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everyone!
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!

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Yes there is a close connection. Suppose $G$ is a group and $k$ is a field. Then there is a 1-1 correspondence between

  1. Linear representations of $G$ on vector spaces over $k$.
  2. 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.

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  • $\begingroup$ I know Group ring.But I think if G is a Lie groups or an algebraic groups,the notation of Group rings looks strange since it is too large and G has some Geometric structure in these cases..... $\endgroup$ – OptimusPrime Nov 8 '16 at 6:39
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    $\begingroup$ Do you know the convolution algebra $L^1(G)$ or the group C*-algebra? For a locally compact group, these give you ways to study the unitary representation theory through representations of associated algebras. $\endgroup$ – Mike F Nov 8 '16 at 6:44

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