[Paper Review] On Antipodes Of Hom-Hopf algebras

Mohammad Hassanzadeh|arXiv (Cornell University)|Mar 5, 2018

Advanced Topics in Algebra20 references3 citations

TL;DR

This paper investigates the antipode in Hom-Hopf algebras under a weakened definition where the antipode is a relative Hom-inverse of the identity under convolution, rather than a strict inverse. It establishes that the antipode is a relative Hom-anti-algebra and anti-coalgebra morphism, and if twisting maps are invertible, it becomes a true anti-algebra and anti-coalgebra map. The key result is that for commutative or cocommutative Hom-Hopf algebras with invertible twisting maps, $ S^2 = \mathrm{id} $, generalizing the classical Hopf algebra property.

ABSTRACT

In the recent definition of Hom-Hopf algebras the antipode S is the relative Hominverse of the identity map with respect to the convolution product. We observe that some fundamental properties of the antipode of Hopf algebras and Hom-Hopf algebras, with the original definition, do not hold generally in the new setting. We show that the antipode is a relative Hom-anti algebra and a relative anti-coalgebra morphism. It is also relative Hom-unital, and relative Hom-counital. Furthermore if the twisting maps of multiplications and comultiplications are invertible then S is an anti-algebra and an anti-coalgebra map. We show that any Hom-bialgebra map between two Hom-Hopf algebras is a relative Hom-morphism of Hom-Hopf alegbras. Specially if the corresponding twisting maps are all invertible then it is a Hom-Hopf algebra map. If the Hom-Hopf algebra is commutative or cocommutative we observe that S^2 is equal to the identity map in some sense. At the end we study the images of primitive and group-like elements under the antipode.

Motivation & Objective

• To investigate whether fundamental antipode properties of classical Hopf algebras persist in the generalized Hom-Hopf algebra setting with relative Hom-invertibility.
• To determine which properties of the antipode can be recovered when only the relative Hom-inverse condition is assumed, without requiring unitality, counitality, or anti-morphism properties.
• To examine the behavior of $ S^2 $ in commutative and cocommutative Hom-Hopf algebras under the new definition.
• To analyze the images of primitive and group-like elements under the antipode in this generalized framework.

Proposed method

• Define a Hom-Hopf algebra via a Hom-bialgebra equipped with a map $ S $ that is a relative Hom-inverse of the identity under convolution, i.e., $ \alpha^k(S \star \mathrm{id}) = \alpha^k(\mathrm{id} \star S) = \eta \circ \varepsilon $ for some $ k \in \mathbb{N} $.
• Establish that $ S $ is a relative Hom-anti-algebra and relative Hom-anti-coalgebra morphism using convolution identities and the Hom-associativity/coassociativity axioms.
• Prove that $ S $ is relative Hom-unital and relative Hom-counital by leveraging the relative inverse condition and the structure of the Hom-bialgebra.
• Show that if the twisting maps $ \alpha $ and $ \beta $ are invertible, then $ S $ becomes a true anti-algebra and anti-coalgebra map, and $ S \circ \alpha = \alpha \circ S $.
• Analyze $ S^2 $ in commutative and cocommutative Hom-Hopf algebras by showing $ S^2 $ is also a relative Hom-inverse of $ S $, leading to $ \alpha^{k+2} \circ S^2 \circ \beta^2 = \alpha^{k+2} \circ \mathrm{id} \circ \beta^2 $.
• Study the action of $ S $ on primitive and group-like elements by applying the relative inverse condition and using the definitions $ \Delta(h) = 1 \otimes h + h \otimes 1 $ and $ \Delta(h) = h \otimes h $.

Experimental results

Research questions

• RQ1 Does the antipode in a Hom-Hopf algebra defined via relative Hom-invertibility retain the anti-algebra and anti-coalgebra morphism properties?
• RQ2 Under what conditions does the antipode become a strict anti-algebra and anti-coalgebra map?
• RQ3 Can the classical result $ S^2 = \mathrm{id} $ be recovered in Hom-Hopf algebras, and if so, under what assumptions?
• RQ4 How does the antipode act on primitive and group-like elements in the relative Hom-inverse setting?
• RQ5 What is the relationship between the antipode and the twisting maps $ \alpha $ and $ \beta $, especially when they are invertible?

Key findings

• The antipode $ S $ is a relative Hom-anti-algebra and relative Hom-anti-coalgebra morphism, even without assuming strict anti-morphism properties.
• If the twisting maps $ \alpha $ and $ \beta $ are invertible, then $ S $ is a true anti-algebra and anti-coalgebra map, and $ S \circ \alpha = \alpha \circ S $.
• For commutative Hom-Hopf algebras with invertible $ \alpha $ and $ \beta $, it holds that $ S^2 = \mathrm{id} $.
• For cocommutative Hom-Hopf algebras with invertible $ \alpha $ and $ \beta $, it also holds that $ S^2 = \mathrm{id} $.
• For any primitive element $ h $, there exists $ k \in \mathbb{N} $ such that $ \alpha^{k+1}(S(h)) = -\alpha^{k+1}(h) $.
• For any group-like element $ h $, there exists $ k \in \mathbb{N} $ such that $ \alpha^k(S(h)h) = \alpha^k(hS(h)) = 1 $, and $ S(h) $ is the relative Hom-inverse of $ h $.