[Paper Review] Left counital Hopf algebra structures on free commutative Nijenhuis algebras

Shanghua Zheng, Li Guo|arXiv (Cornell University)|Nov 13, 2017

Advanced Topics in Algebra32 references3 citations

TL;DR

This paper establishes a left counital right antipode Hopf algebra structure on free commutative Nijenhuis algebras by leveraging a cocycle condition to construct a left counital bialgebra, then proving that connected graded left counital bialgebras admit a right-sided antipode, thereby yielding a left counital Hopf algebra. The key contribution is a new class of Hopf-like structures with relaxed symmetry conditions.

ABSTRACT

Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally apply this result to show that the left counital bialgebra on a free commutative Nijenhuis algebra on a connected left counital bialgebra is connected and graded, hence is a left counital right antipode Hopf algebra.

Motivation & Objective

• To extend Hopf algebra structures from free commutative Rota-Baxter algebras to free commutative Nijenhuis algebras.
• To address the lack of full symmetry in Nijenhuis algebra structures by introducing left counital and right antipode conditions.
• To generalize the classical result that connected graded bialgebras are Hopf algebras to the left counital context.
• To prove that the free commutative Nijenhuis algebra on a connected left counital bialgebra inherits a connected graded structure, enabling a left counital right antipode Hopf algebra.
• To provide a new class of non-standard Hopf algebras with one-sided counit and antipode, distinct from existing one-side Hopf algebra notions.

Proposed method

• Construct free commutative Nijenhuis algebras using a right-shift shuffle product generalization.
• Define a comultiplication Δ_T via a cocycle condition on the underlying left counital bialgebra.
• Verify compatibility of Δ_T and the counit ε_T with the Nijenhuis algebra multiplication.
• Prove coassociativity of Δ_T and left counicity of ε_T using induction on the degree of elements.
• Establish that the resulting bialgebra is connected and graded by defining a natural filtration via tensor powers.
• Apply a generalized version of the classical connected graded bialgebra ⇒ Hopf algebra theorem to deduce the existence of a right-sided antipode.

Experimental results

Research questions

• RQ1 Can free commutative Nijenhuis algebras be endowed with a Hopf algebra-like structure despite the absence of full symmetry?
• RQ2 Does a cocycle condition on the underlying bialgebra structure suffice to lift a left counital bialgebra to a left counital right antipode Hopf algebra?
• RQ3 Is the free commutative Nijenhuis algebra on a connected left counital bialgebra naturally graded and connected?
• RQ4 Can the classical result that connected graded bialgebras are Hopf algebras be extended to the left counital setting?
• RQ5 What is the precise nature of the antipode in such algebras, and why is it only right-sided?

Key findings

• A free commutative Nijenhuis algebra on a left counital bialgebra can be equipped with a comultiplication Δ_T via a cocycle condition, making it a left counital bialgebra.
• The comultiplication Δ_T is coassociative and the counit ε_T is left-sided, satisfying the left counicity condition.
• The free commutative Nijenhuis algebra on a connected left counital bialgebra is naturally connected and graded by tensor degree.
• A general result is proven: every connected graded left counital bialgebra admits a right-sided antipode, making it a left counital right antipode Hopf algebra.
• As a consequence, the free commutative Nijenhuis algebra on a connected left counital bialgebra becomes a left counital right antipode Hopf algebra.
• The construction yields a new class of Hopf-like structures with asymmetric duality, distinct from standard Hopf algebras and prior one-side Hopf algebra notions.