I have known that there are two kinds of representation theory:
The first is "the representation theory of algebas",this kind of representation theory use a lot of homological methods(Representations are exactly modules;representations of quivers......).
The second kind of representation theory is more about "representations of groups"(Algebraic groups,Lie groups,$GL_n(F_q)$.....).This kind of representation theory use a lot of tools in analysis and number theory(L-functions,modular forms,Harmonic analysis.....).It looks like that these two kind of theory are different.So is there any natural connection between them?
Thanks a lot!
Yes there is a close connection. Suppose $G$ is a group and $k$ is a field. Then there is a 1-1 correspondence between
- Linear representations of $G$ on vector spaces over $k$.
- Left $k[G]$-modules, where $k[G]$ is the group ring of $G$ coefficents in $k$. (Note: we should require that $1$ acts as the identity). Since you mentioned algebras, note you can think of $k[G]$ as a $k$-algebra.
The correspondence comes from the copy $G \subset k[G]$. To go from 1 to 2, we extend a representation by linearity. To go from 2 to 1, we just have to restrict.