Intro to Semilinear Maps

I first learned about semilinear maps through one of my mathematical explorations. In this article, we introduce them and prove some basic results about them. We assume some knowledge of abstract algebra and related subjects (vector spaces, modules, universal algebra, category theory, etc.)

(I was still learning about the concept alongside writing this article, so if there are any inaccuracies here, please let me know!)

Introduction

We motivate the concept of a semilinear map by asking the following question: is there a way to generalize the idea of a linear map to vector spaces over different fields? Such a concept could facilitate insights and translations of results between vector spaces over different fields in a way that may not have been as accessible before.

We’ll start by examining the standard definition of linear map:

Definition 1 Let ${V,W}$ be vector spaces over a field ${F}$ . A linear map ${h:V\rightarrow W}$ is a function that satisfies ${h(a+b)=h(a)+h(b)}$ and ${h(sa)=sh(a)}$ for ${a,b\in V}$ and ${s\in F}$ .

We want to generalize this definition so that ${W}$ can be over another field ${G}$ . For this new definition, we note that ${h(a+b)=h(a)+h(b)}$ can translate exactly regardless of the underlying field, while ${h(sa)=sh(a)}$ requires that ${W}$ be over the same field ${F}$ . To allow ${W}$ to be over ${G}$ , we must replace ${sh(a)}$ with ${s'h(a)}$ where ${s'}$ is a corresponding scalar from ${G}$ . Thus, we require a map between the fields, ${\rho:F\rightarrow G}$ , such that ${h(sa)=\rho(s)h(a)}$ . For the sake of compatibility, it seems like we should impose an additional condition on ${\rho}$ , for example that ${\rho}$ be a field homomorphism. However, it turns out that this is not necessary, since we will prove that ${\rho}$ must be a field homomorphism anyway.

Let’s start with our provisional definition:

Definition 2 Let ${V,W}$ be vector spaces over fields ${F,G}$ respectively. A provisional semilinear map is a pair of ${h:V\rightarrow W}$ and ${\rho:F\rightarrow G}$ such that ${h}$ is an additive group homomorphism and ${h(sa)=\rho(s)h(a)}$ for all ${a\in V,s\in F}$ . (Of course, the term “provisional semilinear map” is not standard.)

Note that ${(0,\rho)}$ is always a provisional semilinear map for any ${\rho}$ (not just for ${\rho}$ a homomorphism.) However, we have:

Lemma 3 If ${(h,\rho )}$ is a provisional semilinear map and ${h\neq 0}$ , then ${\rho}$ is a field homomorphism.

Proof: We have, for all ${s,t\in F}$ , ${a\in V}$ ,

$(\rho(s)+\rho(t))h(a) = \rho(s)h(a)+\rho(t)h(a)$
$= h(sa)+h(ta)$
$= h((s+t)a)$
$= \rho(s+t)h(a).$

If there exists ${s,t\in F}$ such that ${\rho(s)+\rho(t)\neq \rho(s+t)}$ , then ${h(a)=0}$ for all ${a}$ , contradiction. Thus, ${\rho(s)+\rho(t)=\rho(s+t)}$ for all ${s,t\in F}$ . Similarly, we have ${(\rho(s)\rho(t))h(a)=\rho(st)h(a)}$ and ${\rho(1)h(a)=h(a)}$ , and analogous logic yields ${\rho(s)\rho(t)=\rho(st)}$ and ${\rho(1)=1}$ . $\Box$

Thus, our provisional concept reduces to:

Definition 4 Let ${V,W}$ be vector spaces over fields ${F,G}$ respectively. A semilinear map is a pair of ${h:V\rightarrow W}$ and ${\rho:F\rightarrow G}$ such that ${h}$ is an additive group homomorphism, ${\rho}$ is a field homomorphism, and ${h(sa)=\rho(s)h(a)}$ for all ${a\in V,s\in F}$ .

This definition is used for example in projective geometry (see the nLab article.)

Note that if we replicate this definition for vector spaces over the same field we don’t actually get back linear maps, since in general $\rho$ could be any field endomorphism, not just the identity. In fact, some authors (e.g., this PlanetMath article) use an even more restrictive definition that is still more general than a linear map:

Definition 5 Let ${V,W}$ be vector spaces over a field ${F}$ . A semilinear map is a pair of ${h:V\rightarrow W}$ and ${\rho:F\rightarrow F}$ such that ${h}$ is an additive group homomorphism, ${\rho}$ is a field automorphism, and ${h(sa)=\rho(s)h(a)}$ for all ${a\in V,s\in F}$ .

This definition is a generalization of a common scenario. For example, over ${\mathbb{C}}$ , it is common to consider maps ${f}$ such that ${f(zv)=\Bar{z}f(v)}$ ; this especially happens in applications such as quantum mechanics. Since complex conjugation is an automorphism, we may view semilinear maps as a generalization of these maps to vector spaces over general fields.

