Braided Hopf structures on exterior algebras

I will explain how the theory of Nichols algebras allows one to endow the exterior algebra of a vector space of dimension greater than one with a one-parameter family of braided Hopf algebra structures. The structure constants with respect to a natural set-theoretic basis can be explicitly computed, although determining the braiding structure in explicit terms is rather challenging. There exists a one-parameter family of diagonal automorphisms, which make it possible to construct solutions to the (constant) Yang--Baxter equation. These solutions conjecturally give rise to the two-variable Links--Gould polynomial invariants associated with the super quantum group $U_q(\mathfrak{gl}(N|1))$, where $N = \dim(V)$. This is an ongoing project in collaboration with Vladimir Mangazeev (ANU, Canberra).