Linear representations in Char 0

[dg0666]

In characteristic zero any linear representation of a reductive group is semisimple. Also in characteristic zero any linear representation of a finite group is semisimple (Maschke's Thm). However is any linear representation of any group semisimple in characteristic zero?

 

[matt grime]

Evidently we must try to think of a group that is not finite, nor an algebraic group (let's avoid all of them). What is there to think of? First we hit the integers under addition - it is not finite, and not defined by a finite number of equations as a subset of R/C/Q/ any field, so we've got a hope. It is generated by a single element so that is good - a rep is just assigning it to a matrix and we're done. From there you should try to think about what it means for a 2-d rep, say (hint!) to be irreducible (again, hint, do all matrices have two eigenvectors?).