Lie algebras play a fundamental role in the representation theory of groups. The connection between real and complex representations for Lie groups and their corresponding Lie algebras is understood classically, but we will be interested in representations over positive characteristic. In this setting it is important to introduce the concept of a restricted Lie algebra. A particular connection I’ve been interested in is how a given group and restricted Lie algebra can have the same abelian category of representations, but with different tensor products. A group or a restricted Lie algebra may appear innocent enough while having wild representation type, making it very difficult to say anything reasonable about these sorts of things. Nevertheless, we will do what we can, even conjecture if we must!
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