In linear algebra, particularly projective geometry, a semilinear transformation between vector spaces V and W over a field K is a function that is a linear transformation "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function that is:
- linear with respect to vector addition:
- semilinear with respect to scalar multiplication: where θ is a field automorphism of K, and means the image of the scalar under the automorphism. There must be a single automorphism θ for T, in which case T is called θ-semilinear.
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 by analogy with and extending the general linear group.
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 (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 SU(n,q) has two extension if n is even and q is odd, and likewise for PSU.
Let K be a field and k its prime subfield. For example, if K is C then k is Q, and if K is the finite field of order then k is
Given a field automorphism of K, a function between two K vector spaces V and W is -semilinear, or simply semilinear, if for all in V and in K it follows:
where denotes the image of under
Note that must be a field automorphism for f to remain additive, for example, must fix the prime subfield as
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 -semilinear map is a k-linear map, where k is the prime subfield of K.
- Let with standard basis Define the map by
- f is semilinear (with respect to the complex conjugation field automorphism) but not linear.
- Let – the Galois field of order p the characteristic. Let By the Freshman's dream it is known that this is a field automorphism. To every linear map between vector spaces V and W over K 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.
General semilinear group
Given a vector space V, the set of all invertible semilinear maps (over all field automorphisms) is the group
Given a vector space V over K, and k the prime subfield of K, then decomposes as the semidirect product
where Gal(K/k) is the Galois group of Similarly, semilinear transforms of other linear groups can be defined as the semidirect product with the Galois group, or more intrinsically as the group of semilinear maps of a vector space preserving some properties.
We identify Gal(K/k) with a subgroup of by fixing a basis B for V and defining the semilinear maps:
for any We shall denoted this subgroup by Gal(K/k)B. We also see these complements to GL(V) in are acted on regularly by GL(V) as they correspond to a change of basis.
Every linear map is semilinear, thus 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 For every 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
The 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 on the associated vector space P(V) yields the projective semilinear group, denoted 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 induces an order-preserving map 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.
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.
- Gruenberg, K. W. and Weir, A.J. Linear Geometry 2nd Ed. (English) Graduate Texts in Mathematics. 49. New York – Heidelberg – Berlin: Springer-Verlag. X, 198 pp. (1977).
- Bray, John N.; Holt, Derek F.; Roney-Dougal, Colva M. (2009), "Certain classical groups are not well-defined", Journal of Group Theory 12 (2): 171–180, doi:10.1515/jgt.2008.069, ISSN 1433-5883, MR 2502211