Decomposition of the Rational Group Algebra into Endomorphisms of irreducible representations I am a third year math undergraduate student. I am currently taking a graduate course in Abstract Analysis and Representation theory and I came across this nice proposition in chapter 3 of Fulton William, Harris Joe : Representation Theory a first course [1]. The proposition (3.29) states that the direct sum of the endomorphisms of the irreducible representations of the group is isomorphic to the complex group algebra CG (it basically gives you the decomposition of the group algebra CG). I was wondering if there is a generalization that allows me to do this for other group algebras where the field is no longer Complex numbers or not even semisimple algebra. Is there a generalization that exists to Rational Numbers?
 A: It should work for any "splitting field" (i.e. as long as the complex irreps are realizable via matrices over a smaller field). It is sufficent that your field $\mathbb{K}$ contains all $e$th roots of unity, where $e$ is the "exponent" of $G$, i.e. the LCM of orders of all $g\in G$. (Note $e$ may be smaller than $n=|G|$.) For smaller fields, as long as $\mathbb{K}[G]$ is still semisimple (i.e. as long as the characteristic of the field does not divide the order of $G$, so Maschke's theorem applies), the algebra $A=\mathbb{K}[G]$ may be written as a direct sum $\bigoplus n_iV_i$ of irreps $V_i$ with multiplicities $n_i$ (i.e. $nV:=\underbrace{V\oplus\cdots\oplus V}_n$ here).
Then the algebra $A$ according to the Artin-Wedderburn theorem is $A\cong \bigoplus \mathrm{End}(n_iV_i)$. Note it is relevant for this that $\hom(V_i,V_j)=0$ for distinct irreps ($i\ne j$). Then, in turn, $\mathrm{End}(nV)$ may be written as $n\times n$ matrices with entries that are themselves from the algebra $D=\mathrm{End}(V)$. When $\mathbb{K}$ is a splitting field, Schur's lemma says every element of $D$ is an isomorphism (of $V$) hence $D=\mathbb{K}$. However, otherwise, $D$ is simply a division algebra over $\mathbb{K}$ (which could be a field extension of $\mathbb{K}$, or a quaternion algebra, or whatever). Thus
$$ A\cong \bigoplus_i M_{n_i}(D_i) $$
where $D_i=\mathrm{End}(V_i)$. (I am sweeping some stuff about "opposite algebras" under the rug.)
