Answer to Question #17183 in Abstract Algebra for Melvin Henriksen

Question

Let k be a field of characteristic zero, and let R be the Weyl algebra A1(k) with generators x, y and relation xy − yx = 1. Let p(y) ∈ k[y] be a fixed polynomial. Show that R → End(Vk) is injective but not an isomorphism.

Answer 1

The natural map α : R → End(Vk) must beinjective since R is a simple ring. However, since Vk is infinite- imensional, End(Vk) is not a simple ring. Therefore, ϕ is not an isomorphism.