# semilinear transformation

Let $K$ be a field and $k$ its prime subfield. For example, if $K$ is $\mathbb{C}$ 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 $f:V\rightarrow W$ between two $K$ vector spaces $V$ and $W$ is $\theta$-semilinear, or simply semilinear, if for all $x,y\in V$ and $l\in K$ it follows: (shown here first in left hand notation and then in the preferred right hand notation.)

1. 1.

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

2. 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

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

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 $(l_{1}+l_{2})^{\theta}=l_{1}^{\theta}+l_{2}^{\theta}$. 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 $K=\mathbb{C}$, $V=\mathbb{C}^{n}$ with standard basis $e_{1},\dots,e_{n}$. Define the map $f:V\rightarrow V$ by

 $f\left(\sum_{i=1}z_{i}e_{i}\right)=\sum_{i=1}^{n}\bar{z}_{i}e_{i}.$

$f$ 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 $f:V\rightarrow W$ between vector spaces $V$ and $W$ over $K$ we can establish a $\theta$-semilinear map

 $\left(\sum_{i=1}l_{i}e_{i}\right)\tilde{f}=\sum_{i=1}^{n}l_{i}^{\theta}e_{i}f.$

$\Box$

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 $\Gamma L(V)$.

###### Proposition 4.

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

 $\Gamma L(V)=GL(V)\rtimes Gal(K/k)$

where $Gal(K/k)$ is the Galois group of $K/k$.

###### Remark 5.

We identify $Gal(K/k)$ with a subgroup of $\Gamma L(V)$ 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 $\sigma\in Gal(K/k)$. We shall denoted this subgroup by $Gal(K/k)_{B}$. We also see these complements to $GL(V)$ in $\Gamma L(V)$ are acted on regularly by $GL(V)$ as they correspond to a change of basis.

###### Proof.

Every linear map is semilinear thus $GL(V)\leq\Gamma L(V)$. 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 $g:V\rightarrow V$ 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 exchange of $V$ and so linear and invertible: $g\in GL(V)$.

Set $h:=g^{-1}f$. For every $v=\sum_{b\in B}l_{b}\neq 0$ in $V$,

 $vh=vg^{-1}f=\sum_{b\in B}l_{b}^{\sigma}b$

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 $\Gamma L(V)=GL(V)\rtimes Gal(K/k)$. ∎

The $\Gamma L(V)$ 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 $f:V\rightarrow W$ induces an order-preserving map $f:PG(V)\rightarrow PG(W)$. 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.

## References

• 1 Gruenberg, K. W. and Weir, A.J. Linear Geometry 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. X, 198 p. DM 29.10; \$ 12.80 (1977).
 Title semilinear transformation Canonical name SemilinearTransformation Date of creation 2013-03-22 15:51:06 Last modified on 2013-03-22 15:51:06 Owner Algeboy (12884) Last modified by Algeboy (12884) Numerical id 20 Author Algeboy (12884) Entry type Definition Classification msc 15A04 Synonym semilinear map Synonym semilinear transform Synonym semi-linear transformation Synonym semi-linear map Related topic ClassicalGroups Related topic ProjectiveSpace Defines semilinear transform Defines Gamma L