On braided Hopf structures on exterior algebras

TL;DR

This work shows that the exterior algebra $igwedge V$ of any vector space with $ ext{dim}(V) vert e 1$ supports a one-parameter family of braided Hopf structures $igwedge_p V$, where the algebra remains undeformed but the braiding is governed by a Hecke-type $R$-matrix. Through MOY calculus, the authors derive a compact, diagrammatic expression for the braiding and compute its matrix coefficients in a natural set-theoretic basis, enabling explicit R-matrix constructions from diagonal automorphisms. They formulate a right generalized Yetter–Drinfel'd module structure that yields constant solutions to the Yang–Baxter equation and relate the resulting knot invariants to the two-variable Links– Gould polynomials for $U_q( ext{gl}(N|1))$, providing detailed low-dimensional computations for $N=2,3$. The results open avenues for deeper connections between Nichols algebras, braided Hopf algebras, and quantum topology, including colored and higher-rank generalizations.

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 $N$.

Theorems & Definitions (61)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Lemma 1
• proof
• Theorem 1
• proof
• Remark 6