Login
Login

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 {{harvs|txt|last=Dieudonné|authorlink=Jean Dieudonné|year=1943}}.

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

For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in K{{hairsp}}×/{{hairsp}}[K{{hairsp}}×, K{{hairsp}}×], of

:\det \left({\begin{array}{*{20}c} a & b \\ c & d \end{array}}\right) =

\left\lbrace{\begin{array}{*{20}c} -cb & \text{if } a = 0 \\ ad - aca^{-1}b & \text{if } a \ne 0. \end{array}}\right.

Properties

Let R be a local ring. There is a determinant map from the matrix ring GL(R{{hairsp}}) to the abelianised unit group R{{hairsp}}×ab with the following properties:Rosenberg (1994) p.64

  • 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{{hairsp}}× 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 center F. The reduced norm gives a homomorphism Nn from GLn(K{{hairsp}}) to F{{hairsp}}×. We also have a homomorphism from GLn(K{{hairsp}}) to F{{hairsp}}× obtained by composing the Dieudonné determinant from GLn(K{{hairsp}}) to K{{hairsp}}×/{{hairsp}}[K{{hairsp}}×, K{{hairsp}}×] with the reduced norm N1 from GL1(K{{hairsp}}) = K{{hairsp}}× to F{{hairsp}}× via the abelianization.

The Tannaka–Artin problem is whether these two maps have the same kernel SLn(K{{hairsp}}). This is true when F is locally compact{{cite journal | zbl=0060.07901 | last1=Nakayama | first1=Tadasi | last2=Matsushima | first2=Yozô | title=Über die multiplikative Gruppe einer p-adischen Divisionsalgebra | language=German | journal=Proc. Imp. Acad. Tokyo | volume=19 | pages=622–628 | year=1943 | doi=10.3792/pia/1195573246| doi-access=free }} but false in general.{{cite journal | zbl=0338.16005 | last=Platonov | first=V.P. | authorlink=Vladimir Platonov | title=The Tannaka-Artin problem and reduced K-theory | language=Russian | journal=Izv. Akad. Nauk SSSR Ser. Mat. | volume=40 | pages=227–261 | year=1976 | issue=2 | doi=10.1070/IM1976v010n02ABEH001686 | bibcode=1976IzMat..10..211P }}

See also

References

{{reflist}}

  • {{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Les déterminants sur un corps non commutatif | mr=0012273 | year=1943 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=71 | pages=27–45 | zbl=0028.33904 | doi=10.24033/bsmf.1345 | doi-access=free }}
  • {{Citation | last1=Rosenberg | first1=Jonathan | authorlink=Jonathan Rosenberg (mathematician) | title=Algebraic K-theory and its applications | url=https://books.google.com/books?id=TtMkTEZbYoYC | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94248-3 | mr=1282290 | zbl=0801.19001 | year=1994 | volume=147}}. [http://www-users.math.umd.edu/~jmr/KThy_errata2.pdf Errata]
  • {{citation | title=Trees | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | publisher=Springer | year=2003 | isbn=3-540-44237-5 | zbl=1013.20001 | page=74 }}
  • {{eom|id=D/d031410|title=Determinant|first=D.A. |last=Suprunenko}}

{{DEFAULTSORT:Dieudonne determinant}}

Category:Linear algebra

Category:Determinants