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:

• additive with respect to vector addition: ${\displaystyle T(v+v')=T(v)+T(v')}$
• there exists a field automorphism θ of K such that ${\displaystyle T(\lambda v)=\lambda ^{\theta }T(v)}$, where ${\displaystyle \lambda ^{\theta }}$ is the image of the scalar ${\displaystyle \lambda }$ under the automorphism. If such an automorphism exists and T is nonzero, it is unique, and T is called θ-semilinear.

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 ${\displaystyle \operatorname {\Gamma L} (V),}$ by analogy with and extending the general linear group. The special case where the field is the complex numbers ${\displaystyle \mathbb {C} }$ and the automorphism is complex conjugation, a semilinear map is called an antilinear map.

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

1. ${\displaystyle f(x+y)=f(x)+f(y),}$
2. ${\displaystyle f(\lambda x)=\sigma (\lambda )f(x).}$

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

1. ${\displaystyle \psi (x+y)=\psi (x)+\psi (y),}$
2. ${\displaystyle \psi (x\lambda )=\sigma (\lambda )\psi (x).}$

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.^[1][2]

The pair (ψ, σ) is referred to as a dimorphism.^[3]

Related

Transpose

Let ${\displaystyle \sigma :R\to S}$ be a ring isomorphism, ${\displaystyle M}$ a right ${\displaystyle R}$-module and ${\displaystyle N}$ a right ${\displaystyle S}$-module, and ${\displaystyle \psi :M\to N}$ a ${\displaystyle \sigma }$-semilinear map. Define the transpose of ${\displaystyle \psi }$ as the mapping ${\displaystyle {}^{t}\psi :N^{*}\to M^{*}}$ that satisfies^[4]

$${\displaystyle \langle y,\psi (x)\rangle =\sigma \left(\left\langle {}^{\text{t}}\psi (y),x\right\rangle \right)\quad {\text{ for all }}y\in N^{*},{\text{ and all }}x\in M.}$$

This is a ${\displaystyle \sigma ^{-1}}$-semilinear map.

Properties

Let ${\displaystyle \sigma :R\to S}$ be a ring isomorphism, ${\displaystyle M}$ a right ${\displaystyle R}$-module and ${\displaystyle N}$ a right ${\displaystyle S}$-module, and ${\displaystyle \psi :M\to N}$ a ${\displaystyle \sigma }$-semilinear map. The mapping

$${\displaystyle M\to R:x\mapsto \sigma ^{-1}(\langle y,\psi (x)\rangle ),\quad y\in N^{*}}$$

defines an ${\displaystyle R}$-linear form.^[5]

Examples

• Let ${\displaystyle K=\mathbf {C} ,V=\mathbf {C} ^{n},}$ with standard basis ${\displaystyle e_{1},\ldots ,e_{n}}$. Define the map ${\displaystyle f\colon V\to V}$ by

  ${\displaystyle f\left(\sum _{i=1}^{n}z_{i}e_{i}\right)=\sum _{i=1}^{n}{\bar {z}}_{i}e_{i}}$

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

• Let ${\displaystyle K=\operatorname {GF} (q)}$ – the Galois field of order ${\displaystyle q=p^{i}}$, p the characteristic. Let ${\displaystyle \ell ^{\theta }=\ell ^{p}}$. By the Freshman's dream it is known that this is a field automorphism. To every linear map ${\displaystyle f\colon V\to W}$ between vector spaces V and W over K we can establish a ${\displaystyle \theta }$-semilinear map

  ${\displaystyle {\widetilde {f}}\left(\sum _{i=1}^{n}\ell _{i}e_{i}\right):=f\left(\sum _{i=1}^{n}\ell _{i}^{\theta }e_{i}\right).}$

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.

• Let ${\displaystyle R}$ be a noncommutative ring, ${\displaystyle M}$ a left ${\displaystyle R}$-module, and ${\displaystyle \alpha }$ an invertible element of ${\displaystyle R}$. Define the map ${\displaystyle \varphi \colon M\to M\colon x\mapsto \alpha x}$, so ${\displaystyle \varphi (\lambda u)=\alpha \lambda u=(\alpha \lambda \alpha ^{-1})\alpha u=\sigma (\lambda )\varphi (u)}$, and ${\displaystyle \sigma }$ is an inner automorphism of ${\displaystyle R}$. Thus, the homothety ${\displaystyle x\mapsto \alpha x}$ need not be a linear map, but is ${\displaystyle \sigma }$-semilinear.^[6]

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, ΓL(V) decomposes as the semidirect product

${\displaystyle \operatorname {\Gamma L} (V)=\operatorname {GL} (V)\rtimes \operatorname {Aut} (K),}$

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:

${\displaystyle \sum _{b\in B}\ell _{b}b\mapsto \sum _{b\in B}\ell _{b}^{\sigma }b}$

for any ${\displaystyle \sigma \in \operatorname {Aut} (K)}$. 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.

Applications

See also

• Antilinear map
• Complex conjugate vector space

References

• Assmus, E.F.; Key, J.D. (1994), Designs and Their Codes, Cambridge University Press, p. 93, ISBN 0-521-45839-0
• Bourbaki, Nicolas (1989) [1970]. Algebra I Chapters 1-3 [Algèbre: Chapitres 1 à 3] (PDF). Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64243-5. OCLC 18588156.
• 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
• Faure, Claude-Alain; Frölicher, Alfred (2000), Modern Projective Geometry, Kluwer Academic Publishers, ISBN 0-7923-6525-9
• Gruenberg, K.W.; Weir, A.J. (1977), Linear Geometry, Graduate Texts in Mathematics, vol. 49 (1st ed.), Springer-Verlag New York
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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