# Tagged Questions

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### In a semi-simple module, any submodule is a direct factor?

I need help understanding the following : In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple) (i) How is this equivalent to the ...
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### Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L$ and $M$ in ...
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### “twisted” powers in symmetric monoidal categories

Suppose $C$ is a symmetric monoidal category with monoidal product $\wedge$, $X$ is a $G$-object for some finite group $G$ (say), and $T$ is a finite $G$-set of size $n$. The $n$-fold monoidal power ...
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### ~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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### Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
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### Isomorphism $\text {Rep}_{G,k}\cong \space \text {Mod}_{k[G]}$

Let $G$ be a group and $k$ a field. Proposition: There is an isomorphism of categories $F:\text {Rep}_{G,k}\rightarrow \space \text {Mod}_{k[G]}$. I begun by proving that there is a functor ...
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### Category of Lie group representations equivalent to the category of representations of their Lie algebra

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra. Consider the category $Rep(G)$ of finite dimensional representations of $G$ and the category $Rep(\mathfrak{g})$ of finite dimensional ...
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