Question #23881

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?

Expert's answer

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*^{−}^{1}*v*).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 *not*a 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.

G-fixed points in W.

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