nLab antilinear map

Idea

An antilinear map or conjugate linear map is much like a linear map, but instead of commuting with “scalar multiplication” it “anti-commutes” with it, in that multiplication by a scalar $c$ is mapped to multiplication by the scalar’s conjugate $\overline{c}$.

In order to make sense of this notion, the ground ring of the modules (or ground field of the vector spaces) that serve as the map’s domain and codomain must have the additional structure of an involution, to serve as the conjugation map $c \mapsto \overline{c}$.

An antilinear map has a central role in the concept of star-algebra. Conversely, an antilinear map can be seen as built on a star-algebra, in that the involution makes the ground ring into a star-algebra over itself.

Definition

Given a commutative ring (often a field, or possibly just a rig) $K$, equipped with an involution $x \mapsto \overline{x}$, meaning an endomorphism with $\overline{\overline{x}} = x$ for all $x \in K$.

Then for $K$-modules (or $K$-linear spaces) $V, W$, a $K$-antilinear map is a function $T \colon V \to W$ such that for all $x, y \in V$ and $r \in K$,

$$T(r \cdot x + y) \;=\; \overline{r} \cdot T(x) + T(y) \,.$$

This differs from the definition of a linear map in the appearance of $\overline{(-)}$ on the right-hand side.

Examples

Simple general examples

Every $K$-linear map is also a $K$-antilinear map, for $K$ regarded as equipped with the identity involution.

Any involution $\overline{(-)} \colon K \to K$ is itself an antilinear map.

Complex vector spaces

A motivating class of examples is when $K = \mathbb{C}$ is the complex numbers, and $\overline{(-)}$ is complex conjugation.

In particular, the Hermitian adjoint is an antilinear map from a space of $\mathbb{C}$-linear operators to itself.

Further examples

A $\*$-algebra requires by definition its anti-involution to be antilinear.

Related concepts

• real structure

• antiunitary operator

• star-algebra

References

See also

• Wikipedia, Antilinear map

and see also the references at Wigner's theorem.