sl2-crystals and duality in monoidal categories ( joint w. T. Zorman)
Sebastian Halbig (Marburg)

Abstract: A classical result of representation theory states that for a module M over a commutative ring R the following are equivalent:

(a) M is finitely-generated projective,

(b) M has a dual M* together with a canonical pairing and a "dual basis" subject to natural snake identities, and

(c) tensoring with M is left adjoint to tensoring with an object M*.

This close relationship between rigidity (condition (b)) and tensor-representability of the internal-hom (condition (c)) prompted Heunen to ask whether tensor-representability and rigidity are always equivalent. We answer the question in the negative by studying sl2-crystals. Moreover, we show that these categories admit a more flexible notion of dualisability known as *-autonomy or Grothendieck–Verdier duality. We will show that this notion is abundant in representation theory and occurs for example for representations of quiver algebras, Mackey-functors, and crossed modules.