Dieudonné determinant
In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné (1943).
If K is a division ring, then the Dieudonné determinant is a
abelianization
K ×/ [K ×, K ×] of the multiplicative group K × of K.
For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in K ×/ [K ×, K ×], of
Properties
Let R be a local ring. There is a determinant map from the
unit group R ×ab with the following properties:[1]
- The determinant is invariant under elementary row operations
- The determinant of the identity matrix is 1
- If a row is left multiplied by a in R × then the determinant is left multiplied by a
- The determinant is multiplicative: det(AB) = det(A)det(B)
- If two rows are exchanged, the determinant is multiplied by −1
- If R is commutative, then the determinant is invariant under transposition
Tannaka–Artin problem
Assume that K is finite over its
reduced norm
gives a homomorphism Nn from GLn(K ) to F ×. We also have a homomorphism from GLn(K ) to F × obtained by composing the Dieudonné determinant from GLn(K ) to K ×/ [K ×, K ×] with the reduced norm N1 from GL1(K ) = K × to F × via the abelianization.
The Tannaka–Artin problem is whether these two maps have the same kernel SLn(K ). This is true when F is locally compact[2] but false in general.[3]
See also
- Moore determinant over a division algebra
References
- ^ Rosenberg (1994) p.64
- Zbl 0060.07901.
- Zbl 0338.16005.
- Zbl 0028.33904
- Zbl 0801.19001. Errata
- Zbl 1013.20001
- Suprunenko, D.A. (2001) [1994], "Determinant", Encyclopedia of Mathematics, EMS Press