[Paper Review] Nakayama Automorphism and Rigidity of Dual Reflections Group Coactions

Ellen Kirkman, James Kuzmanovich+1 more|arXiv (Cornell University)|May 6, 2016

Algebraic structures and combinatorial models26 references3 citations

TL;DR

This paper investigates the homological rigidity of dual reflection group coactions on Artin-Schelter regular algebras, focusing on the role of the Nakayama automorphism and the mass element derived from the homological codeterminant. It establishes that the mass element corresponds to the unique maximal-length group element in a generating set, and proves that certain noncommutative algebras—such as the universal enveloping algebra of a semisimple Lie algebra, the Rees ring of the Weyl algebra, and non-PI Sklyanin algebras—are rigid under finite group coactions, meaning their fixed subrings are never AS regular unless trivial.

ABSTRACT

We study homological properties and rigidity of group coactions on Artin-Schelter regular algebras.

Motivation & Objective

• To understand the homological properties of algebras admitting dual reflection group coactions, particularly focusing on the structure of fixed subrings.
• To characterize the Nakayama automorphism in the context of smash products arising from group coactions on Artin-Schelter regular algebras.
• To establish rigidity results by proving that certain noncommutative AS regular algebras do not admit nontrivial finite group coactions with AS regular fixed subrings.
• To link the homologically defined mass element to combinatorial data such as group length functions and cyclotomic polynomials.
• To extend classical invariants like the homological codeterminant and Nakayama automorphism to noncommutative invariant theory in the context of Hopf algebra coactions.

Proposed method

• Utilizes the equivalence between left $ H $-actions and right $ K $-coactions, where $ H = \Bbbk^G $ and $ K = \Bbbk G $, to translate group coaction problems into algebraic structures.
• Applies the homological codeterminant and its inverse, the mass element $ m $, defined via the ratio of Hilbert series $ \mathfrak{p}(t) = H_A(t) H_{A^H}(t)^{-1} $, to detect group-theoretic properties.
• Employs the length function $ l_\Re(g) $ with respect to a generating set $ \Re \subset G \setminus \{e\} $ to relate degree of homogeneous generators $ f_g $ to group word length.
• Uses the Nakayama automorphism $ \mu_A $ to analyze the structure of the smash product $ A \# \Bbbk G $, linking it to Poincaré duality and Radford's formula.
• Applies results from noncommutative algebraic geometry, such as projective simplicity of twisted homogeneous coordinate rings, to rule out normal elements in non-PI Sklyanin algebras.
• Employs reduction to the commutative case via specialization $ t \mapsto 1 $, mapping to Weyl algebras and using their simplicity to derive contradictions.

Experimental results

Research questions

• RQ1 What is the relationship between the homological codeterminant and the group-theoretic structure of a dual reflection group coacting on an AS regular algebra?
• RQ2 How does the Nakayama automorphism of the fixed subring relate to the group coaction and the structure of the algebra?
• RQ3 Under what conditions is the fixed subring of an AS regular algebra under a finite group coaction also AS regular?
• RQ4 Can the mass element be characterized combinatorially as the unique element of maximal length in a generating set?
• RQ5 Are certain noncommutative AS regular algebras, such as non-PI Sklyanin algebras or Rees rings of Weyl algebras, rigid under finite group coactions?

Key findings

• The mass element $ m \in G $ is the unique element of maximal length with respect to the reduced word length function $ l_\Re $, and $ \deg f_m = l_\Re(m) $, linking homological invariants to group combinatorics.
• The Hilbert series ratio $ \mathfrak{p}(t) = H_A(t) H_{A^H}(t)^{-1} $ is a product of cyclotomic polynomials, satisfies $ \mathfrak{p}(1) = |G| $, and has degree equal to $ l_\Re(m) $, confirming deep arithmetic structure.
• For the universal enveloping algebra of a finite-dimensional semisimple Lie algebra, no nontrivial finite group coaction yields an AS regular fixed subring, implying rigidity of the algebra.
• The Rees ring of the Weyl algebra $ A_n(\Bbbk) $ admits an AS regular fixed subring only for $ G = \mathbb{Z}/(2) $, and even then, the fixed ring is not isomorphic to the original algebra.
• In non-PI Sklyanin algebras of global dimension at least 3, no finite set of degree-one elements has a normal product, which implies no nontrivial group coaction can yield an AS regular fixed subring.
• The Nakayama automorphism of the smash product $ A \# \Bbbk G $ is determined by the group coaction and the mass element, and its structure reflects the homological duality of the algebra.