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](https://en.wikipedia.org/wiki/Character_theory#Properties) , 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?



Character theory of representations of infinite groups
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.


