Algebras with a bilinear form, and Idempotent endomorphisms

TL;DR

This paper establishes an equivalence between the category of algebras with bilinear forms and a specific class of unital algebras with compatible bilinear forms, revealing structural insights and categorical relationships.

Contribution

It introduces a categorical equivalence linking algebras with bilinear forms to unital algebras with compatible bilinear forms, clarifying their structural relationship.

Findings

Category of algebras with bilinear forms is equivalent to a category of unital algebras with compatible forms.

Weak augmentations correspond to certain splitting monomorphisms in the module category.

Structural isomorphism between categories clarifies algebraic and categorical properties.

Abstract

The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is a splitting monomorphism in the category of $k$-modules. Call the left inverses of this splitting monomorphism "weak augmentations" of the algebra. There is a category isomorphism between the category of $k$-algebras with a weak augmentation and the category of unital $k$-algebras $(A,b_A)$ with a bilinear form $b_A$ compatible with the multiplication of $A$, i.e., such that $b_A(x,y)=b_A(z,w)$ for all $x,y,z,w\in A$ for which $xy=zw$.