Semilinear map - Knowlpedia

Let $V$ be a vector space over a field $k$ , let $W$ be a vector space over a field $\ell$ , and let $\sigma:k\to\ell$ be a unital field homomorphism. A map $T:V\to W$ is $\sigma$ -semilinear if

T(v+w)=T(v)+T(w),\qquad T(av)=\sigma(a)T(v)

for all $v,w\in V$ and $a\in k$ .

Automorphism convention

When $k=\ell$ and $\sigma$ is a field automorphism, one simply calls $T$ semilinear.

Relation to linear maps

For $\sigma=\operatorname{id}_k$ , semilinearity is ordinary linearity. More generally, if ${}_{\sigma}W$ denotes $W$ with scalar action $a\cdot_\sigma w=\sigma(a)w$ , then a $\sigma$ -semilinear map $V\to W$ is the same additive function as a $k$ -linear map $V\to{}_{\sigma}W$ .

For an automorphism $\sigma:k\to k$ , the inverse of a bijective $\sigma$ -semilinear self-map is $\sigma^{-1}$ -semilinear. Composing a $\sigma$ -semilinear map with a $\tau$ -semilinear map gives a $\tau\circ\sigma$ -semilinear map.

Projective significance

A semilinear isomorphism sends linear subspaces to linear subspaces and therefore induces a collineation of projective spaces. The fundamental theorem of projective geometry says that, in projective dimension at least two, every collineation arises this way.

Examples

Coordinatewise complex conjugation on $\mathbb C^n$ is semilinear for the automorphism $z\mapsto\overline z$ , but it is not $\mathbb C$ -linear. Over a field with no nontrivial automorphisms, every semilinear self-map is linear.

References

Emil Artin, Geometric Algebra, Interscience, 1957. Relevant: Chapter II, §§3–4, semilinear transformations and projective geometry.
Peter J. Cameron, Projective and Polar Spaces, Queen Mary and Westfield College, 1992. Author-maintained text. Relevant: Chapter 1, semilinear maps and collineations.