Exercise 26, § 1

(a) Let K be a field of characteristic $\neq 2$, E an n-dimensional vector space over K and $\Phi$ a non-degenerate symmetric bilinear form on E. Suppose that E has an orthogonal basis with respect to $\Phi$ and let R denote the (diagonal) matrix of $\Phi$ with respect to this basis. We know (Algebra, Chapter IX, § 4, Exercise 11) that every matrix (relative to the basis in question ) $U$ of the orthogonal group $\mathbf{G}(\Phi)$ such that $\det(I + U) \neq 0$ can be written

where $S = R(U - I)(U + I)^{-1}$ is an alternating matrix such that $\det(I - R^{-1}S) \neq 0$; conversely, for every matrix $S$ satisfying these conditions, $U = (I - R^{-1}S)^{-1}(I + R^{-1}S)$ is a matrix of $\mathbf{G}(\Phi)$ such that $\det(I + U) \neq 0$. Let $S_{ij}, S'_{ij}, S''_{ij}$ be $3n(n-1)/2$ indeterminates $(1 \leq i < j \leq n)$ and let L be a field containing the ring of formal power series K[[S_{ij}, S'_{ij}, S''_{ij}]]. If we denote by S, S', S'' the matrices

then det(I - R^{-1}S) \neq 0 and the analogues for S' and S'' hold; if we write

then also det(I + UU') \neq 0. We write

the f_{ij}(S, S') belong to the ring of formal power series K[[S_{ij}, S'_{ij}]] and, by considering F(S, S') as a formal power series with coefficients in the algebra of matrices of order n over K, we can write:

where in the elements of the matrix o_2(S, S') the terms are all of degree \geq 1 in the S_{ij} and of degree \geq 1 in the S'_{ij}; then similarly:

the elements of o'_2(S) being formal power series of order \geq 2. Show that F(S, S') is a formal group law over K of dimension n(n - 1)/2; this formal group is called the formal orthogonal group (corresponding to \Phi) and is denoted by G(\Phi) by an abuse of notation.

(b) If we write M(S) = (m_{ij}(S)) = (I - R^{-1}S)^{-1}(I + R^{-1}S), M is a formal homomorphism of G(\Phi) into GL(n, K). Let \mathfrak{o}(\Phi) be the Lie algebra of G(\Phi) and let (T_{ij})_{i < j} denote the basis of this algebra such that T^{(e)}_{ij} = D^{(e)}_{ij} (cf. Exercise 24 (d)); if an element D = \sum_{i < j} c_{ij} T_{ij} of \mathfrak{o}(\Phi) is identified with the matrix $D = \sum_{i < j} c_{ij}(E_{ij} - E_{ji})$, show that (in the notation of Exercise 24 (d) with the identification made in Exercise 25 of the Lie algebra of GL(n, K) with gl(n, K)):

Deduce that M* is an isomorphism of \mathfrak{o}(\Phi) onto the subalgebra of gl(n, K) consisting of the matrices X such that ^tXR + RX = 0 (for every ordered pair (a, b) of elements of E, identified with matrices with one column, consider \Phi(a + M(S).a, b + M(S).b) = ^ta.(I + ^tM(S))R(I + M(S)).b as a formal power series reduced to its constant term and use the fact that the subalgebra of X \in gl(n, K) such that ^tXR + RX = 0 is of dimension n(n - 1)/2).

(c) Define similarly the formal symplectic group in 2n variables over an arbitrary (commutative) field and show that its Lie algebra is identified with the subalgebra of $\mathfrak{gl}(2n, \mathbf{K})$ consisting of the matrices $X$ such that