Semilinear map

In linear algebra, particularly projective geometry, a semilinear map between vector spaces "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:
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 formula_5 by analogy with and extending the general linear group. The special case where the field is the complex numbers and the automorphism is complex conjugation, a semilinear map is called an antilinear map. 
Similar notation (replacing Latin characters with Greek) are used for semilinear analogs of more restricted linear transform; formally, the semidirect product of a linear group with the Galois group of field automorphism. For example, PΣU is used for the semilinear analogs of the projective special unitary group PSU. Note however, that it is 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.
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
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
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.
Related.
Transpose.
Let be a ring isomorphism, a right -module and a right -module, and a -semilinear map. We define the transpose of as the mapping that satisfies
This is a -semilinear map.
Properties.
Let be a ring isomorphism, a right -module and a right -module, and a -semilinear map. The mapping
defines an -linear form.
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
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:
for any formula_35. 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.
Proof.
Every linear map is semilinear, thus formula_36. Fix a basis "B" of "V". Now given any semilinear map "f" with respect to a field automorphism , then define by
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 formula_38. For every formula_39 in "V",
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 .
Applications.
Projective geometry.
The formula_41 groups extend the typical classical groups in GL("V"). The importance in considering such maps follows from the consideration of projective geometry. The induced action of formula_41 on the associated projective space P("V") yields the , denoted formula_43, 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 formula_19 induces an order-preserving map formula_45. 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.
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.