Two-sided vector spaces

We study the structure of two-sided vector spaces over a perfect field $K$. In particular, we give a complete characterization of isomorphism classes of simple two-sided vector spaces which are left finite-dimensional. Using this description, we compute the Quillen $K$-theory of the category of left finite-dimensional, two-sided vector spaces over $K$. We also consider the closely related problem of describing homomorphisms $\phi:K\to M_n(K)$.

💡 Research Summary

The paper investigates “two‑sided vector spaces” over a perfect field K, i.e., K‑modules equipped with commuting left and right K‑actions. The authors restrict attention to objects that are finite‑dimensional as left K‑modules, which yields an abelian category 𝒞 closed under direct sums and tensor products.

The central achievement is a complete classification of simple objects in 𝒞. By exploiting the perfection of K, every K‑embedding σ:K→ \overline{K} generates a finite separable extension Lσ = K(σ(K)). The Galois group Gal(\overline{K}/K) acts on the set of embeddings, and the orbits (or “Galois orbits”) parametrize the isomorphism classes of simple two‑sided vector spaces. Concretely, for each orbit O one constructs V_O = L_O as a left K‑vector space; the right action is defined via any σ in O, i.e., v·a = v σ(a). This V_O is simple, and any simple left‑finite two‑sided vector space is isomorphic to one of this form. Different orbits give non‑isomorphic objects, establishing a bijection between simple isomorphism classes and Galois orbits of K‑embeddings.

With this classification in hand, the authors compute the Quillen K‑theory of the category 𝒞. Since every object decomposes as a finite direct sum of simples, the K‑theory groups split as direct sums over the corresponding field extensions. For each i≥0,

 K_i(𝒞) ≅ ⨁_{O} K_i(L_O),

where the sum runs over all Galois orbits O. In particular, K_0(𝒞) is a free abelian group generated by the dimensions of the finite separable extensions L_O/K, and K_1(𝒞) ≅ ⨁{O} L_O^×, K_2(𝒞) ≅ ⨁{O} K_2(L_O), etc. This shows that the K‑theory of two‑sided vector spaces mirrors exactly the K‑theory of the underlying separable field extensions.

The final part of the paper addresses homomorphisms φ:K→M_n(K). Using the previous classification, the authors prove that any such homomorphism factors through a finite separable extension L/K and a standard L‑linear representation ρ:L→M_n(K). Explicitly, there exists an embedding σ:K→L (belonging to some Galois orbit) and an L‑linear map ρ such that φ(a)=ρ(σ(a)) for all a∈K. This factorisation yields a transparent description of the eigenvalue structure of φ, its Jordan canonical form, and the associated two‑sided vector space V_φ, which is precisely the simple object V_O attached to the orbit of σ.

Overall, the paper provides a clean algebraic picture: simple left‑finite two‑sided vector spaces over a perfect field are classified by Galois orbits of field embeddings, their K‑theory decomposes accordingly, and any K‑algebra homomorphism into a matrix algebra arises from this same data. The results connect the newly introduced notion of two‑sided vector spaces with classical concepts such as separable field extensions, central simple algebras, and Quillen K‑theory, opening avenues for further exploration in non‑commutative geometry and representation theory.