nLab antilinear map



Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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see also algebraic topology



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Basic facts


Complex geometry



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 cc is mapped to multiplication by the scalar’s conjugate c¯\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 cc¯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.


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

Then for KK-modules (or KK-linear spaces) V,WV, W, a KK-antilinear map is a function T:VWT \colon V \to W such that for all x,yVx, y \in V and rKr \in K,

T(rx+y)=r¯T(x)+T(y). 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.


Simple general examples

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

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

Complex vector spaces

A motivating class of examples is when K=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.


See also

and see also the references at Wigner's theorem.

Last revised on October 23, 2023 at 08:03:44. See the history of this page for a list of all contributions to it.