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 T : V → W that is:
Contents
- Definition
- Transpose
- Properties
- Examples
- General semilinear group
- Proof
- Projective geometry
- Mathieu group
- References
Where the domain and codomain are the same space (i.e. T : V → V), 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
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 extensions if n is even and q is odd, and likewise for PSU.
Definition
A map f : V → W for vector spaces V and W over fields K and L respectively is σ-semilinear, or simply semilinear, if there exists a field homomorphism σ : K → L such that for all x, y in V and λ in K it holds that
-
f ( x + y ) = f ( x ) + f ( y ) , -
f ( λ x ) = σ ( λ ) f ( x ) .
A given embedding σ of a field K in L allows us to identify K with a subfield of L, making a σ-semilinear map a K-linear map under this identification. However, a map that is τ-semilinear for a distinct embedding τ ≠ σ will not be K-linear with respect to the original identification σ, unless f is identically zero.
More generally, a map ψ : M → N between a right R-module M and a left S-module N is σ-semilinear if there exists a ring antihomomorphism σ : R → S such that for all x, y in M and λ in R it holds that
-
ψ ( x + y ) = ψ ( x ) + ψ ( y ) , -
ψ ( x λ ) = σ ( λ ) ψ ( x ) .
The term semilinear applies for any combination of left and right modules with suitable adjustment of the above expressions, with σ being a homomorphism as needed.
The pair (ψ, σ) is referred to as a dimorphism.
Transpose
Let σ : R → S be a ring isomorphism, M a right R-module and N a right S-module, and ψ : M → N a σ-semilinear map. We define the transpose of ψ as the mapping tψ : N∗ → M∗ that satisfies
This is a σ−1-semilinear map.
Properties
Let σ : R → S be a ring isomorphism, M a right R-module and N a right S-module, and ψ : M → N a σ-semilinear map. The mapping
defines an R-linear form.
Examples
General semilinear group
Given a vector space V, the set of all invertible semilinear transformations V → V (over all field automorphisms) is the group ΓL(V).
Given a vector space V over K, and k the prime field of K, then ΓL(V) decomposes as the semidirect product
where Aut(K/k) is the automorphisms of K as a field over 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/k) with a subgroup of ΓL(V) by fixing a basis B for V and defining the semilinear maps:
for any
Proof
Every linear map is semilinear, thus
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: g ∈ GL(V).
Set
thus h is in the Aut(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 Aut(K/k)B, so ΓL(V) = GL(V) ⋊ Aut(K/k).
Projective geometry
The
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
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.