I am a math student just starting to know group algebra. Let $FG$ be a group algebra in which $F$ is a field and $G$ is a group. I am particularly interested in the results of the decomposition of $FG$. An exciting result I know is that $\mathbb{C}G$ is a direct sum of matrix rings over the complex number $\mathbb{C}$ when $G$ is a finite group. Please let me know any similar results if you know, especially sources citing.
Any comments are all appreciated and thank you in advance.