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:''"Witt's theorem" or "the Witt theorem" may also refer to the [[Bourbaki–Witt theorem|Bourbaki–Witt fixed point theorem]] of order theory.'' | |||
In mathematics, '''Witt's theorem''', named after [[Ernst Witt]], is a basic result in the algebraic theory of [[quadratic form]]s: any [[Isometry (quadratic forms)|isometry]] between two subspaces of a nonsingular [[quadratic space]] over a [[field (algebra)|field]] ''k'' may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian [[bilinear form]]s over arbitrary fields. The theorem applies to classification of quadratic forms over ''k'' and in particular allows one to define the [[Witt group]] ''W''(''k'') which describes the "stable" theory of quadratic forms over the field ''k''. | |||
== Statement of the theorem == | |||
Let (''V'', ''b'') be a finite-dimensional vector space over an arbitrary [[field (algebra)|field]] ''k'' together with a nondegenerate symmetric or skew-symmetric [[bilinear form]]. If ''f'': ''U''→''U' '' is an [[isometry]] between two subspaces of ''V'' then ''f'' extends to an isometry of ''V''. | |||
Witt's theorem implies that the dimension of a maximal [[isotropic subspace]] of ''V'' is an invariant, called the '''index''' or '''{{visible anchor|Witt index}}''' of ''b'', and moreover, that the [[isometry group]] of (''V'', ''b'') [[group action|acts]] transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and [[group representation|representation theory]] of the isometry group and in the theory of [[reductive dual pair]]s. | |||
== Witt's cancellation theorem == | |||
Let (''V'', ''q''), (''V''<sub>1</sub>, ''q''<sub>1</sub>), (''V''<sub>2</sub>, ''q''<sub>2</sub>) be three quadratic spaces over a field ''k''. Assume that | |||
: <math> (V_1,q_1)\oplus(V,q) \simeq (V_2,q_2)\oplus(V,q).</math> | |||
Then the quadratic spaces (''V''<sub>1</sub>, ''q''<sub>1</sub>) and (''V''<sub>2</sub>, ''q''<sub>2</sub>) are isometric: | |||
: <math> (V_1,q_1)\simeq (V_2,q_2).</math> | |||
In other words, the direct summand (''V'', ''q'') appearing in both sides of an isomorphism between quadratic spaces may be "cancelled". | |||
== Witt's decomposition theorem == | |||
Let (''V'', ''q'') be a quadratic space over a field ''k''. Then | |||
it admits a '''Witt decomposition''': | |||
: <math>(V,q)\simeq (V_0,0)\oplus(V_a, q_a)\oplus (V_h,q_h),</math> | |||
where ''V''<sub>0</sub>=ker ''q'' is the [[Radical of a quadratic space|radical]] of ''q'', (''V''<sub>''a''</sub>, ''q''<sub>''a''</sub>) is an [[anisotropic quadratic space]] and (''V''<sub>''h''</sub>, ''q''<sub>''h''</sub>) is a [[split quadratic space]]. Moreover, the anisotropic summand, termed the '''core form''', and the hyperbolic summand in a Witt decomposition of (''V'', ''q'') are determined uniquely up to isomorphism.<ref>Lorenz (2008) p.30</ref> | |||
Quadratic forms with the same core form are said to be ''similar'' or '''Witt equivalent'''. | |||
== References == | |||
{{reflist}} | |||
* {{cite book | first=O. Timothy | last=O'Meara | authorlink=O. Timothy O'Meara | title=Introduction to Quadratic Forms | publisher=[[Springer-Verlag]] | year=1973 | series=Die Grundlehren der mathematischen Wissenschaften | volume=117 | zbl=0259.10018 }} | |||
* {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=[[Springer-Verlag]] | isbn=978-0-387-72487-4 | pages=15–27 | zbl=1130.12001 }} | |||
[[Category:Theorems in algebra]] | |||
[[Category:Quadratic forms]] |
Revision as of 09:21, 15 March 2013
- "Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.
In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms over arbitrary fields. The theorem applies to classification of quadratic forms over k and in particular allows one to define the Witt group W(k) which describes the "stable" theory of quadratic forms over the field k.
Statement of the theorem
Let (V, b) be a finite-dimensional vector space over an arbitrary field k together with a nondegenerate symmetric or skew-symmetric bilinear form. If f: U→U' is an isometry between two subspaces of V then f extends to an isometry of V.
Witt's theorem implies that the dimension of a maximal isotropic subspace of V is an invariant, called the index or Template:Visible anchor of b, and moreover, that the isometry group of (V, b) acts transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and representation theory of the isometry group and in the theory of reductive dual pairs.
Witt's cancellation theorem
Let (V, q), (V1, q1), (V2, q2) be three quadratic spaces over a field k. Assume that
Then the quadratic spaces (V1, q1) and (V2, q2) are isometric:
In other words, the direct summand (V, q) appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".
Witt's decomposition theorem
Let (V, q) be a quadratic space over a field k. Then it admits a Witt decomposition:
where V0=ker q is the radical of q, (Va, qa) is an anisotropic quadratic space and (Vh, qh) is a split quadratic space. Moreover, the anisotropic summand, termed the core form, and the hyperbolic summand in a Witt decomposition of (V, q) are determined uniquely up to isomorphism.[1]
Quadratic forms with the same core form are said to be similar or Witt equivalent.
References
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- ↑ Lorenz (2008) p.30