Stable flatness of nonarchimedean hyperenveloping algebras

TL;DR

This paper proves that the hyperenveloping algebra of a p-adic Lie algebra is stably flat, ensuring its cohomology aligns with the Lie algebra cohomology, which advances understanding of nonarchimedean algebraic structures.

Contribution

It establishes the stable flatness of hyperenveloping algebras over p-adic fields, linking their cohomology to Lie algebra cohomology, a novel result in nonarchimedean analysis.

Findings

Hyperenveloping algebra F(g) is stably flat over its universal enveloping algebra.

Relative cohomology of F(g) matches Lie algebra cohomology.

Provides foundational results for nonarchimedean Lie theory.

Abstract

Let L be a p-adic local field and g a finite dimensional Lie algebra over L. We show that its hyperenveloping algebra F(g) is a stably flat completion of its universal enveloping algebra. As a consequence the relative cohomology for the locally convex algebra F(g) coincides with the underlying Lie algebra cohomology. Final version. Some minor items corrected. Appeared in Journal of Algebra (2010).