Hopf coactions on commutative algebras generated by a quadratically independent comodule

TL;DR

This paper investigates conditions under which a Hopf algebra coacting on a commutative algebra generated by a quadratically independent subspace must be commutative, revealing new structural constraints.

Contribution

It proves that Hopf algebras coacting inner-faithfully on such algebras are necessarily commutative under specific conditions, extending understanding of symmetry in algebraic structures.

Findings

Q must be commutative if it preserves a non-degenerate bilinear form on V.

Q is commutative if it is co-semisimple, finite-dimensional, and char(k)=0.

The results impose structural restrictions on Hopf coactions in algebraic geometry.

Abstract

Let A be a commutative unital algebra over an algebraically closed field k of characteristic not equal to 2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let Q be a Hopf algebra that coacts on A inner-faithfully, while leaving V invariant. We prove that Q must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) Q is co-semisimple, finite-dimensional, and char(k)=0.