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In [[linear algebra]], particularly [[projective geometry]], a '''semilinear map''' between [[vector space]]s ''V'' and ''W'' over a field ''K'' is a function that is a [[linear map]] "up to a twist", hence ''semi''-linear, where "twist" means "[[field automorphism]] of ''K''". Explicitly, it is a function {{nowrap|''T'' : ''V'' → ''W''}} that is:
* [[Additive map|additive]] with respect to vector addition: <math>T(v+v') = T(v)+T(v')</math>
* there exists a field automorphism ''&theta;'' of ''K'' such that <math>T(\lambda v) = \theta(\lambda) T(v)</math>. If such an automorphism exists and ''T'' is nonzero, it is unique, and ''T'' is called ''&theta;''-semilinear.

Where the domain and codomain are the same space (i.e. {{nowrap|''T'' : ''V'' → ''V''}}), it may be termed a '''semilinear transformation'''.  The invertible semilinear transforms of a given vector space ''V'' (for all choices of field automorphism) form a group, called the '''general semilinear group''' and denoted <math>\operatorname{\Gamma L}(V),</math> by analogy with and extending the [[general linear group]]. The special case where the field is the complex numbers <math>\mathbb{C}</math> and the automorphism is [[Complex conjugate|complex conjugation]], a semilinear map is called an '''[[antilinear map]]'''. 

Similar notation (replacing Latin characters with Greek ones) is used for semilinear analogs of more restricted linear transformations; formally, the [[semidirect product]] of a linear group with the Galois group of field automorphisms. For example, PΣU is used for the semilinear analogs of the [[projective special unitary group]] PSU.  Note, however, that it was only recently noticed that these generalized semilinear groups are not well-defined, as pointed out in <!-- the review of -->{{harv|Bray|Holt|Roney-Dougal|2009}} – isomorphic classical groups ''G'' and ''H'' (subgroups of SL) may have non-isomorphic semilinear extensions. At the level of semidirect products, this corresponds to different actions of the Galois group on a given abstract group, a semidirect product depending on two groups and an action. If the extension is non-unique, there are exactly two semilinear extensions; for example, symplectic groups have a unique semilinear extension, while {{nowrap|SU(''n'', ''q'')}} has two extensions if ''n'' is even and ''q'' is odd, and likewise for PSU.

== Definition ==
A map {{math|''f'' : ''V'' → ''W''}} for vector spaces {{math|''V''}} and {{math|''W''}} over fields {{math|''K''}} and {{math|''L''}} respectively is {{math|''σ''}}-semilinear, or simply '''semilinear''', if there exists a field homomorphism {{math|''σ'' : ''K'' → ''L''}} such that for all {{math|''x''}}, {{math|''y''}} in {{math|''V''}} and {{math|''λ''}} in {{math|''K''}} it holds that
# <math>f(x+y)=f(x)+f(y),</math>
# <math>f(\lambda x)=\sigma(\lambda) f(x).</math>

A given [[Embedding#Field theory|embedding]] {{math|''σ''}} of a field {{math|''K''}} in {{math|''L''}} allows us to identify {{math|''K''}} with a subfield of {{math|''L''}}, making a {{math|''σ''}}-semilinear map a ''K''-[[linear map]] under this identification. However, a map that is {{math|''τ''}}-semilinear for a distinct embedding {{math|''τ'' ≠ ''σ''}} will not be ''K''-linear with respect to the original identification {{math|''σ''}}, unless {{math|''f''}} is identically zero.

More generally, a map {{math|''ψ'' : ''M'' → ''N''}} between a right {{math|''R''}}-[[Module (mathematics)|module]] {{math|''M''}} and a left {{math|''S''}}-module {{math|''N''}} is {{math|''σ''}}-'''semilinear''' if there exists a ring [[antihomomorphism]] {{math|''σ'' : ''R'' → ''S''}} such that for all {{math|''x''}}, {{math|''y''}} in {{math|''M''}} and {{math|''λ''}} in {{math|''R''}} it holds that
# <math>\psi(x + y) = \psi(x) + \psi(y) ,</math>
# <math>\psi(x \lambda) = \sigma(\lambda) \psi(x) .</math>
The term ''semilinear'' applies for any combination of left and right modules with suitable adjustment of the above expressions, with {{math|''σ''}} being a homomorphism as needed.{{refn|{{citation|author=Ian R. Porteous|year=1995|title=Clifford Algebras and the Classical Groups|publisher=[[Cambridge University Press]]}}}}{{refn|{{citation|author=Bourbaki|year=1989|title=Algebra I|edition=2nd|publisher=Springer-Verlag|page=223}}}}

