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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]]].
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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
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<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>
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Any isometry between two subspaces  $  F _ {1} $
 +
and  $  F _ {2} $
 +
of a finite-dimensional [[Vector space|vector space]]  $  V $,
 +
defined over a field  $  k $
 +
of characteristic different from 2 and provided with a metric structure induced from a non-degenerate symmetric or skew-symmetric [[Bilinear form|bilinear form]]  $  f $,
 +
may be extended to a metric automorphism of the entire space  $  V $.  
 +
The theorem was first obtained by E. Witt [[#References|[1]]].
  
(property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809020.png" />). Property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809021.png" /> holds if, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809022.png" /> is a Hermitian form and the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809023.png" /> is different from 2, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809024.png" /> is an alternating form. Witt's theorem is also valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809025.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809026.png" /> is the symmetric bilinear form associated with a non-degenerate [[Quadratic form|quadratic form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809028.png" />. It follows from Witt's theorem that the group of metric automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809029.png" /> transitively permutes the totally-isotropic subspaces of the same dimension and that all maximal totally-isotropic subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809030.png" /> have the same dimension (the Witt index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809031.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809032.png" /> with direct orthogonal sum form a monoid with cancellation; the canonical mapping of this monoid into its [[Grothendieck group|Grothendieck group]] is injective. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809033.png" /> is called the Witt–Grothendieck group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809035.png" />; the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098090/w09809036.png" /> [[#References|[7]]].
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Witt's theorem may also be proved under wider assumptions on  $  k $
 +
and  $  f $[[#References|[2]]], [[#References|[3]]]. In fact, the theorem remains valid if  $  k $
 +
is a skew-field,  $  V $
 +
is a finite-dimensional left  $  k $-
 +
module and  $  f $
 +
is a non-degenerate  $  \epsilon $-
 +
Hermitian form (with respect to some fixed involutory [[anti-automorphism]]  $  \sigma $
 +
of  $  k $,
 +
cf. [[Hermitian form|Hermitian form]]) satisfying the following condition: For any  $  v \in V $
 +
there exists an element  $  \alpha \in k $
 +
such that
 +
 
 +
$$
 +
f ( v, v)  =  \alpha + \epsilon \alpha  ^  \sigma
 +
$$
 +
 
 +
(property $  ( T) $).  
 +
Property $  ( T) $
 +
holds if, for example, $  f $
 +
is a Hermitian form and the characteristic of $  k $
 +
is different from 2, or if $  f $
 +
is an alternating form. Witt's theorem is also valid if $  k $
 +
is a field and $  f $
 +
is the symmetric bilinear form associated with a non-degenerate [[Quadratic form|quadratic form]] $  Q $
 +
on $  V $.  
 +
It follows from Witt's theorem that the group of metric automorphisms of $  V $
 +
transitively permutes the totally-isotropic subspaces of the same dimension and that all maximal totally-isotropic subspaces in $  V $
 +
have the same dimension (the Witt index of $  f  $).  
 +
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 $  k $
 +
with direct orthogonal sum form a monoid with cancellation; the canonical mapping of this monoid into its [[Grothendieck group|Grothendieck group]] is injective. The group $  \mathop{\rm WG} ( k) $
 +
is called the Witt–Grothendieck group $  \mathop{\rm WG} ( k) $
 +
of $  k $;  
 +
the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of $  k $[[#References|[7]]].
  
 
For other applications of Witt's theorem see [[Witt decomposition|Witt decomposition]]; [[Witt ring|Witt ring]].
 
For other applications of Witt's theorem see [[Witt decomposition|Witt decomposition]]; [[Witt ring|Witt ring]].

Revision as of 08:29, 6 June 2020


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

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

$$ f ( v, v) = \alpha + \epsilon \alpha ^ \sigma $$

(property $ ( T) $). Property $ ( T) $ holds if, for example, $ f $ is a Hermitian form and the characteristic of $ k $ is different from 2, or if $ f $ is an alternating form. Witt's theorem is also valid if $ k $ is a field and $ f $ is the symmetric bilinear form associated with a non-degenerate quadratic form $ Q $ on $ V $. It follows from Witt's theorem that the group of metric automorphisms of $ V $ transitively permutes the totally-isotropic subspaces of the same dimension and that all maximal totally-isotropic subspaces in $ V $ have the same dimension (the Witt index of $ f $). 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 $ k $ with direct orthogonal sum form a monoid with cancellation; the canonical mapping of this monoid into its Grothendieck group is injective. The group $ \mathop{\rm WG} ( k) $ is called the Witt–Grothendieck group $ \mathop{\rm WG} ( k) $ of $ k $; the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of $ k $[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=49234
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article