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Dieudonné determinant

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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).

Contents

If K is a division ring, then the Dieudonné determinant is a homomorphism of groups from the group GLn(K) of invertible n by n matrices over K onto the abelianization K*/[K*, K*] of the multiplicative group K* of K.

For example, the Dieudonné determinant for a 2-by-2 matrix is

det ( a b c d ) = { − c b if  a = 0 a d − a c a − 1 b if  a ≠ 0 .

Properties

Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group R∗ab with the following properties:

  • The determinant is invariant under elementary row operations
  • The determinant of the identity 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 centre F. The 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 but false in general.

    References

    Dieudonné determinant Wikipedia
    (Text) CC BY-SA


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