Steinberg groups for Jordan pairs
Given a locally finite root system \(\phi\), for any two elements \(\alpha\) and \(\beta\) in \(\phi\), the authors define \(((\alpha,\beta)):=\phi\cap(\mathbb N_+\alpha+\mathbb N_+\beta)\) where \(\mathbb N_+:=\{x\in\mathbb N: x>0\}\). A pair \((\alpha, \beta)\) of roots is said to be nilpotent if \(p\alpha+q\beta\neq 0\) for all \(p,q\in\mathbb N_+\). A \textsl{group with commutator relations} of type \(\phi\) is a group \(G\) provided with a family of subgroups \((U_\alpha)_{\alpha\in\phi}\) (called the \textsl{root groups}) which generate \(G\) and for which the commutator relations \(((U_\alpha,U_\beta))\subset U_{((\alpha,\beta))}\) hold for all nilpotent pairs \((\alpha,\beta)\). The notations \(U_{((\alpha,\beta))}\) stands for the subgroup of \(G\) generated by all \(U_\gamma\) where \(\gamma\in((\alpha,\beta))\) (plus some additional condition if \(\phi\) is not reduced). Given any two groups with \(\phi\)-commutator relations, the notion of a morphism \(\phi: (G,U_\alpha)\to(\tilde G, {\tilde U}_\alpha)\) is the obvious one: root groups are \(\phi\)-invariant. In the paper, the authors use the notion of \textsl{strong morphism} which roughly speaking amounts to requiring that each restriction \(\phi: U_\alpha\to {\tilde U}_\alpha\) is bijective and that \(G\) has the same generators and commutator relations than \(\tilde G\). For a fixed group with \(\phi\)-commutator relations \((\tilde G, {\tilde U}_\alpha)\) we can consider the category whose objects are the pairs \((G,s)\) where \(G=(G,U_\alpha)\) is a group with \(\phi\)-commutator relations and \(s: G\to \tilde G\) is a strong morphism. An initial element for this category is called a \textsl{Steinberg group} of \(\tilde G\). Consider a Jordan pair \(V\) over a commutative (associative) unitary ring \(k\) and its Tits-Kantor-Koecher algebra \(g(V)=g=g_{-1}\oplus g_0\oplus g_1\) (where \(g_{\pm 1}=V^{\pm}\) as \(k\)-modules.) Since \(\text{ad}(x)^3=0\) for \(x\in g_{\pm}\) we can exponentiate \(\text{ad}(x)\) and so define the \textsl{projective elementary group} of \(V\) by \(\text{PE}(V)=(\exp_+(V^+),\exp_-(V^-))\), a subgroup of the automorphism group of \(g\). The paper announces results on projective elementary groups and on Steinberg groups associated to Jordan pairs \(V\) with a grading by locally finite 3-graded root systems \(\phi\): the projective elementary group \(\text{PE}(V)\) of \(V\) is a group with \(\phi\)-commutator relations with respect to suitable root subgroups. Under certain conditions the Steinberg group associated to \(\text{PE}(V)\) covers all central extensions of \(\text{PE}(V)\) and is the universal central extension of \(\text{PE}(V)\) if \(\phi\) is irreducible and has infinite rank.
scientific article; zbMATH DE number 5779540