Linear maps on k^I, and homomorphic images of infinite direct product algebras

TL;DR

This paper investigates the structure of linear maps on infinite product spaces over an infinite field and shows that under certain size constraints, homomorphisms from infinite product algebras essentially factor through finitely many components, with specific implications for Lie algebras.

Contribution

It establishes conditions under which kernels of linear maps contain elements with finitely many nonzero components and demonstrates that homomorphisms from infinite product algebras factor through finitely many factors under size constraints.

Findings

Kernels of linear maps contain elements with finitely many nonzero components.

Homomorphisms from infinite product algebras factor through finitely many factors.

Results have specific implications for Lie algebras.

Abstract

Let k be an infinite field, I an infinite set, V a k-vector-space, and g:k^I\to V a k-linear map. It is shown that if dim_k(V) is not too large (under various hypotheses on card(k) and card(I), if it is finite, respectively countable, respectively < card(k)), then ker(g) must contain elements (u_i)_{i\in I} with all but finitely many components u_i nonzero. These results are used to prove that any homomorphism from a direct product \prod_I A_i of not-necessarily-associative algebras A_i onto an algebra B, where dim_k(B) is not too large (in the same senses) must factor through the projection of \prod_I A_i onto the product of finitely many of the A_i, modulo a map into the subalgebra \{b\in B | bB=Bb=\{0\}\}\subseteq B. Detailed consequences are noted in the case where the A_i are Lie algebras.