Semilinear transformation
Encyclopedia
In linear algebra
, particularly projective geometry
, a semilinear transformation between vector space
s 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:
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 – 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.
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: (shown here first in left hand notation and then in the preferred right hand notation.)
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
Also
so
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
-semilinear map is a k 
-linear map, where k is the prime subfield of K.
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.
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.
Fix a basis B of V. Now given any semilinear map f with respect to a field automorphism 
then define 
by
As (B)f 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
groups extend the typical classical group
s 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 , 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.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, particularly projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, a semilinear transformation between vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s V and W over a field K is a function that is a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
"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 groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
.
Similar notation (replacing Latin characters with Greek) are used for semilinear analogs of more restricted linear transform; formally, the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct 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 – 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.
Definition
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 fieldFinite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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: (shown here first in left hand notation and then in the preferred right hand notation.)
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 asAlso
so
Finally,Every linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
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.Examples
- 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 dreamFreshman's dreamThe freshman's dream is a name sometimes given to the error n = xn + yn, where n is a real number . Beginning students commonly make this error in computing the exponential of a sum of real numbers...
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
- Let
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 groupGiven a vector space V over K, and k the prime subfield of K, then
decomposes as the semidirect productSemidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
where Gal(K/k) is the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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.Proof
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 byAs (B)f 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
Projective geometry
The groups extend the typical classical groupClassical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s in GL(V). The importance in considering such maps follows from the consideration of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
. The induced action of
on the associated vector space P(V) yields the , denoted extending the projective linear groupProjective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...
, 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.