On braided Hopf structures on exterior algebras

Abstract

We show that the exterior algebra of a vector space V of dimension greater than one admits a one-parameter family of braided Hopf algebra structures, arising from its identification with a Nichols algebra. We explicitly compute the structure constants with respect to a natural set-theoretic basis.

A one-parameter family of diagonal automorphisms exists, which we use to construct solutions to the (constant) Yang–Baxter equation. These solutions are conjectured to give rise to the two-variable Links–Gould polynomial invariants associated with the super-quantum group $U_q\left(\mathrm{gl}\right(N\left|1\right))$, where $N=\dim \left(V\right)$. We support this conjecture through computations for small values of N.