nLab
sesquilinear form

Context

Higher linear algebra

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

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

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

On a complex vector space a sesquilinear map is a function of two arguments which is a linear function in one argument and complex anti-linear in the other.

More generally:

Let AA be a star algebra. Then every left AA-module ρ l:AVV\rho_l \colon A \otimes V \longrightarrow V canonically becomes a right module ρ r:VAA\rho_r \colon V \otimes A \longrightarrow A by setting

ρ r(v,a)a *v \rho_r(v,a) \coloneqq a^\ast v

and vice versa.

With this operation understood to turn a left module VV into a right module, then a sesquilinear form on VV is simply an element in the tensor product of modules

V * AV * V^\ast \otimes_A V^\ast

of the AA-linear dual V *V^\ast of VV with itself.

References

Last revised on September 6, 2016 at 09:51:05. See the history of this page for a list of all contributions to it.