semilinear transformation

Let K be a field and k its prime subfieldMathworldPlanetmath. For example, if K is then k is , and if K is the finite fieldMathworldPlanetmath of order q=pi, then k is p.

Definition 1.

Given a field automorphism θ of K, a function f:VW between two K vector spacesMathworldPlanetmath V and W is θ-semilinear, or simply semilinear, if for all x,yV and lK it follows: (shown here first in left hand notation and then in the preferred right hand notation.)

  1. 1.

    f(x+y)=f(x)+f(y), (in right hand notation: (x+y)f=xf+yf.)

  2. 2.

    f(lx)=lθf(x), (in right hand notation: (lx)f=lθxf.)

where lθ denotes the image of l under θ.

Remark 2.

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




so (l1+l2)θ=l1θ+l2θ. Finally,


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 θ-semilinear map is a k-linear map, where k is the prime subfield of K.


  • Let K=, V=n with standard basis e1,,en. Define the map f:VV by


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

  • Let K=GF(q) – the Galois field of order q=pi, p the characteristicPlanetmathPlanetmath. Let lθ=lp, for lK. By the Freshman’s dream it is known that this is a field automorphism. To every linear map f:VW between vector spaces V and W over K we can establish a θ-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 V, the set of all invertiblePlanetmathPlanetmath semilinear maps (over all field automorphisms) is the group ΓL(V).

Proposition 4.

Given a vector space V over K, and k the prime subfield of K, then ΓL(V) decomposes as the semidirect productMathworldPlanetmath


where Gal(K/k) is the Galois groupMathworldPlanetmath of K/k.

Remark 5.

We identify Gal(K/k) with a subgroupMathworldPlanetmathPlanetmath of ΓL(V) by fixing a basis B for V and defining the semilinear maps:


for any σGal(K/k). We shall denoted this subgroup by Gal(K/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.


Every linear map is semilinear thus GL(V)ΓL(V). Fix a basis B of V. Now given any semilinear map f with respect to a field automorphism σGal(K/k), then define g:VV by


As (B)f is also a basis of V, it follows g is simply a basis exchange of V and so linear and invertible: gGL(V).

Set h:=g-1f. For every v=bBlb0 in V,


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 ΓL(V)=GL(V)Gal(K/k). ∎

The ΓL(V) groups extend the typical classical groups in GL(V). The importance in considering such maps follows from the consideration of projective geometryMathworldPlanetmath.

The projective geometry of a vector space V, denoted PG(V), is the lattice of all subspacesPlanetmathPlanetmathPlanetmath of V. Although the typical semilinear map is not a linear map, it does follow that every semilinear map f:VW induces an order-preserving map f:PG(V)PG(W). That is, every semilinear map induces a projectivityMathworldPlanetmath. The converse of this observation is the Fundamental Theorem of Projective GeometryMathworldPlanetmath. Thus semilinear maps are useful because they define the automorphism groupMathworldPlanetmath of the projective geometry of a vector space.


  • 1 Gruenberg, K. W. and Weir, A.J. Linear GeometryMathworldPlanetmath 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. X, 198 p. DM 29.10; $ 12.80 (1977).
Title semilinear transformation
Canonical name SemilinearTransformation
Date of creation 2013-03-22 15:51:06
Last modified on 2013-03-22 15:51:06
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 20
Author Algeboy (12884)
Entry type Definition
Classification msc 15A04
Synonym semilinear map
Synonym semilinear transform
Synonym semi-linear transformation
Synonym semi-linear map
Related topic ClassicalGroups
Related topic ProjectiveSpace
Defines semilinear transform
Defines Gamma L