In the other direction, we can generalize the definition of semilinear map further to modules over rings. The only difference in the statement of this definition is with making the equations make sense with left and right module multiplications:

Definition 6 Let ${R,S}$ be rings and let ${M}$ be a right ${R}$ -module and ${N}$ be a left ${S}$ -module. A semilinear map is a pair of ${h:M\rightarrow N}$ and ${\rho:R\rightarrow S}$ such that ${h}$ is an additive group homomorphism, ${\rho}$ is a ring antihomomorphism, and ${h(ra)=\rho(r)h(a)}$ for all ${a\in V,r\in R}$ .

In this case, we cannot omit the requirement that ${\rho}$ be an antihomomorphism by replicating our earlier proof for vector spaces, since that proof relied on field properties (specifically, ability to multiply by an inverse of a scalar to get that ${sv=0\rightarrow s=0 \lor v=0}$ .)

In fact, we can generalize the definition of semilinear map to any combination of ${R}$ and ${S}$ being left or right modules, requiring that ${\rho}$ be a homomorphism or antihomomorphism appropriately (see Bourbaki, Algebra I Chapters 1-3; I used a version translated into English by Hermann in 1974.)

We will henceforth discuss semilinear maps over different fields.

Semilinear Maps Over Different Fields

Generalizations of concepts like linear maps (which are just homomorphisms for vector spaces) are fundamental to the subjects of universal algebra and category theory. How do semilinear maps compare to these other generalizations?

For universal algebra, semilinear maps cannot be “naturally” expressed as homomorphisms of suitable algebras (in the universal algebraic sense of course), since these maps are between vector spaces, which when viewed as algebras have signatures that are dependent on their underlying fields, and homomorphisms are not defined for algebras which do not have the same signature.

On the other hand, for category theory, semilinear maps indeed are the morphisms of a suitable category, namely the category of pairs of vector spaces and their underlying fields; the composition of morphisms is given “naturally” by ${(h,\rho)\circ (k,\sigma)=(h\circ k,\rho\circ\sigma)}$ .

We now compare semilinear and linear maps.

In the case that ${F=G}$ , every linear map results in a semilinear map by taking ${\rho}$ as the identity. However, semilinear maps are not equivalent to linear maps even if ${F=G}$ , in fact even if ${V=W=F=G}$ . To show this, we note that, if semilinear maps were equivalent to linear maps when ${V=W=F=G}$ , then since every linear map results in a semilinear map with ${\rho}$ as the identity, every semilinear map would need to be produced in this way, and thus have ${\rho}$ as the identity. We produce a counterexample: let ${V=W=F=G=\mathbb{C}}$ , then we may take ${h,\rho}$ both to be complex conjugation.

Let ${(h,\rho )}$ be a semilinear map with ${h\neq 0}$ . Since ${\rho}$ is a field homomorphism, ${\rho}$ must be an embedding. (If you are unfamiliar with this result, one possible proof is to see that the kernel of a homomorphism is an ideal, then to show that the only ideals of a field ${F}$ are ${0}$ and ${F}$ , and finally to account for the fact that the homomorphism must send 1 to 1.)

Then, let ${s\in F}$ , ${w\in W}$ , and define the scalar multiplication ${sw:=\rho(s)w}$ , so that ${W}$ is a vector space over ${F}$ . The equations ${h(a+b)=h(a)+h(b)}$ and ${h(sa)=\rho(s)h(a)}$ show that ${h}$ is a linear map with ${V,W}$ viewed as vector spaces over ${F}$ .

Conversely, given a field embedding ${\rho}$ , let ${h:V\rightarrow W}$ be a linear map with both viewed as vector spaces over ${F}$ . Then, ${h(a+b)=h(a)+h(b)}$ and ${h(sa)=sh(a)=\rho(s)h(a)}$ imply that ${(h,\rho )}$ is a semilinear map.

Thus, semilinear maps with nonzero ${h}$ are equivalent to pairs of field embeddings and nonzero linear maps between ${V}$ and ${W}$ with both viewed as vector spaces over ${F}$ .

Equivalently, given ${\rho}$ , let ${F'}$ be the image of ${\rho}$ ; ${F'}$ is a subfield of ${G}$ , so ${W}$ is a vector space over ${F'}$ . Let ${s\in F',v\in V}$ , and define the scalar multiplication ${sv:=\rho^{-1}(s)v}$ , so that ${V}$ is a vector space over ${F'}$ . By similar logic as above, we can show that semilinear maps with nonzero ${h}$ are equivalent to nonzero linear maps between ${V}$ and ${W}$ with both viewed as vector spaces over ${F'}$ : if ${h:V\rightarrow W}$ is a linear map, then ${h(\rho^{-1}(s)a)=h(sa)=sh(a)}$ , so letting ${t=\rho^{-1}(s)}$ , ${s=\rho(t)}$ and ${h(ta)=\rho(t)h(a)}$ . This is true for all ${t\in F}$ since ${\text{Im }\rho^{-1}=F}$ , so ${(h,\rho)}$ is a semilinear map. The converse is similarly proven.

Written July and August 2021. Edited March 2025.

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