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
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) |
Comments
Instead of adjoint module and adjoint automorphism one also uses dual module and dual automorphism (cf. Adjoint module; Automorphism).
Contragredient automorphism. V.L. Popov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contragredient_automorphism&oldid=19201