) is not white. A white (resp., grey, black) point is a 1-dimensional subspace of \(\mathbb{J}\) spanned by a white (resp., grey, black) vector.\N\NIn the paper, the authors are mostly interested in the action of \( SE_{6} ( F ) \) on white points. In particular, it is shown that the action on white points is primitive. Also it is proved (Theorem 9.1) that if \( \mathbb{O} \) is a split octonion algebra, then the stabilizer of a white point in \( SE_{6} ( F ) \) is isomorphic to \( F^{16} : \mathrm{Spin}_{10}^{+} ( F )\! \,. F^{\times} \), where \( F^{\times} \) is the multiplicative group of the field \(F\), \( A : B \) denotes a semidirect product with a normal subgroup \(A\) and a subgroup \(B\), and \( A.B \) is an unspecified group extension.\N\NThe authors suggest a way which allows them to encode elements of \( SE_{6} ( F ) \) by \( 3\times 3 \) matrices over \( \mathbb{O} \). This, together with the action on white points, leads to obtaining the fact that the group \( E_{6} ( F ) \) is simple. In the final two sections of the paper, some of geometric assertions which involve white points are listed (Section 11) and facts dealing with the situation when the underlying octonion algebra is non-split are given (Section 12).