Let be a group and an algebraically closed field of characteristic . If is a finite-dimensional representation of over , then by the classical Krull–Schmidt theorem, the th tensor power of can be uniquely decomposed into a direct sum of indecomposable representations. But we know very little about this decomposition, even for very small groups, such as for or for .

For example, what can we say about the number of summands with dimension coprime to ? It is easy to show that there is a finite limit , but what kind of number is this? Is it algebraic or transcendental? Until recently, there were no techniques to solve such questions (and in particular the same question about the sum of dimensions of these summands is still wide open). Remarkably, a new subject which may be called "Lie theory in tensor categories" gives methods to show that is indeed an algebraic number, which moreover has the form where is a natural number, is a particular root of unity, and is a -number. Moreover, and , so is a character of the Green ring of over . Finally, , for some , and we can give lower bounds for . In the talk, I will explain what Lie theory in tensor categories is and how it can be applied to such problems. This is joint work with K. Coulembier and V. Ostrik.