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 $U_q(\mathfrak{gl}(N|1))$, where $N=\dim (V)$. We support this conjecture through computations for small values of $N$.

27 pages, this is a major revision with all proofs and new results added

References in corpus (3)

• Spiders for rank 2 Lie algebras
• Webs and quantum skew Howe duality
• Nichols algebras that are quantum planes