Tensor triangulated categories arise across many areas in mathematics. Examples of tensor triangulated categories include the derived category of perfect complexes of a suitably nice scheme X, the stable category of kG-modules where G is a finite group and k is a field of characteristic dividing the order of G, and many more. In his 2005 paper titled The spectrum of prime ideals in tensor triangulated categories, Paul Balmer associates a topological space, now called the Balmer spectrum, to tensor triangulated categories in a manner analogous to the spectrum of a commutative ring.
In the early 2000s Boe, Kujawa, and Nakano published a series of papers studying Lie superalgebras. They show that for classical Lie superalgebras 𝔤 admitting stable and polar actions of the algebraic group G0, there exist interesting subalgebras 𝔣 which ‘detect’ the cohomology of 𝔤 relative to its even part. They also consider the stable category of 𝔤 modules which are completely reducible over 𝔤0. This is a TTC, and in special cases, including for the detecting subalgebras and the Lie superalgebra 𝔤𝔩(m|n), they give a concrete description of the Balmer spectrum using methods from geometric invariant theory. In our talk we discuss how to recover this result for the detecting subalgebras using a different approach. Namely, we will discuss some circumstances in which we have a stratifying ring.
This video was part of the Southeastern Lie Theory Workshop XIII.
