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Semisimplicity and separability for pseudocompact algebras

Kostiantyn Iusenko, John MacQuarrie

TL;DR

The paper develops a robust theory of semisimple and separable pseudocompact algebras, extending finite-dimensional Artin–Wedderburn structure to the topological setting. It centers on the topological Jacobson radical $J(A)$, establishing equivalent characterizations of topological semisimplicity and showing that semisimple pseudocompact algebras are precisely products of matrix algebras over finite division algebras. It then defines separable pseudocompact algebras and proves a Wedderburn–Malcev splitting: if $A/J(A)$ is separable, then $A$ splits as $A = S \oplus J(A)$ with splittings unique up to conjugation by $1+\omega$, $\omega\in J(A)$. The results unify and extend the finite-dimensional theory, offering structural tools for pseudocompact modules and completed group algebras, and linking to coalgebra duality and representation theory of profinite groups.

Abstract

In this expository article, we give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such algebras analogous to those for finite dimensional algebras. We give a self-contained proof of the Wedderburn-Malcev Theorem for pseudocompact algebras.

Semisimplicity and separability for pseudocompact algebras

TL;DR

The paper develops a robust theory of semisimple and separable pseudocompact algebras, extending finite-dimensional Artin–Wedderburn structure to the topological setting. It centers on the topological Jacobson radical , establishing equivalent characterizations of topological semisimplicity and showing that semisimple pseudocompact algebras are precisely products of matrix algebras over finite division algebras. It then defines separable pseudocompact algebras and proves a Wedderburn–Malcev splitting: if is separable, then splits as with splittings unique up to conjugation by , . The results unify and extend the finite-dimensional theory, offering structural tools for pseudocompact modules and completed group algebras, and linking to coalgebra duality and representation theory of profinite groups.

Abstract

In this expository article, we give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such algebras analogous to those for finite dimensional algebras. We give a self-contained proof of the Wedderburn-Malcev Theorem for pseudocompact algebras.
Paper Structure (9 sections, 16 theorems, 50 equations)

This paper contains 9 sections, 16 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Jacobson radical and semisimple pseudocompact algebras
  4. The Jacobson radical
  5. Topologically semisimple algebras
  6. Separable pseudocompact algebras and the Wedderburn-Malcev Theorem
  7. Separable pseudocompact algebras
  8. The Wedderburn splitting theorem
  9. The Malcev uniqueness theorem

Key Result

Lemma 2.7

Let $U$ be a pseudocompact vector space. Then $U$ is linearly compact, meaning that if $\mathcal{X}$ is a collection of closed affine subspaces of $U$ and if $\bigcap_{X\in \mathcal{X}}X = \varnothing$, then there is a finite subcollection $X_1,\hdots,X_n\in \mathcal{X}$ such that $X_1\cap \hdots\ca

Theorems & Definitions (46)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8: adaptation of Proposition 0.3.3 from wilson
  • proof
  • ...and 36 more