The pair {{math|(''ψ'', ''σ'')}} is referred to as a '''dimorphism'''.{{refn|{{citation|author=Bourbaki|year=1989|title=Algebra I|edition=2nd|publisher=Springer-Verlag|page=223}}}}

==Related==

===Transpose===

Let <math>\sigma : R \to S</math> be a ring isomorphism, <math>M</math> a right <math>R</math>-module and <math>N</math> a right <math>S</math>-module, and <math>\psi : M \to N</math> a <math>\sigma</math>-semilinear map.  Define the '''transpose''' of <math>\psi</math> as the mapping <math>{}^t\psi : N^* \to M^*</math> that satisfies{{refn|{{citation|author=Bourbaki|year=1989|title=Algebra I|edition=2nd|publisher=Springer-Verlag|page=236}}}}
<math display="block">\langle y , \psi(x) \rangle = \sigma\left(\left\langle {}^\text{t} \psi(y), x \right\rangle\right) \quad \text{ for all } y \in N^*, \text{ and all } x \in M.</math>
This is a <math>\sigma^{-1}</math>-semilinear map.

===Properties===

Let <math>\sigma : R \to S</math> be a ring isomorphism, <math>M</math> a right <math>R</math>-module and <math>N</math> a right <math>S</math>-module, and <math>\psi : M \to N</math> a <math>\sigma</math>-semilinear map.  The mapping
<math display="block">M \to R : x \mapsto \sigma^{-1}(\langle y, \psi(x)\rangle), \quad y \in N^*</math>
defines an <math>R</math>-linear form.{{refn|{{citation|author=Bourbaki|year=1989|title=Algebra I|edition=2nd|publisher=Springer-Verlag|page=236}}}}

== Examples ==

* Let <math>K=\mathbf{C}, V=\mathbf{C}^n,</math> with standard basis <math>e_1,\ldots, e_n</math>. Define the map <math>f\colon V \to V</math> by
*:<math>f\left(\sum_{i=1}^n z_i e_i \right) = \sum_{i=1}^n \bar z_i e_i</math> 
:''f'' is semilinear (with respect to the complex conjugation field automorphism) but not linear.
* Let <math>K=\operatorname{GF}(q)</math> – the Galois field of order <math>q=p^i</math>, ''p'' the characteristic. Let <math>\ell^\theta = \ell^p</math>. By the [[Freshman's dream]] it is known that this is a field automorphism. To every linear map <math>f\colon V \to W</math> between vector spaces ''V'' and ''W'' over ''K'' we can establish a <math>\theta</math>-semilinear map
*:<math>\widetilde{f} \left( \sum_{i=1}^n \ell_i e_i\right) := f \left( \sum_{i=1}^n \ell_i^\theta e_i \right) .</math>
: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.
* Let <math>R</math> be a noncommutative ring, <math>M</math> a left <math>R</math>-module, and <math>\alpha</math> an invertible element of <math>R</math>.  Define the map <math>\varphi\colon M\to M \colon x \mapsto\alpha x</math>, so <math>\varphi(\lambda u)=\alpha \lambda u = (\alpha \lambda \alpha^{-1}) \alpha u = \sigma(\lambda) \varphi(u)</math>, and <math>\sigma</math> is an inner automorphism of <math>R</math>.  Thus, the [[Homothetic transformation|homothety]] <math>x\mapsto\alpha x</math> need not be a linear map, but is <math>\sigma</math>-semilinear.{{refn|{{citation|author=Bourbaki|year=1989|title=Algebra I|edition=2nd|publisher=Springer-Verlag|page=223}}}}

== General semilinear group ==
Given a vector space ''V'', the set of all invertible semilinear transformations {{nowrap|''V'' → ''V''}} (over all field automorphisms) is the group ΓL(''V'').

Given a vector space ''V'' over ''K'', ΓL(''V'') decomposes as the [[semidirect product]]
:<math>\operatorname{\Gamma L}(V) = \operatorname{GL}(V) \rtimes \operatorname{Aut}(K) ,</math> 
where Aut(''K'') is the automorphisms of ''K''. Similarly, semilinear transforms of other linear groups can be ''defined'' as the semidirect product with the automorphism group, or more intrinsically as the group of semilinear maps of a vector space preserving some properties.

