Namespaces
Variants
Actions

Difference between revisions of "Witt theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (link)
Line 1: Line 1:
 
Any isometry between two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980902.png" /> of a finite-dimensional [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980903.png" />, defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980904.png" /> of characteristic different from 2 and provided with a metric structure induced from a non-degenerate symmetric or skew-symmetric [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980905.png" />, may be extended to a metric automorphism of the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980906.png" />. The theorem was first obtained by E. Witt [[#References|[1]]].
 
Any isometry between two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980902.png" /> of a finite-dimensional [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980903.png" />, defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980904.png" /> of characteristic different from 2 and provided with a metric structure induced from a non-degenerate symmetric or skew-symmetric [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980905.png" />, may be extended to a metric automorphism of the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980906.png" />. The theorem was first obtained by E. Witt [[#References|[1]]].
  
Witt's theorem may also be proved under wider assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980908.png" /> [[#References|[2]]], [[#References|[3]]]. In fact, the theorem remains valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980909.png" /> is a skew-field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809010.png" /> is a finite-dimensional left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809011.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809012.png" /> is a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809013.png" />-Hermitian form (with respect to some fixed involutory anti-automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809015.png" />, cf. [[Hermitian form|Hermitian form]]) satisfying the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809016.png" /> there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809017.png" /> such that
+
Witt's theorem may also be proved under wider assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980908.png" /> [[#References|[2]]], [[#References|[3]]]. In fact, the theorem remains valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w0980909.png" /> is a skew-field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809010.png" /> is a finite-dimensional left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809011.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809012.png" /> is a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809013.png" />-Hermitian form (with respect to some fixed involutory [[anti-automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809015.png" />, cf. [[Hermitian form|Hermitian form]]) satisfying the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809016.png" /> there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809017.png" /> such that
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809018.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809018.png" /></td> </tr></table>

Revision as of 17:12, 9 October 2016

Any isometry between two subspaces and of a finite-dimensional vector space , defined over a field of characteristic different from 2 and provided with a metric structure induced from a non-degenerate symmetric or skew-symmetric bilinear form , may be extended to a metric automorphism of the entire space . The theorem was first obtained by E. Witt [1].

Witt's theorem may also be proved under wider assumptions on and [2], [3]. In fact, the theorem remains valid if is a skew-field, is a finite-dimensional left -module and is a non-degenerate -Hermitian form (with respect to some fixed involutory anti-automorphism of , cf. Hermitian form) satisfying the following condition: For any there exists an element such that

(property ). Property holds if, for example, is a Hermitian form and the characteristic of is different from 2, or if is an alternating form. Witt's theorem is also valid if is a field and is the symmetric bilinear form associated with a non-degenerate quadratic form on . It follows from Witt's theorem that the group of metric automorphisms of transitively permutes the totally-isotropic subspaces of the same dimension and that all maximal totally-isotropic subspaces in have the same dimension (the Witt index of ). A second consequence of Witt's theorem may be stated as follows: The isometry classes of non-degenerate symmetric bilinear forms of finite rank over with direct orthogonal sum form a monoid with cancellation; the canonical mapping of this monoid into its Grothendieck group is injective. The group is called the Witt–Grothendieck group of ; the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of [7].

For other applications of Witt's theorem see Witt decomposition; Witt ring.

References

[1] E. Witt, "Theorie der quadratischen formen in beliebigen Körpern" J. Reine Angew. Math. , 176 (1937) pp. 31–44 Zbl 0015.05701 Zbl 62.0106.02
[2] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , Elements of mathematics , 1 , Addison-Wesley (1974) pp. Chapts. 1–2 (Translated from French) MR0354207
[3] J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056
[4] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[5] E. Artin, "Geometric algebra" , Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[6] J.-P. Serre, "A course in arithmetic" , Springer (1973) (Translated from French) MR0344216 Zbl 0256.12001
[7] J. Milnor, "Algebraic -theory and quadratic forms" Invent. Math. , 9 (1969/70) pp. 318–344
How to Cite This Entry:
Witt theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Witt_theorem&oldid=39392
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article