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For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in the complex representation theory of finite groups $G$.

Every now and then, I encountered rings whose structures are more loose. For example, $\mathbb{F}[G]$, where the field isn't a nice as $\mathbb{C}$. Another example is the Borel-Moore homology $H(Z;\mathbb{C})$ of the Steinberg variety $Z$, which by Deligne's decomposition theorem is semisimple modulo its nilradical [1, (8.6.11)].

I thus wish to expand my knowledge beyond A-W theorem, but I'm also aware that the theory could be less organized. Would you mind providing some pointers, or even better a road map? It'd be great if the abstract theory is accompany with some applications to important works like the above.

Reference

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    $\begingroup$ For finite dimensional algebras up to morita equivalence you can assume your algebra comes from a quiver with relations. And this is a big subject. Ralf Schiffler has a nice introduction $\endgroup$
    – Benjamin Steinberg
    Commented Jan 10, 2021 at 17:16
  • $\begingroup$ The introduction referenced by @BenjaminSteinberg (probably): Ralf Schiffler - Quiver representations. $\endgroup$
    – LSpice
    Commented Jan 10, 2021 at 20:21
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    $\begingroup$ @LSpice, yes thanks. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 10, 2021 at 20:44
  • $\begingroup$ For semisimple algebras over a field $\mathbb F$, like $\mathbb F[G]$ (at least when the order of $G$ is nonzero in $F$), the key thing to generalize the Artin-Wedderburn picture is to understand the division algebras over the field, i.e. the central simple algebras over the extension fields. These are classified by the Brauer group of their extension fields. These concepts are presumably all points on the road map... $\endgroup$
    – Will Sawin
    Commented Jan 10, 2021 at 22:05

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