Answer to Question #23881 in Abstract Algebra for Irvin

Question

Let U, V be simple modules of infinite k-dimensions over a group algebra kG (where k is a field), and let W = U ⊗k V , viewed as a kG-module under the diagonal G-action. Does W have finite length?

Answer 1

The answer is “no”, by acounterexample. Let G be the multiplicative group of any infinite fieldextension K of k. Let U (resp. V ) be a copy of K(as k-vector space) with G-action defined by g ∗ u = gu (resp. g ∗ v = g^−1v).Clearly, U, V are simple kG-modules. If we view K asa kG-module with the trivial G-action, the k-map
ϕ : U ⊗k V → K induced by u ⊗ v → uv is easily checked to be a kG-epimorphism.Since K has infinite length as a kG-module, so does W = U⊗k V . We finish by checking that W is nota semisimple kG-module. Indeed, if W is semisimple, ϕ would be a split kG-epimormorphism,so W would have a submodule with trivial G-action mappingisomorphically onto K. We have contradiction there are actually nononzero
G-fixed points in W.