# Witt theorem

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 $K$-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=49265
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article