Homomorphic images of pro-nilpotent algebras

TL;DR

This paper proves that finite-dimensional homomorphic images of inverse limits of nilpotent algebras are nilpotent, extending to algebras over rings, and explores related properties for solvable Lie algebras.

Contribution

It establishes that such images are nilpotent or solvable, generalizing previous results and analyzing the behavior of multiplication algebras in this context.

Findings

Finite-dimensional homomorphic images of inverse limits of nilpotent algebras are nilpotent.

Finite-dimensional homomorphic images of inverse limits of solvable Lie algebras are solvable.

Infinite-dimensional images can have properties far from nilpotency, not residually nilpotent.

Abstract

It is shown that any finite-dimensional homomorphic image of an inverse limit of nilpotent not-necessarily-associative algebras over a field is nilpotent. More generally, this is true of algebras over a general commutative ring k, with "finite-dimensional" replaced by "of finite length as a k-module". These results are obtained by considering the multiplication algebra M(A) of an algebra A (the associative algebra of k-linear maps A -> A generated by left and right multiplications by elements of A), and its behavior with respect to nilpotence, inverse limits, and homomorphic images. As a corollary, it is shown that a finite-dimensional homomorphic image of an inverse limit of finite-dimensional solvable Lie algebras over a field of characteristic 0 is solvable. Examples are given showing that infinite-dimensional homomorphic images of inverse limits of nilpotent algebras can have properties far from those of nilpotent algebras; in particular, properties that imply that they are not residually nilpotent. Several open questions and directions for further investigation are noted.