Semilinear map

 ( Functions And Mappings ) 

In linear algebra, particularly projective geometry, a semilinear map between 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 that is:

• Additive map with respect to vector addition: $T(v+v\text{'})\; =\; T(v)+T(v\text{'})$
• there exists a field automorphism θ of K such that $T(\backslash lambda\; v)\; =\; \backslash theta(\backslash lambda)\; T(v)$. If such an automorphism exists and T is nonzero, it is unique, and T is called θ-semilinear.

Where the domain and codomain are the same space (i.e. ), 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 $\backslash operatorname\{\backslash Gamma\; L\}(V),$ by analogy with and extending the general linear group. The special case where the field is the complex numbers $\backslash mathbb\{C\}$ and the automorphism is 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 – 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 has two extensions if n is even and q is odd, and likewise for PSU.

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Definition A map for vector spaces and over fields and respectively is -semilinear, or simply semilinear, if there exists a field homomorphism such that for all , in and in it holds that

1. $f(x+y)=f(x)+f(y),$
2. $f(\backslash lambda\; x)=\backslash sigma(\backslash lambda)\; f(x).$

A given embedding of a field in allows us to identify with a subfield of , making a -semilinear map a K-linear map under this identification. However, a map that is -semilinear for a distinct embedding will not be K-linear with respect to the original identification , unless is identically zero.

More generally, a map between a right -module and a left -module is - semilinear if there exists a ring antihomomorphism such that for all , in and in it holds that

1. $\backslash psi(x\; +\; y)\; =\; \backslash psi(x)\; +\; \backslash psi(y)\; ,$
2. $\backslash psi(x\; \backslash lambda)\; =\; \backslash sigma(\backslash lambda)\; \backslash psi(x)\; .$

The term semilinear applies for any combination of left and right modules with suitable adjustment of the above expressions, with being a homomorphism as needed.

The pair is referred to as a dimorphism.

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Related

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Transpose Let $\backslash sigma\; :\; R\; \backslash to\; S$ be a ring isomorphism, $M$ a right $R$-module and $N$ a right $S$-module, and $\backslash psi\; :\; M\; \backslash to\; N$ a $\backslash sigma$-semilinear map. Define the transpose of $\backslash psi$ as the mapping $\{\}^t\backslash psi\; :\; N^*\; \backslash to\; M^*$ that satisfies $$\backslash langle\; y\; ,\; \backslash psi(x)\; \backslash rangle\; =\; \backslash sigma\backslash left(\backslash left\backslash langle\; \{\}^\backslash text\{t\}\; \backslash psi(y),\; x\; \backslash right\backslash rangle\backslash right)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; y\; \backslash in\; N^*,\; \backslash text\{\; and\; all\; \}\; x\; \backslash in\; M.$$ This is a $\backslash sigma^\{-1\}$-semilinear map.

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Properties Let $\backslash sigma\; :\; R\; \backslash to\; S$ be a ring isomorphism, $M$ a right $R$-module and $N$ a right $S$-module, and $\backslash psi\; :\; M\; \backslash to\; N$ a $\backslash sigma$-semilinear map. The mapping $$M\; \backslash to\; R\; :\; x\; \backslash mapsto\; \backslash sigma^\{-1\}(\backslash langle\; y,\; \backslash psi(x)\backslash rangle),\; \backslash quad\; y\; \backslash in\; N^*$$ defines an $R$-linear form.

