Contragredient automorphism

to an automorphism  of a right module  over a ring

The automorphism  of the left -module  (* denotes taking the dual or adjoint module) that is adjoint to the inverse automorphism to . More generally, if  is an automorphism between a right -module  and a right -module , then the contragredient isomorphism to  is the isomorphism of the left -module  onto the left -module  that is adjoint to the inverse of the isomorphism . Let  and  be the canonical bilinear forms on  and . Then  is defined by the following identity with respect to , :

If  and  have finite bases, then  is the isomorphism contragredient to .

Let  be a ring with an identity, let  be a right -module with a finite basis, let  be an automorphism of , and let  be the matrix of  in a fixed basis (this matrix is invertible). Then in the dual basis, the matrix of the contragredient automorphism  has the form

(here  denotes the transpose). The matrix  is called the matrix contragredient to the invertible matrix .

#### References[edit]

[1] |  N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)

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#### Comments[edit]

Instead of adjoint module and adjoint automorphism one also uses dual module and dual automorphism (cf. Adjoint module; Automorphism).