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The aim is to explore the structure and properties of weakly complete universal enveloping algebras of Lie algebras.

March 29, 2026Journal of Lie theoryOpen Access

On Weakly Complete Universal Enveloping Algebras: A Poincaré-Birkhoff-Witt Theorem

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The aim is to explore the structure and properties of weakly complete universal enveloping algebras of Lie algebras.
Introduced and analyzed the weakly complete universal enveloping algebra U(g) of profinite-dimensional Lie algebra g.
Showed that U(|g|) is a dense subalgebra generated by g in U(g).
Established the preservation of projective limits in U(g) through algebraic analysis.
Demonstrated U(g) forms a weakly complete symmetric Hopf algebra with primitive elements.
Identified a canonical pro-Lie subgroup * (g) corresponding to G(U(g)) with Lie algebra isomorphic to g.
Derived a morphism connecting weakly complete symmetric Hopf algebras U(g) and R[G] in relation to compact groups.

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This research shows the structure of weakly complete universal enveloping algebras, indicating new aspects of Lie's Third Theorem.

Hofmann et al. (2022) studied this question.

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