Stably flat completions of universal enveloping algebras
Volume 441 / 2006
Abstract
\def\Aut{\mathop{\rm Aut}\nolimits}% \def\fg{{\got g}}% \def\wh{\widehat}% \def\A{{\scr A}}% \def\Ptens{\mathop{\widehat\otimes}}Let $\fg$ be a complex Lie algebra, and let $U(\fg)$ be its universal enveloping algebra. We study homological properties of topological Hopf algebras containing $U(\fg)$ as a dense subalgebra. Specifically, let $\theta\colon U(\fg)\to H$ be a homomorphism to a topological Hopf algebra $H$. Assuming that $H$ is either a nuclear Fréchet space or a nuclear (DF)-space, we formulate conditions on the dual algebra, $H'$, that are sufficient for $H$ to be stably flat over $U(\fg)$ in the sense of A.~Neeman and A.~Ranicki (2001) (i.e., for $\theta$ to be a localization in the sense of J.~L.~Taylor (1972)). As an application, we prove that the Arens–Michael envelope, $\wh{U}(\fg)$, of $U(\fg)$ is stably flat over $U(\fg)$ provided $\fg$ admits a positive grading. We also show that R.~Goodman's (1979) weighted completions of $U(\fg)$ are stably flat over $U(\fg)$ for each nilpotent Lie algebra $\fg$, and that P.~K.~Rashevskii's (1966) hyperenveloping algebra is stably flat over $U(\fg)$ for any $\fg$. Finally, the algebra $\A(G)$ of analytic functionals (introduced by G.~L.~Litvinov (1969)) on the corresponding connected, simply connected complex Lie group $G$ is shown to be stably flat over $U(\fg)$ precisely when $\fg$ is solvable.
