Kbilinear

Kbilinear, often written as K-bilinear, is a term used in algebra to describe maps that are bilinear over a fixed commutative ring K. If V, W, and U are modules over K, a map f: V × W → U is Kbilinear when it is linear in each argument with respect to K: for all k in K, v in V, w in W, f(k v, w) = k f(v, w) and f(v, k w) = k f(v, w). In other words, f is linear separately in v and in w when scalars come from K.

Kbilinear maps form the central concept behind the tensor product. There is a universal property: corresponding to every Kbilinear map f: V × W → U, there exists a unique K-linear map φ: V ⊗_K W → U such that f = φ ∘ t, where t: V × W → V ⊗_K W is the canonical bilinearization. This correspondence establishes an isomorphism Hom_K(V ⊗_K W, U) ≅ Bil_K(V, W; U), making tensor products the natural recipients of bilinear data.

Examples include the standard dot product on K^n when K is a field, which is a symmetric bilinear form, and the multiplication map K × K → K in a field, which is bilinear over K. More generally, any bilinear form, such as a pairing into K, is Kbilinear.

Variants and properties of Kbilinear maps include symmetry, alternation, and nondegeneracy, which give rise to bilinear forms with geometric or algebraic significance. Kbilinear maps are foundational in areas such as linear algebra, representation theory, and algebraic geometry, and underpin constructions of tensor products and multilinear maps.

See also: Bilinear form, Bilinear map, Tensor product, Multilinear map, Module.