Character theory of representations of infinite groups I saw that "Over an algebraically closed field of characteristic 0, semisimple representations are isomorphic if and only if they have the same character" in the Wikipedia page , which does not mention the condition that the group is finite.
However, I can only find reference for the result about representations of finite group. 
Does anyone know any reference for infinite group or could someone give a proof or disproof of this argument, please?
 A: Yes, it is true, and one doesn't even need to assume the field $k$ is algebraically closed.   Section 7 of Lam's "A First Course in Noncommutative Rings" is a good reference for character theory for $k$-algebras.  In fact, Theorem 7.19 says exactly what you want:  If $M$ and $M'$ are finite-dimensional semisimple representations of a $k$-algebra $R$ with the same character, then they are isomorphic.  The only assumption on $k$ is that it has characteristic 0.  
Additionally, one sees there that in positive characteristic, over an algebraically closed field, the same result is true for simple representations (Corollary 7.21), although not for semisimple representations in general.
A: I think it is true.  First note it is well known if $A$ is a finite dimensional algebra over an algebraically closed field  $K$ of characteristic 0 then two semisimple modules are isomorphic if and only if they have the same character by Wedderburn theory.  Basically if $e$ is the primitive idempotent corresponding to the projective cover of a simple its trace will tell you the number of copies of the simple in a semisimple module. 
Now let $V,W$ be finite dimensional  semisimple $KG$-modules with the same character and let $I$ be the intersection of the annihilator ideals of these modules. Then $A=KG/I$ is finite dimensional and $V,W$ are semisimple $A$-modules with the same character so isomorphic. 
