semilinear

Semilinear describes a type of map between vector spaces over a division ring or field that is additive and scales by a field automorphism: f(x+y)=f(x)+f(y) and f(a x)=φ(a) f(x) for some field automorphism φ. If φ is the identity, f is linear.

In linear algebra over noncommutative division rings, semilinear maps generalize linear maps. Composition of semilinear maps with automorphisms yields semilinear maps with composed automorphisms. A common intuition is that semilinear maps preserve addition but scale scalars by a nontrivial automorphism of the underlying field.

A standard example is complex conjugation on a complex vector space, which is conjugate-linear: f(a x)=ā f(x) for a in C. More generally, a map f is semilinear with respect to φ if f(z x)=φ(z) f(x). When φ is complex conjugation, such maps are called conjugate-linear.

In projective geometry and related areas, semilinear automorphisms of a vector space V over F preserve the structure of the projective space up to a field automorphism. The corresponding group of bijections is the projective semilinear group PΓL, which contains the linear group PGL as well as automorphisms of the underlying field.

In the analysis of differential equations, semilinear refers to equations in which the highest-order derivatives appear linearly but nonlinear terms depend on the unknown function (and possibly its lower-order derivatives). A typical example is the semilinear elliptic equation Δu + f(u) = 0, where the Laplacian acts linearly on u while the nonlinearity involves u itself.

See also: linear map, semilinear transformation, conjugate-linear, PΓL, automorphism, field automorphism.