Let be a field and its prime subfield. For example, if is then is , and if is the finite field of order , then is .

Definition 1   Given a field automorphism of , a function between two vector spaces and is -semilinear, or simply semilinear, if for all and it follows: (shown here first in left hand notation and then in the preferred right hand notation.)

1. , (in right hand notation: .)
2. , (in right hand notation: .)

where denotes the image of under .

Remark 2   must be a field automorphism for to remain additive, for example, must fix the prime subfield as

Also

so . Finally,

Every linear transformation is semilinear, but the converse is generally not true. If we treat and as vector spaces over , (by considering as vector space over first) then every -semilinear map is a -linear map, where is the prime subfield of .

Example

• Let , with standard basis . Define the map by
  
  is semilinear (with respect to the complex conjugation field automorphism) but not linear.
• Let - the Galois field of order , the characteristic. Let , for . By the Freshman's dream it is known that this is a field automorphism. To every linear map between vector spaces and over we can establish a -semilinear map
  

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 , the set of all invertible semilinear maps (over all field automorphisms) is the group .

Proposition 4   Given a vector space over , and the prime subfield of , then decomposes as the semidirect product

where is the Galois group of .

Remark 5   We identify with a subgroup of by fixing a basis for and defining the semilinear maps:

for any . We shall denoted this subgroup by . We also see these complements to in are acted on regularly by as they correspond to a change of basis.

Proof. Every linear map is semilinear thus . Fix a basis of . Now given any semilinear map with respect to a field automorphism , then define by

As is also a basis of , it follows is simply a basis exchange of and so linear and invertible: .

Set . For every in ,

thus is in the subgroup relative to the fixed basis . This factorization is unique to the fixed basis . Furthermore, is normalized by the action of , so .

The groups extend the typical classical groups in . The importance in considering such maps follows from the consideration of projective geometry.

The projective geometry of a vector space , denoted , is the lattice of all subspaces of . Although the typical semilinear map is not a linear map, it does follow that every semilinear map induces an order-preserving map . 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.

Bibliography

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).