Homomorphic images of pro-nilpotent algebras

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.

It is also shown by example that infinite-dimensional homomorphic images of pro-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.