Everybody knows what a normal gabi-algebra is

TL;DR

This paper characterizes normal gabi-algebras as precisely those algebras whose module categories are closed and have a forgetful functor to modules, showing they are automatically Hopf algebras, thus connecting categorical properties to algebraic structures.

Contribution

It introduces the concept of gabi-algebras based on categorical lifting conditions and proves that normal gabi-algebras are necessarily Hopf algebras, establishing a new characterization.

Findings

Normal gabi-algebras are automatically Hopf algebras.

Conditions for lifting closed structures lead to the notion of gabi-algebras.

The result applies in both set-theoretic and linear contexts.

Abstract

Let A be a k-algebra over a commutative ring k. By the renowned Tannaka-Krein reconstruction, liftings of the monoidal structure from k-modules to A-modules correspond to bialgebra structures on A and liftings of the closed monoidal structure correspond to Hopf algebra structures on A. In this paper, we determine conditions on A that correspond to liftings of the closed structure alone, i.e. without considering the monoidal one, which lead to the notion of what we call a gabi-algebra. First, we tackle the question from the general perspective of monads, then we focus on the set-theoretic and the linear setting. Our main and most surprising result is that a normal gabi-algebra, that is an algebra A whose category of modules is (associative and unital normal) closed with closed forgetful functor to k-modules, is automatically a Hopf algebra (thus justifying our title).