Homomorphic images of pro-nilpotent algebras

Details Homomorphic images of pro-nilpotent algebras Homomorphic images of pro-nilpotent algebras

2009-11-30

arxiv.org/abs/0912.0020v1

Mathematics

Rings and Algebras

18 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy

Scientific paper

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.

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