We identify Aut(''K'') with a subgroup of ΓL(''V'') by fixing a basis ''B'' for ''V'' and defining the semilinear maps:
:<math>\sum_{b\in B} \ell_b b \mapsto \sum_{b \in B} \ell_b^\sigma b</math>
for any <math>\sigma \in \operatorname{Aut}(K)</math>. We shall denoted this subgroup by Aut(''K'')<sub>''B''</sub>. We also see these complements to GL(''V'') in ΓL(''V'') are acted on regularly by GL(''V'') as they correspond to a [[change of basis]].

=== Proof ===
Every linear map is semilinear, thus <math>\operatorname{GL}(V) \leq \operatorname{\Gamma L}(V)</math>. Fix a basis ''B'' of ''V''. Now given any semilinear map ''f'' with respect to a field automorphism {{nowrap|''σ'' ∈ Aut(''K'')}}, then define {{nowrap|''g'' : ''V'' → ''V''}} by
:<math>g \left(\sum_{b \in B} \ell_b b\right)
:= \sum_{b \in B}f \left(\ell_b^{\sigma^{-1}} b\right)
= \sum_{b \in B} \ell_b f (b)
</math>
As ''f''(''B'') is also a basis of ''V'', it follows that ''g'' is simply a basis exchange of ''V'' and so linear and invertible: {{nowrap|''g'' ∈ GL(''V'')}}.

Set <math>h:=f g^{-1}</math>. For every <math>v=\sum_{b \in B} \ell_b b</math> in ''V'',
:<math>hv=fg^{-1}v=\sum_{b \in B} \ell_b^\sigma b</math> 
thus ''h'' is in the Aut(''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 Aut(''K'')<sub>''B''</sub>, so {{nowrap|1=ΓL(''V'') = GL(''V'') ⋊ Aut(''K'')}}.

== Applications ==

=== Projective geometry ===
The <math>\operatorname{\Gamma L}(V)</math> groups extend the typical [[classical group]]s in GL(''V''). The importance in considering such maps follows from the consideration of [[projective geometry]]. The induced action of <math>\operatorname{\Gamma L}(V)</math> on the associated projective space P(''V'') yields the '''{{visible anchor|projective semilinear group}}''', denoted <math>\operatorname{P\Gamma L}(V)</math>, extending the [[projective linear group]], PGL(''V'').

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 <math>f\colon V \to W</math> induces an order-preserving map <math>f\colon \operatorname{PG}(V) \to \operatorname{PG}(W)</math>. That is, every semilinear map induces a [[projectivity]]. The converse of this observation (except for the projective line) 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.

=== Mathieu group ===
{{main|Mathieu group}}
The group PΓL(3,4) can be used to construct the Mathieu group M<sub>24</sub>, which is one of the [[sporadic simple group]]s; PΓL(3,4) is a maximal subgroup of M<sub>24</sub>, and there are many ways to extend it to the full Mathieu group.

== See also ==

* [[Antilinear map]]
* [[Complex conjugate vector space]]

== References ==
{{reflist}}
* {{citation | last1=Assmus | first1=E.F. | last2=Key | first2=J.D. | year=1994 | title=Designs and Their Codes | publisher=[[Cambridge University Press]] | page=93 | isbn=0-521-45839-0 }}
* {{Bourbaki Algebra I Chapters 1-3 Springer}} <!-- {{sfn | Bourbaki | 1989 | p=}} -->
* {{citation | last1=Bray | first1=John N. | last2=Holt | first2=Derek F. | last3=Roney-Dougal | first3=Colva M. |author3-link= Colva Roney-Dougal | title=Certain classical groups are not well-defined |mr=2502211 | year=2009 | journal=Journal of Group Theory | issn=1433-5883 | volume=12 | issue=2 | pages=171–180 | doi=10.1515/jgt.2008.069| doi-access=free }}
* {{citation | last1=Faure | first1=Claude-Alain | last2=Frölicher | first2=Alfred | year=2000 | title=Modern Projective Geometry | publisher=[[Kluwer Academic Publishers]] | isbn=0-7923-6525-9 }}
* {{citation | last1=Gruenberg | first1=K.W. | last2=Weir | first2=A.J. | year=1977 | title=Linear Geometry | series=Graduate Texts in Mathematics | volume=49 | edition=1st | publisher=Springer-Verlag New York }}
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->

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[[Category:Functions and mappings]]
[[Category:Linear algebra]]
[[Category:Linear operators]]
[[Category:Projective geometry]]