semilinear transformation

Let $K$ be a field and $k$ its prime subfield. For example, if $K$ is $ℂ$ then $k$ is $\mathbb{Q}$, and if $K$ is the finite field of order $q=p^i$, then $k$ is ${\mathbb{Z}}_p$.

Definition 1.

Given a field automorphism $\theta$ of $K$, a function$f\mathrm{:}V\mathrm{\to }W$ between two $K$ vector spaces $V$ and $W$ is$\theta$-semilinear, or simply semilinear, if for all $x\mathrm{,}y\mathrm{\in }V$ and $l\mathrm{\in }K$it follows: (shown here first in left hand notation and then in the preferred right hand notation.)

1. $f(x+y)=f(x)+f(y)$, (in right hand notation: $(x+y)f=xf+yf$.)

2. $f(lx)=l^{\theta }f(x)$, (in right hand notation: $(lx)f=l^{\theta }xf$.)

where $l^{\theta }$ denotes the image of $l$ under $\theta$.

Remark 2.

$\theta$ must be a field automorphism for $f$ to remain additive, for example,$\theta$ must fix the prime subfield as

$$n^{\theta }xf=(nx)f=(x+\mathrm{\cdots }+x)f=n(xf).$$

Also

$${(l_1+l_2)}^{\theta }xf=((l_1+l_2)x)f=(l_{1}x)f+(l_{2}x)f=(l_1^{\theta }+l_2^{\theta })xf$$

so ${\mathrm{(}{l}_{\mathrm{1}}\mathrm{+}{l}_{\mathrm{2}}\mathrm{)}}^{\theta }\mathrm{=}{l}_{\mathrm{1}}^{\theta }\mathrm{+}{l}_{\mathrm{2}}^{\theta }$. Finally,

$${(l_1{l}_2)}^{\theta }xf=((l_1{l}_{2}x)f=l_1^{\theta }(l_{2}x)f=l_1^{\theta }{l}_2^{\theta }xf.$$

Every linear transformation is semilinear, but the converse is generally not true. If we treat $V$ and $W$ as vector spaces over $k$, (by considering $K$ as vector space over $k$ first) then every $\theta$-semilinear map is a $k$-linear map, where $k$ is the prime subfield of $K$.

Example

• Let $K=ℂ$, $V={ℂ}^n$ with standard basis $e_1,\mathrm{\dots },e_n$. Define the map $f:V\to V$ by

  $$f\left(\sum _{i=1}{z}_i{e}_i\right)=\sum _{i=1}^n{\overline{z}}_i{e}_i.$$

  $f$ is semilinear (with respect to the complex conjugation field automorphism)but not linear.

• Let $K=GF(q)$ – the Galois field of order $q=p^i$, $p$ the characteristic.Let $l^{\theta }=l^p$, for $l\in K$. By the Freshman’s dream it is known that this is a field automorphism. To every linear map $f:V\to W$ betweenvector spaces $V$ and $W$ over $K$ we can establish a $\theta$-semilinear map

  $$\left(\sum _{i=1}{l}_i{e}_i\right)\tilde{f}=\sum _{i=1}^n{l}_i^{\theta }{e}_{i}f.$$

$\mathrm{\square }$

Indeed every linear map can be converted into a semilinear map in such a way.This is part of a general observation collected into the following result.

Definition 3.

Given a vector space $V$, the set of all invertible semilinear maps (over all field automorphisms) is the group $\mathrm{\Gamma }\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$.

Proposition 4.

Given a vector space $V$ over $K$, and $k$ the prime subfield of $K$, then$\mathrm{\Gamma }\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$ decomposes as the semidirect product

$$\mathrm{\Gamma }L(V)=GL(V)⋊Gal(K/k)$$

where $G\mathit{}a\mathit{}l\mathit{}\mathrm{(}K\mathrm{/}k\mathrm{)}$ is the Galois group of $K\mathrm{/}k$.

Remark 5.

We identify $G\mathit{}a\mathit{}l\mathit{}\mathrm{(}K\mathrm{/}k\mathrm{)}$ with a subgroup of $\mathrm{\Gamma }\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$ by fixing a basis$B$ for $V$ and defining the semilinear maps:

$$\sum _{b\in B}{l}_{b}b\mapsto \sum _{b\in B}{l}_b^{\sigma }b$$

for any $\sigma \mathrm{\in }G\mathit{}a\mathit{}l\mathit{}\mathrm{(}K\mathrm{/}k\mathrm{)}$. We shall denoted this subgroup by $G\mathit{}a\mathit{}l\mathit{}{\mathrm{(}K\mathrm{/}k\mathrm{)}}_B$. We also see these complements to $G\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$ in $\mathrm{\Gamma }\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$ are acted on regularly by $G\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$ as they correspond to a change of basis.

Proof.

Every linear map is semilinear thus $GL(V)\le \mathrm{\Gamma }L(V)$. Fix a basis$B$ of $V$. Now given any semilinear map $f$ with respect to a field automorphism $\sigma \in Gal(K/k)$, then define $g:V\to V$ by

$$\left(\sum _{b\in B}{l}_{b}b\right)g=\sum _{b\in B}(l_b^{{\sigma }^{-1}}b)f=\sum _{b\in B}{l}_b(b)f.$$

As $(B)f$ is also a basis of $V$, it follows $g$ is simply a basis exchangeof $V$ and so linear and invertible: $g\in GL(V)$.

Set $h:=g^{-1}f$. For every $v={\sum }_{b\in B}{l}_b\ne 0$ in $V$,

$$vh=v{g}^{-1}f=\sum _{b\in B}{l}_b^{\sigma }b$$

thus $h$ is in the $Gal(K/k)$ subgroup relative to the fixed basis $B$.This factorization is unique to the fixed basis $B$. Furthermore, $GL(V)$ is normalized by the action of $Gal{(K/k)}_B$, so $\mathrm{\Gamma }L(V)=GL(V)⋊Gal(K/k)$.∎

The $\mathrm{\Gamma }L(V)$ groups extend the typical classical groups in $GL(V)$. The importance in considering such maps follows from the consideration of projective geometry.

The projective geometry of a vector space $V$, denoted $PG(V)$, is the lattice of all subspaces of $V$. Although the typical semilinear map is not a linear map, it does follow that every semilinear map $f:V\to W$ induces an order-preserving map $f:PG(V)\to PG(W)$. That is, every semilinear map induces a projectivity. The converse of this observation is the Fundamental Theorem of Projective Geometry. Thus semilinear maps are useful because they define the automorphism group of the projective geometry of a vector space.