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