[Paper Review] Hopf actions and Nakayama automorphisms

Kenneth Chan, Chelsea Walton+1 more|arXiv (Cornell University)|Oct 24, 2012

Algebraic structures and combinatorial models12 references5 citations

TL;DR

This paper establishes a deep connection between the Nakayama automorphism of an N-Koszul Artin-Schelter regular algebra A and the square of the antipode S² of a Hopf algebra H coacting on A. Under inner-faithful coaction and finite-dimensionality, it proves that conjugation by the homological codeterminant D satisfies η_D ∘ S² = η_{μ_A^τ}, linking homological invariants of A to Hopf algebra structure. This result characterizes the Hopf algebras acting on such algebras, particularly showing that finite-dimensional Hopf actions on non-PI AS regular algebras of dimension two must be group algebras when char(k)=0.

ABSTRACT

Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin-Schelter regular algebra. We study connections between the Nakayama automorphism of A and S^2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given.

Motivation & Objective

• To understand the relationship between the Nakayama automorphism of a noncommutative algebra and the antipode of a Hopf algebra coacting on it.
• To generalize known results on Nakayama automorphisms in Frobenius and AS regular algebras to the setting of Hopf coactions.
• To classify finite-dimensional Hopf algebras that act inner-faithfully on Artin-Schelter regular algebras of global dimension two.
• To explore whether the connection between S² and the Nakayama automorphism can be unified across different algebraic settings.
• To determine the structure of Hopf algebras coacting on specific non-PI AS regular algebras, such as skew polynomial algebras.

Proposed method

• Use Manin’s construction of quantum linear groups to relate the coaction of a Hopf algebra H on a graded algebra A to the structure of H.
• Define the homological codeterminant D of the H-coaction on A as a central element in H, which controls the twist in the automorphism relation.
• Construct automorphisms η_D and η_{μ_A^τ} on H via conjugation by D and by the transpose of the matrix of the Nakayama automorphism μ_A, respectively.
• Establish the key identity η_D ∘ S² = η_{μ_A^τ} using Ext-algebra coaction and matrix coordinate computations in the Hopf algebra.
• Apply Radford’s theorem on the order of S² and use characteristic 0 to force vanishing of primitive elements, leading to group algebra structure.
• Use the inner-faithful coaction condition to ensure the generators of H span the entire algebra, enabling structural classification.

Experimental results

Research questions

• RQ1 How does the Nakayama automorphism of an N-Koszul AS regular algebra relate to the square of the antipode of a coacting Hopf algebra?
• RQ2 Can the identity η_D ∘ S² = η_{μ_A^τ} be extended beyond N-Koszul algebras to general AS regular algebras?
• RQ3 What is the structure of a finite-dimensional Hopf algebra coacting inner-faithfully on a non-PI AS regular algebra of global dimension two?
• RQ4 Is there a finite-dimensional noncommutative Hopf algebra of Gelfand-Kirillov dimension 1 coacting inner-faithfully on a skew polynomial algebra k_J[x₁,x₂]?
• RQ5 Can the relation between S² and the Nakayama automorphism be unified with the known formula μ_K = S² ∘ Ξ^{l}_{∫^l} for AS regular Hopf algebras?

Key findings

• The key identity η_D ∘ S² = η_{μ_A^τ} holds for any finite-dimensional Hopf algebra H coacting inner-faithfully on an N-Koszul AS regular algebra A with bijective antipode.
• When char(k) = 0, any finite-dimensional Hopf algebra coacting inner-faithfully on the non-PI algebra A_J = k_J[x₁,x₂] must be isomorphic to a group algebra kG for some cyclic group G.
• The structure of H is constrained by the matrix of the Nakayama automorphism μ_A and the homological codeterminant D, with η_D ∘ S² acting as conjugation by the transpose of μ_A’s matrix.
• In the case of A = k_p[x₁,x₂] with p not a root of unity, the same conclusion holds: H must be a group algebra, as shown via Theorem 0.4.
• The absence of non-trivial primitive elements in finite-dimensional Hopf algebras over characteristic 0 fields forces the generators to be group-like, leading to the group algebra conclusion.
• The result confirms that non-PI AS regular algebras of dimension two admit only group algebra coactions under inner-faithful, finite-dimensional Hopf coactions in characteristic 0.