Algebras with a bilinear form, and idempotent endomorphisms

Abstract

The category k-BFAlg of all k-algebras with a bilinear form, whose objects are all pairs (R,b) where R is a k-algebra and b:R×R→k is a bilinear mapping, is equivalent to the category of unital k-algebras A for which the canonical homomorphism (k,1)→(A,1A) of unital k-algebras is a splitting monomorphism in k-Mod. 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,bA) with a bilinear form bA compatible with the multiplication of A, i.e., such that bA(x,y)=bA(z,w) for all x,y,z,w∈A for which xy=zw.