:''"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
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time.
Biography
Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...
, is a basic result in the algebraic theory of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s: any
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
between two subspaces of a nonsingular
quadratic space
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''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
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
s over arbitrary fields. The theorem applies to classification of quadratic forms over ''k'' and in particular allows one to define the
Witt group
In mathematics, a Witt group of a field (mathematics), field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear form, symmetric bilinear forms over the field.
Definition
Fix a field ''k'' of characte ...
''W''(''k'') which describes the "stable" theory of quadratic forms over the field ''k''.
Statement
Let be a finite-dimensional vector space over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''k'' of
characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
. If is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
between two subspaces of ''V'' then ''f'' extends to an isometry of ''V''.
Witt's theorem implies that the dimension of a maximal
totally isotropic subspace (null space) of ''V'' is an invariant, called the index or of ''b'', and moreover, that the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of the isometry group and in the theory of
reductive dual pair In the mathematical field of representation theory, a reductive dual pair is a pair of subgroups (''G'', ''G''′) of the isometry group Sp(''W'') of a symplectic vector space ''W'', such that ''G'' is the centralizer of ''G''′ in Sp(''W'') and vi ...
s.
Witt's cancellation theorem
Let , , be three quadratic spaces over a field ''k''. Assume that
:
Then the quadratic spaces and are isometric:
:
In other words, the direct summand appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".
Witt's decomposition theorem
Let be a quadratic space over a field ''k''. Then
it admits a Witt decomposition:
:
where is the
radical
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
of ''q'', is an
anisotropic quadratic space
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector sp ...
and is a
split quadratic space
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector sp ...
. Moreover, the anisotropic summand, termed the core form, and the hyperbolic summand in a Witt decomposition of are determined uniquely up to isomorphism.
Quadratic forms with the same core form are said to be ''similar'' or Witt equivalent.
Citations
References
*
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
(1957
''Geometric Algebra'', page 121via
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
*
*
* {{citation , first=O. Timothy , last=O'Meara , authorlink=O. Timothy O'Meara , year=1973 , title=Introduction to Quadratic Forms , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, series=Die Grundlehren der mathematischen Wissenschaften , volume=117 , zbl=0259.10018
Theorems in linear algebra
Quadratic forms