Witt decomposition

of a vector space

A decomposition of the space into a direct sum of three subspaces having certain properties. More exactly, let be a vector space over a field with characteristic different from 2, provided with a metric structure by means of a symmetric or skew-symmetric bilinear form . The direct decomposition

is said to be a Witt decomposition of if and are totally isotropic, while is anisotropic and is orthogonal to with respect to . The Witt decomposition plays an important role in the study of the structure of the form and in problems of classification of bilinear forms.

Let be a non-degenerate bilinear form and let be finite-dimensional. Then any maximal totally isotropic subspace in may be included in a Witt decomposition of as or . For any Witt decomposition , and for any basis in , there exists a basis in such that ( are the Kronecker symbols). For any two Witt decompositions

the condition , is necessary and sufficient for the existence of a metric automorphism of such that

A non-degenerate symmetric or skew-symmetric bilinear form on is said to be neutral if is finite-dimensional and has a Witt decomposition with . In this case the symmetric form is said to be a hyperbolic form, while is said to be a hyperbolic space. An orthogonal direct sum of neutral forms is neutral. The matrix of a neutral form (in the basis of the space described above) looks like

where is the identity matrix of order , while for a symmetric form and for a skew-symmetric form. Two neutral forms are isometric if and only if they have the same rank. The class of neutral symmetric bilinear forms is the zero (i.e. the neutral element for addition) in the Witt ring of the field . Neutral forms and only such forms have Witt index . A skew-symmetric form on a finite-dimensional space is neutral.

If is a non-degenerate symmetric bilinear form on a finite-dimensional space and is a Witt decomposition in which is equal to the Witt index of , the restriction of to is a definite, or anisotropic, bilinear form, i.e. is such that for all non-zero . This form is independent (apart from an isometry) of the choice of the Witt decomposition of . In the set of definite bilinear forms it is possible to introduce an addition, converting it into an Abelian group — the Witt group of (cf. Witt ring).

Let be bases in , , such that ; the union of these bases with an arbitrary basis in yields a basis in in which the matrix of looks like

For symmetric bilinear forms there exists an orthogonal basis in , i.e. a basis in which the matrix of the form is diagonal. If the field is algebraically closed, there even exists an orthonormal basis (a basis in which the matrix of the form is the identity), and for this reason two non-degenerate symmetric bilinear forms of finite rank over are isometric if and only if they have the same rank. In the general case the classification of such forms substantially depends on the arithmetical properties of the field .

The study and classification of degenerate symmetric and skew-symmetric forms can be reduced to the study of non-degenerate forms (the restriction of the form to a subspace which is complementary to the kernel of the form).

All what has been said above permits a generalization to the case of -Hermitian forms over a skew-field with property (cf. Witt theorem), and also to the case of symmetric bilinear forms associated with quadratic forms, without restrictions on the characteristic of the field.

References

[1]	N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Addison-Wesley (1974) pp. Chapts. 1–2 (Translated from French) MR0354207 Zbl 0281.00006
[2]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[3]	E. Artin, "Geometric algebra" , Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[4]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056

Comments

A vector space with a neutral non-degenerate bilinear form on it is called split or metabolic.

A different form of the Witt decomposition theorem gives a decomposition of a quadratic space (i.e. a vector space with a quadratic form on it) into an orthogonal sum

(*)

with totally isotropic, hyperbolic and anisotropic. Moreover, the isometry classes of , and are uniquely determined by that of .

In this decomposition, is the radical of , , where is the symmetric bilinear form on associated to :

The uniqueness of the factors in the Witt decomposition (*) follows from the Witt cancellation theorem, which says that if is isometric to , then and are isometric.

References

[a1]	J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) pp. 16 MR0506372 Zbl 0292.10016
[a2]	T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973) MR0396410 Zbl 0259.10019