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Distributive laws and Hopf quasigroups

Ramón González Rodríguez

Abstract

In this paper we introduce the notion of $a$-monoidal distributive law between two Hopf quasigroups $A$ and $H$. We prove that every $a$-monoidal distributive law induce a product on $A\otimes H$, called the wreath product, thanks to which $A\otimes H$ becomes in a Hopf quasigroup. Finally, using this construction, we show that double cross products of Hopf quasigroups, cross products of Hopf quasigroups with a skew pairing between them, Hopf quasigroups defined by the twisted double method, smash products of Hopf quasigroups and twisted smash products of Hopf quasigroups are examples of wreath products associated to $a$-monoidal distributive laws.

Distributive laws and Hopf quasigroups

Abstract

In this paper we introduce the notion of -monoidal distributive law between two Hopf quasigroups and . We prove that every -monoidal distributive law induce a product on , called the wreath product, thanks to which becomes in a Hopf quasigroup. Finally, using this construction, we show that double cross products of Hopf quasigroups, cross products of Hopf quasigroups with a skew pairing between them, Hopf quasigroups defined by the twisted double method, smash products of Hopf quasigroups and twisted smash products of Hopf quasigroups are examples of wreath products associated to -monoidal distributive laws.
Paper Structure (3 sections, 1 theorem, 103 equations)

This paper contains 3 sections, 1 theorem, 103 equations.

Table of Contents

  1. Introduction
  2. Distributive laws between Hopf quasigroups
  3. The wreath product magma

Key Result

Theorem 3.1

Let $A$ and $H$ be Hopf quasigroups. Let $\Psi:H\otimes A\rightarrow A\otimes H$ be an $a$-comonoidal distributive law of $H$ over $A$. Then the wreath product $A\otimes_{\Psi} H$ built on $A\otimes H$ with tensor coproduct and unit, where the product and antipode are defined by (wpa1) and respectively, is a Hopf quasigroup.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 6 more