Sesquilinear Form

• Note taken on [2025-03-13 Thu 23:12]
  Should this be over a pair of modules?

A sesquilinear function on a Left Module $M$ over a Involutive Ring $R$ is a function $\langle-,-\rangle:M\times M\to R$ that is Linear in one variable and Semilinear in the other.

Explicitly:

• $\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$
• $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$
• $\langle cx,y\rangle=c^{*}\langle x,y\rangle$
• $\langle x,cy\rangle=\langle x,y\rangle c$

We can obtain the corresponding notion for Right Modules by replacing the final two equations with

• $\langle xc,y\rangle=c^{*}\langle x,y\rangle$
• $\langle x,yc\rangle=\langle x,y\rangle c$

When the involution $(-)^{*}$ is the identity function, then sesquilinearity is equivalent to Bilinearity.