Lie algebra of the DSER elementary orthogonal group (Q6572371)

Let \(R\) be a commutative ring and \((Q,q)\) be a quadratic space over \(R\). For a finitely generated projective \(R\)-module \(P\), let \({\mathbb H}(P)\) denote the hyperbolic space of \(P\), that is, \((P\oplus P^*, p)\) with \(p((x,f))=f(x)\). \N\N\textit{A. Roy} [J. Algebra 10, 286--298 (1968; Zbl 0181.04302)] defines the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal \(EO_R(Q,{\mathbb H}(P))\) as the subgroup of the orthogonal group \(O_R(Q\perp{\mathbb H}(P))\) generated by orthogonal transformations \(E_\alpha\) and \(E_\beta\) of \(Q\perp{\mathbb H}(P)\). \N\NIn this paper, the Lie algebra of \(EO_R(Q,{\mathbb H}(P))\) is computed in the case of \(R={\mathbb C}\). Since \(Q\) and \(P\) are isomorphic to \({\mathbb R}^n\) and \({\mathbb R}^m\), respectively, the Lie algebra consists of \((n+2m)\times(n+2m)\) matrices \(X\) satisfying \(\exp(tX)\in EO_R(Q,{\mathbb H}(P))\) for all \(t\in{\mathbb R}\). The explicit forms of generators of the Lie algebra, that is, \(\log\) for \(E_\alpha\) and \(E_\beta\), are obtained.