Continuity of Homomorphisms on Pro-nilpotent Algebras

Let V be a variety of not necessarily associative algebras, and A an inverse limit of nilpotent algebras Ai ∈V, such that some finitely generated subalgebra S ⊆A is dense in A under the inverse limit of the discrete topologies on the Ai. A sufficient condition on V is obtained for all algebra homomorphisms from A to finite-dimensional algebras B to be continuous; in other words, for the kernels of all such homomorphisms to be open ideals. This condition is satisfied, in particular, if V is the variety of associative, Lie, or Jordan algebras. Examples are given showing the need for our hypotheses, and some open questions are noted. 1. Background: From pro-p groups to pro-nilpotent algebras A result of Serre’s on topological groups says that if G is a pro-p group (an inverse limit of finite p-groups) which is topologically finitely generated (i.e., has a finitely generated subgroup which is dense in G under the inverse limit topology), then any homomorphism from G to a finite group H is continuous ([6, Theorem 1.17], [14, Section I.4.2, Exercises 5–6, p. 32]). Two key steps in the proof are that (i) every finite homomorphic image of a pro-p group G is a p-group, and (ii) for G a topologically finitely generated pro-p group, its subgroup G[G,G] is closed. In [2], we obtained a result similar to (i), namely, that if A is a pronilpotent (not necessarily associative) algebra over a field k, then every finitedimensional homomorphic image of A is nilpotent. The analog of (ii) would Received July 1, 2010; received in final form August 23, 2010. Any updates, errata, related references, etc., learned of after publication of this note will be recorded at http://math.berkeley.edu/~gbergman/papers/. 2010 Mathematics Subject Classification. Primary 17A01, 18A30, 49S10. Secondary 16W80, 17B99, 17C99. 749 c ©2013 University of Illinois