The Arens-Michael envelope of a solvable Lie algebra is a homological epimorphism

Technical Summary: The Arens–Michael Envelope of a Solvable Lie Algebra is a Homological Epimorphism

Problem Statement
The paper addresses a question regarding the homological properties of the Arens–Michael envelope of the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional complex Lie algebra $\mathfrak{g}$. Specifically, it investigates under what conditions the canonical homomorphism $\iota: U(\mathfrak{g}) \to \widehat{U}(\mathfrak{g})$ (where $\widehat{U}(\mathfrak{g})$ denotes the Arens–Michael envelope) constitutes a homological epimorphism.

A homological epimorphism is a concept introduced by J. Taylor as a weakened form of flatness (often termed "absolute localization" or "stably flat"). Its primary significance lies in the preservation of Hochschild homologies and cohomologies. While Taylor established that this property holds for abelian Lie algebras and fails for semisimple ones, and Pirkovskii later proved that solvability is a necessary condition, the sufficiency of solvability remained an open problem for general solvable (non-nilpotent) Lie algebras, with only specific low-dimensional cases previously resolved.

Methodology
The author employs a combination of functional analytic techniques and relative homological algebra to prove the sufficiency of solvability. The core strategy involves an inductive approach based on the structural decomposition of solvable Lie algebras.

1. Structural Decomposition: The proof relies on a decomposition of the solvable Lie algebra $\mathfrak{g}$ into an iterated semidirect sum of one-dimensional subalgebras. This induces a corresponding decomposition of the universal enveloping algebra $U(\mathfrak{g})$ into an iterated smash product (specifically, an iterated Ore extension structure).
2. Analytic Smash Products: Instead of relying solely on analytic Ore extensions (as used in previous work on low-dimensional cases), the author utilizes analytic smash products ($A \mathbin{\hat{\#}} H$) involving Hopf $\hat{\otimes}$-algebras. The Arens–Michael envelope $\widehat{U}(\mathfrak{g})$ is shown to be isomorphic to an iterated analytic smash product of specific algebras $A_s$ (completions of polynomial rings with specific seminorms).
3. Relative Homological Epimorphisms: The paper adapts a technique developed by Pirkovskii involving relative homological epimorphisms. The goal is to show that the map between the iterated smash products of the enveloping algebra and its envelope is a homological epimorphism at each step of the induction.
4. Key Technical Tools:
   • Property (UDE): The "Unique Extension Property" for derivations is central. The paper proves that certain homomorphisms satisfy this property and that it is preserved under pushouts and analytic smash products.
   • Pushout Diagrams: The author utilizes pushouts in the category of $\hat{\otimes}$-algebras to establish that the property of being a homological epimorphism is preserved when moving from the algebraic level to the analytic completion level.
   • Relatively Quasi-Free Algebras: The proof leverages the fact that Ore extensions are relatively quasi-free, which simplifies the verification of homological conditions.

Key Contributions and Results
The primary result of the paper is Theorem 4.1, which states:

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. Then the Arens–Michael envelope homomorphism $U(\mathfrak{g}) \to \widehat{U}(\mathfrak{g})$ is a homological epimorphism if and only if $\mathfrak{g}$ is solvable.

This result completes the classification initiated by Taylor and Pirkovskii. The specific technical contributions include:

• Generalization of Low-Dimensional Cases: The paper extends the results previously known for the two-dimensional non-abelian algebra and the three-dimensional Heisenberg algebra to all finite-dimensional solvable Lie algebras.
• Theorem 4.2: A new result establishing that the analytic smash product of two homological epimorphisms (with dense range) is itself a homological epimorphism. This generalizes previous theorems regarding Ore extensions.
• Theorem 4.3: A generalization of Pirkovskii's Theorem 1.2, providing sufficient conditions for a homomorphism to be a two-sided relative homological epimorphism based on the Unique Extension Property (UDE) and relative quasi-freeness, without requiring the map to be an Arens–Michael envelope at intermediate steps.
• Preservation of (UDE): The paper proves that the Unique Extension Property for derivations is preserved under pushouts and analytic smash products (Theorem 3.1), a crucial step for the inductive argument.

Significance and Claims
The author claims that this work provides the final answer to the question of when the Arens–Michael envelope is a homological epimorphism for universal enveloping algebras. The sufficiency of solvability is established through a rigorous inductive argument that overcomes the limitations of previous methods, which could not directly handle the general high-dimensional solvable case due to the failure of intermediate completions to be analytic Ore extensions.

The paper positions itself as a component of a larger project concerning completions of universal enveloping algebras and algebras of analytic functionals. While the current text focuses on the homological properties of the enveloping algebra, the author notes that the decomposition techniques (iterated analytic smash products) developed here are applicable to other completions and will be utilized in future work regarding the homological properties of algebras of analytic functionals.

The author acknowledges that a previous proof attempt (announced in 2018) contained a gap, and the current text presents a different, corrected argument relying on the interplay between analytic smash products and relative homological epimorphisms.