Division, adjoints, and dualities of bilinear maps

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, nonassociative division rings can be studied within this framework. This also corrects an error in an earlier pre-print; see Remark 2.11.

💡 Research Summary

The paper develops a categorical framework for bilinear maps (bimaps) by introducing three distinct types of morphisms—homotopisms, adjoint‑morphisms, and non‑degenerate adjoint‑morphisms—and studies the structural properties of the resulting categories.

First, the author recalls the definition of an (R,S)-bimap B : U × V → W, emphasizing its left‑ and right‑linearity and the distributive laws. Homotopisms, originally introduced by Albert for non‑associative rings, are triples (φ,γ;κ) that simultaneously act on the left, right, and target modules and preserve the bimap. This yields a homotopism category that contains familiar sub‑categories such as pseudo‑isometries and principal homotopisms, and it is shown to have kernels, quotients, and an isomorphism theorem.

The core of the paper is the adjoint‑morphism category Adj(W). Objects are all (R,S)-bimaps with a fixed codomain W; a morphism (μ,ν) from B : U×V→W to C : U′×V′→W satisfies u μ C v = u B ν v for all u∈U, v∈V′. The author proves that Adj(W) is pre‑additive, complete, cocomplete, and abelian; kernels and images of adjoint‑morphisms are described in terms of orthogonal operators K and J, which give rise to a Galois connection between submodules of U and V. Theorem 2.27 and 2.31 establish the existence of projectives and injectives, making Adj(W) a well‑behaved abelian category.

A crucial observation is that Adj(W) possesses a contravariant transpose functor t, sending a bimap B to its transpose Bᵗ : V×U→W, and sending (μ,ν) to (ν,μ). This yields a duality between Adj(W) and Adj(Wᵒᵖ). However, Theorem 2.10 shows that Adj(W) cannot be equivalent to any full module category nor to its dual, because module categories are not self‑dual and a Grothendieck category that is also co‑Grothendieck would be trivial. Remark 2.11 corrects an earlier mistake in a pre‑print and clarifies that Adj(W) is not equivalent to the subcategory of W‑reflexive modules, even over a field.

The paper then examines the relationship between adjoint‑morphisms and tensor products. An (R,S)-bimap B is A‑midlinear when U is an (R,A)-bimodule, V an (A,S)-bimodule, and the scalar action of A can be moved across B. The adjoint ring Adj(B) is universal for this property: any representation of an algebra A that makes B A‑midlinear embeds into Adj(B). Theorem 2.8 establishes a Galois connection between bimaps on U×V and algebra representations in End_R U × End_S V, showing that Adj(B) captures the “best” ring for tensoring.

Section 3 focuses on non‑degenerate adjoint‑morphisms, i.e., those whose kernels and images are trivial. By restricting to KJ‑ and JK‑stable submodules, the author obtains a genuine duality between the corresponding lattices. Proposition 3.3 characterizes non‑degenerate adjoint‑simple bimaps as minimal non‑degenerate elements in these lattices. The main result, Theorem 3.13, proves that a bimap is non‑degenerate adjoint‑simple precisely when it is a division bimap: u D v = 0 implies u = 0 or v = 0. This formalizes the intuition that the “atoms” of linear geometry are algebraic objects without zero‑divisors. Moreover, adjoint‑isomorphism coincides with principal isotopism, allowing the study of non‑associative division rings within this categorical setting.

Throughout, the paper provides concrete examples (e.g., group commutator maps, families of bilinear forms) to illustrate how homotopism and adjoint categories interact, and how the Galois connection simplifies problems such as determining automorphism groups of p‑groups or intersections of classical subgroups of GL(V). The author also discusses forgetful functors from Adj(W) to the module categories, their adjoints (called “versors”), and how these interact with the transpose duality.

In summary, Wilson constructs a robust categorical environment for bilinear maps that is richer than traditional module categories. The adjoint‑morphism category is abelian, possesses projectives and injectives, and enjoys a natural transpose duality, yet it remains fundamentally distinct from any module category. Non‑degenerate adjoint‑simple bimaps are identified with division bimaps, linking the theory to non‑associative division algebras. The work corrects earlier inaccuracies and opens new avenues for applying categorical methods to non‑associative algebra, tensor products, and linear geometry.