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:

• additive

• there exists a field automorphism of K. If such an automorphism exists and T is nonzero, it is unique, and T is called -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 by analogy with and extending the general linear group 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. However, 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 SU(n, q) 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

-semilinear map. Define the transpose that satisfies-semilinear map.

Properties

-linear form.

Examples

• by

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

• , p. By the Freshman's dream between vector spaces V and W over K-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.

• is an inner automorphism. Thus, the homothety-semilinear.

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:

. 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

. 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: .

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

groups extend the typical classical groups in GL(V). The importance in considering such maps follows from the consideration of projective geometry on the associated projective space P(V) yields the , 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. 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.

See also

• Complex conjugate vector space

References