Witt's theorem

"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.

Witt 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 controls 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 Witt index 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

$(V_1,q_1)\oplus(V,q) \simeq (V_2,q_2)\oplus(V,q).$

Then the quadratic spaces (V₁, q₁) and (V₂, q₂) are isometric:

$(V_1,q_1)\simeq (V_2,q_2).$

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:

$(V,q)\simeq (V_0,0)\oplus(V_a, q_a)\oplus (V_h,q_h),$

where V₀=ker q is the radical of q, (V_a, q_a) is an anisotropic quadratic space and (V_h, q_h) is a hyperbolic quadratic space. Moreover, the anisotropic summand and the hyperbolic summand in a Witt decomposition of (V, q) are determined uniquely up to isomorphism.

References

• O. Timothy O'Meara, Introduction to Quadratic Forms, Springer-Verlag, 1973