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semilinear transformation
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(Definition)
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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.)
- $f(x+y)=f(x)+f(y)$ , (in right hand notation: $(x+y)f=xf+yf$ .)
- $f(lx)=l^\theta f(x)$ , (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 $$ (l_1+l_2)^\theta xf=((l_1+l_2)x)f=(l_1 x)f+(l_2 x)f=(l_1^\theta + l_2^\theta)x $$ 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 $$ \left(\sum_{i=1}l_i e_i\right)\tilde{f}=\sum_{i=1}^n l_i^\theta e_i f $$
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: $$ \sum_{b\in B} l_b b\mapsto \sum_{b\in B} l_b^\sigma $$ 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 $V$ , $$ vh=vg^{-1}f=\sum_{b\in B} l_b^{\sigma} $$ 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.
- 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).
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See Also: classical groups, projective space
| Other names: |
semilinear map, semilinear transform, semi-linear transformation, semi-linear map |
| Also defines: |
semilinear transform, Gamma L |
| Keywords: |
field automorphism, linear, Gamma L |
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Cross-references: automorphism group, useful, fundamental theorem of projective geometry, projectivity, order-preserving map, induces, subspaces, lattice, projective geometry, classical groups, action, change of basis, complements, basis, subgroup, Galois group, semidirect product, group, invertible, freshman's dream, characteristic, complex conjugation, standard basis, map, converse, linear transformation, fix, additive, image, right, vector spaces, function, automorphism, order, finite field, prime subfield, field
There are 6 references to this entry.
This is version 17 of semilinear transformation, born on 2006-04-15, modified 2006-06-19.
Object id is 7835, canonical name is SemilinearTransformation.
Accessed 6481 times total.
Classification:
| AMS MSC: | 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations) |
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Pending Errata and Addenda
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