Sesquilinear Form - Generalization

Generalization

A generalization called a semi-bilinear form was used by Reinhold Baer to characterize linear manifolds that are dual to eachother in chapter 5 of his book Linear Algebra and Projective Geometry (1952). For a field F and A linear over F he requires

A pair consisting of an anti-automorphism α of the field F and a function f:A×A→F satisfying

for all a,b,c ∈ A and

for all t ∈ F, all x,y ∈ A (page 101)

(The "transformation exponential notation" is adopted in group theory literature.)

Baer calls such a form an α-form over A. The usual sesquilinear form has complex conjugation for α. When α is the identity, then f is a bilinear form.

In the algebraic structure called a *-ring the anti-automorphism is denoted by * and forms are constructed as indicated for α. Special constructions such as skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms are all considered in the broader context.

Particularly in L-theory, one also sees the term ε-symmetric form, where, to refer to both symmetric and skew-symmetric forms.