Separable cowreaths in higher dimension

TL;DR

This paper constructs an infinite family of higher-dimensional separable cowreaths using Clifford algebras and generalized Hopf algebras, extending previous results to more complex algebraic structures.

Contribution

It introduces a method to generate higher-dimensional separable cowreaths by combining Clifford algebras with generalized Hopf algebras, expanding the class of known examples.

Findings

Constructed infinite families of separable cowreaths in higher dimensions.

Extended previous results from 4-dimensional cases to 2^{n+1}-dimensional cases.

Derived conditions on scalars for separability morphisms in these structures.

Abstract

In this paper we present an infinite family of (h-)separable cowreaths with increasing dimension. Menini and Torrecillas proved in [20] that for $A=Cl(\alpha,\beta, \gamma)$, a four-dimensional Clifford algebra, and $H=H_4$, Sweedler's Hopf algebra, the cowreath $(A \otimes H^{op},H, \psi)$ is always (h-)separable. We show how to produce similar examples in higher dimension by considering a $2^{n+1}$-dimensional Clifford algebra $A=Cl(\alpha,\beta_i,\gamma_i,\lambda_{ij})$ and $H=E(n)$, a suitable pointed Hopf algebra that generalizes $H_4$. We adopt the approach pursued in [19], requiring that the separability morphism be of a simplified form, which in turn forces the defining scalars $\alpha,\beta_i,\gamma_i,\lambda_{ij}$ to satisfy further conditions.