Normality of Vaserstein group
Ruddarraju Amrutha, Pratyusha Chattopadhyay
TL;DR
This work extends the Suslin–Kopeiko normality program to the Vaserstein group $V(P,IP,\langle,\rangle)$ on symplectic $R$-modules with Pfaffian-$1$ skew-symmetric forms. It develops the symplectic analogue of the polynomial-extension framework, establishes a Local-Global principle for the relative case when the underlying module is free, and proves a broad normality result: $V(P,IP,\langle,\rangle)$ is normal in $ ext{Sp}(P,\langle,\rangle)$. The approach unifies absolute and relative settings and leverages dilation-type arguments and local-global reductions to relate $V$ to the relative elementary symplectic group $ ext{ESp}_\varphi(R,I)$. Together, these results generalize classical normality theorems to projective-module contexts and contribute to K1-type questions in Serre-type conjectures.
Abstract
A.A. Suslin proved a normality theorem for an elementary linear group and V.I. Kopeiko extended this result of Suslin for a symplectic group defined with respect to the standard skew-symmetric matrix of even size. We generalized the result of Kopeiko for a symplectic group defined with respect to any invertible skew-symmetric matrix of Pfaffian one. Vaserstein group is an extension of a symplectic group defined with respect to any invertible skew-symmetric matrix of Pfaffian one in the set up of projective modules. Here we prove a normality theorem for Vaserstein group.