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Examples

• Let $K=\backslash mathbf\{C\},\; V=\backslash mathbf\{C\}^n,$ with standard basis $e\_1,\backslash ldots,\; e\_n$. Define the map $f\backslash colon\; V\; \backslash to\; V$ by
• :$f\backslash left(\backslash sum\_\{i=1\}^n\; z\_i\; e\_i\; \backslash right)\; =\; \backslash sum\_\{i=1\}^n\; \backslash bar\; z\_i\; e\_i$

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

• Let $K=\backslash operatorname\{GF\}(q)$ – the Galois field of order $q=p^i$, p the characteristic. Let $\backslash ell^\backslash theta\; =\; \backslash ell^p$. By the Freshman's dream it is known that this is a field automorphism. To every linear map $f\backslash colon\; V\; \backslash to\; W$ between vector spaces V and W over K we can establish a $\backslash theta$-semilinear map
• :$\backslash widetilde\{f\}\; \backslash left(\; \backslash sum\_\{i=1\}^n\; \backslash ell\_i\; e\_i\backslash right)\; :=\; f\; \backslash left(\; \backslash sum\_\{i=1\}^n\; \backslash ell\_i^\backslash theta\; e\_i\; \backslash right)\; .$

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 $R$ be a noncommutative ring, $M$ a left $R$-module, and $\backslash alpha$ an invertible element of $R$. Define the map $\backslash varphi\backslash colon\; M\backslash to\; M\; \backslash colon\; x\; \backslash mapsto\backslash alpha\; x$, so $\backslash varphi(\backslash lambda\; u)=\backslash alpha\; \backslash lambda\; u\; =\; (\backslash alpha\; \backslash lambda\; \backslash alpha^\{-1\})\; \backslash alpha\; u\; =\; \backslash sigma(\backslash lambda)\; \backslash varphi(u)$, and $\backslash sigma$ is an inner automorphism of $R$. Thus, the homothety $x\backslash mapsto\backslash alpha\; x$ need not be a linear map, but is $\backslash sigma$-semilinear.

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General semilinear group Given a vector space V, the set of all invertible semilinear transformations (over all field automorphisms) is the group ΓL( V).

Given a vector space V over K, ΓL( V) decomposes as the semidirect product

$\backslash operatorname\{\backslash Gamma\; L\}(V)\; =\; \backslash operatorname\{GL\}(V)\; \backslash rtimes\; \backslash operatorname\{Aut\}(K)\; ,$

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:

$\backslash sum\_\{b\backslash in\; B\}\; \backslash ell\_b\; b\; \backslash mapsto\; \backslash sum\_\{b\; \backslash in\; B\}\; \backslash ell\_b^\backslash sigma\; b$

for any $\backslash sigma\; \backslash in\; \backslash operatorname\{Aut\}(K)$. We shall denoted this subgroup by Aut( K) _B. 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.

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Proof Every linear map is semilinear, thus $\backslash operatorname\{GL\}(V)\; \backslash leq\; \backslash operatorname\{\backslash Gamma\; L\}(V)$. Fix a basis B of V. Now given any semilinear map f with respect to a field automorphism , then define by

$g\; \backslash left(\backslash sum\_\{b\; \backslash in\; B\}\; \backslash ell\_b\; b\backslash right)$

= \sum_{b \in B}f \left(\ell_b^{\sigma^{-1}} b\right)

= \sum_{b \in B} \ell_b f (b) 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: .

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

$hv=fg^\{-1\}v=\backslash sum\_\{b\; \backslash in\; B\}\; \backslash ell\_b^\backslash sigma\; b$

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) _B, so .

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Applications

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Projective geometry The $\backslash operatorname\{\backslash Gamma\; L\}(V)$ groups extend the typical in GL( V). The importance in considering such maps follows from the consideration of projective geometry. The induced action of $\backslash operatorname\{\backslash Gamma\; L\}(V)$ on the associated projective space P( V) yields the , denoted $\backslash operatorname\{P\backslash Gamma\; L\}(V)$, 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 $f\backslash colon\; V\; \backslash to\; W$ induces an order-preserving map $f\backslash colon\; \backslash operatorname\{PG\}(V)\; \backslash to\; \backslash operatorname\{PG\}(W)$. 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.

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Mathieu group The group PΓL(3,4) can be used to construct the Mathieu group M₂₄, which is one of the sporadic simple groups; PΓL(3,4) is a maximal subgroup of M₂₄, and there are many ways to extend it to the full Mathieu group.

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See also

• Antilinear map
• Complex conjugate vector space

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