On braided Hopf structures on exterior algebras

TL;DR

This paper introduces a family of braided Hopf algebra structures on exterior algebras, computes their structure constants, and explores their connection to solutions of the Yang--Baxter equation and link invariants.

Contribution

It explicitly constructs braided Hopf structures on exterior algebras and links these to solutions of the Yang--Baxter equation and link invariants.

Findings

Existence of a one-parameter family of braided Hopf structures

Explicit computation of structure constants

Conjectured relation to Links--Gould polynomial invariants

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 $N$.