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 group homomorphism 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 the residue class, in K ×/ [K ×, K ×], of
- [math]\displaystyle{ \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. }[/math]
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:[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 center 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[2] but false in general.[3]
See also
- Moore determinant over a division algebra
References
- ↑ Rosenberg (1994) p.64
- ↑ Nakayama, Tadasi; Matsushima, Yozô (1943). "Über die multiplikative Gruppe einer p-adischen Divisionsalgebra" (in German). Proc. Imp. Acad. Tokyo 19: 622–628. doi:10.3792/pia/1195573246.
- ↑ Platonov, V.P. (1976). "The Tannaka-Artin problem and reduced K-theory" (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 40 (2): 227–261. doi:10.1070/IM1976v010n02ABEH001686. Bibcode: 1976IzMat..10..211P.
- Dieudonné, Jean (1943), "Les déterminants sur un corps non commutatif", Bulletin de la Société Mathématique de France 71: 27–45, doi:10.24033/bsmf.1345, ISSN 0037-9484
- Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, 147, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94248-3, https://books.google.com/books?id=TtMkTEZbYoYC. Errata
- Serre, Jean-Pierre (2003), Trees, Springer, p. 74, ISBN 3-540-44237-5
- Hazewinkel, Michiel, ed. (2001), "Determinant", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=D/d031410
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