nLab sesquilinear form

Idea

On a complex vector space $V$ a sesquilinear map is a function of two arguments

$$\langle -,-\rangle \;\colon\; V \times V \longrightarrow \mathbb{C}$$

which is a linear function in one argument (say the second) and complex anti-linear in the other.

Definition

A sesquilinear form $\langle -,-\rangle$ is called

$$\left. \array{ \text{positive definite} &if& \langle v^\ast,v \rangle \gt 0 \\ \text{negative definite} &if& \langle v^\ast,v \rangle \lt 0 \\ \text{positive semi-definite} &if& \langle v^\ast,v \rangle \geq 0 \\ \text{negative semi-definite} &if& \langle v^\ast,v \rangle \leq 0 } \right\rbrace \;\; \text{for all}\; v \neq 0 \,.$$

Finally, it is called indefinite if it is neither positive nor negative semi-definite.

More generally:

Let $A$ be a star algebra. Then every left $A$-module $\rho_l \colon A \otimes V \longrightarrow V$ canonically becomes a right module $\rho_r \colon V \otimes A \longrightarrow A$ by setting

$$\rho_r(v,a) \coloneqq a^\ast v$$

and vice versa.

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

$$V^\ast \otimes_A V^\ast$$

of the $A$-linear dual $V^\ast$ of $V$ with itself.

Related concepts

• bilinear form, quadratic form

• symplectic form, Kähler form, Hermitian form

References

See also:

• Brian Osserman, A primer on sesquilinear forms (pdf)

• Wikipedia, Sesquilinear form