Structure and presentations of Lie-type groups (Q2766372)

The purpose of this work is a generalization of the class of Chevalley groups and twisted Chevalley groups in such a way that finite-dimensional classical groups over division rings, simple algebraic groups and groups of mixed type (in the terminology of \textit{J. Tits} [``Buildings of spherical type and finite BN-pairs'', Lect. Notes Math. 386, Springer (1974; Zbl 0295.20047)]) are included.NEWLINENEWLINENEWLINELet \(\mathcal B\) be a spherical, thick, irreducible building over \(I=\{1,\dots,\ell\}\), \(\ell\geq 3\), or a Moufang polygon (i.e. \(\ell=2\)). Let \(\Phi\) be a root system for a fixed apartment \(\mathcal A\) of \(\mathcal B\). For \(r\in\Phi\) denote by \(A_r\) the root subgroup on \(B\) with respect to \(r\). Then the author calls \(G=\langle A_r\mid r\in\Phi\rangle\leq\Aut({\mathcal B})\) the Lie group type \(\mathcal B\). Fundamental is the author's theory on abstract root subgroups developed in [Adv. Math. 142, No. 1, 1-150 (1999; Zbl 0934.20025)]. This leads to a uniform simplicity proof for groups of Lie type. The author defines an extension \(\widetilde\Phi\) of the root system by \(\widetilde\Phi=\Phi\cup\{2r\mid r\in\Phi,\;1\neq A_r'\}\) and defines more root subgroups with the property \(A_r'\leq A_{2r}\leq Z(A_r)\). This leads to the following generalization of the Chevalley commutator formula:NEWLINENEWLINENEWLINELet \(r,s\in\widetilde\Phi\) such that \(s\not\in\mathbb{Z} r\) then \([A_r,A_s]\leq\langle A_{\lambda r+\mu s}\mid\lambda r+\mu s\in\widetilde\Phi;\;\lambda,\mu\in\mathbb{N}\rangle\).NEWLINENEWLINENEWLINEWith the help of this commutator formula this theory is developed further in analogy to the theory of Chevalley groups: a pair of groups \(B,N\) and parabolic subgroups are introduced having similar properties as the usual BN-pairs and parabolic groups in Chevalley groups.NEWLINENEWLINENEWLINEOne of the main results of this paper is a kind of generalization of work of \textit{G. M. Seitz} [Ann. Math. (2) 97, 27-56 (1973; Zbl 0338.20052)] about flag transitive subgroups of Chevalley groups. One obtains a classification of groups \(L\) with the property \(G\leq L\leq\Aut({\mathcal B})\). For a second result the author generalizes the notion of a parabolic system and classifies Lie type groups as groups with (certain) parabolic systems. In a third result the author generalizes the so-called Curtis-Tits theorem [\textit{C. W. Curtis}, J. Reine Angew. Math. 220, 174-185 (1965; Zbl 0137.25701)]. He obtains a description of a Lie type group as \(G=\langle X_1,\dots,X_\ell\rangle\) where one assumes that \(\langle X_i,X_j\rangle=X_i*X_j\) or that \(\langle X_i,X_j\rangle\) is a central extension of a Lie type group associated with a Moufang plane or a Moufang quadrangle.