On Lie Theory in the Verlinde Category

Abstract

A symmetric tensor category arises by axiomatizing the basic properties of a representation category of a finite group A famous theorem of Deligne states that, in characteristic p = 0, any symmetric tensor category of moderate growth is essentially a representation category of an affine supergroup scheme. This is not true in positive characteristic, with the most fundamental counterexamples being the Verlinde category Verₚ and its higher analogs Verₚn. It seems these categories will play a role in generalizing Deligne's theorem, and therefore, to understand symmetric tensor categories of moderate growth in general, it is important to study affine group schemes in these categories. The first part of the thesis reviews this theory.

In the remainder of the thesis, we approach the study of Verₚ by considering two perspectives: the first perspective is that because these categories do not fiber over the category of supervector spaces, these categories provide examples of new phenomena which do not arise out of (super)algebra or (super)geometry. In particular, we explain how the Verlinde category can be used to provide new constructions of Lie superalgebras, and in particular exceptional simple Lie superalgebras in low characteristic. We also show that in characteristic 5 a new algebraic structure we call a "weak Jordan algebra" arises. Finally, we classify bilinear forms in the Verlinde category Ver₄⁺ and discuss the associated Witt semi-ring, which is a new algebraic structure.

The second perspective is that these categories actually contain the category of supervector spaces, so they must generalize what is already known. We extend the theory of Frobenius kernels to the Verlinde category and use it to prove an analog of the Steinberg tensor product theorem for the group scheme GL(X).