Symmetric bilinear forms
In this chapter, we describe the foundations of the theory of non-degenerate symmetric bilinear forms on finite-dimensional vector spaces and their orthogonal groups. Among the highlights of this discussion are the Cartan–Dieudonné Theorem, which states t
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Symmetric bilinear forms
In this chapter we will describe the foundations of the theory of non-degenerate symmetric bilinear forms on finite-dimensional vector spaces and their orthogonal groups. Among the highlights of this discussion are the Cartan–Dieudonné Theorem, which states that any orthogonal transformation is a finite product of reflections, and Witt’s Theorem giving a partial normal form for quadratic forms. The theory of split symmetric bilinear forms is found to have many parallels to the theory of symplectic forms, and we will give a discussion of the Lagrangian Grassmannian for this case. Throughout K will denote a ground field of characteristic = 2. We are mainly interested in the cases K = R or C, and sometimes we specialize to those two cases.
1.1 Quadratic vector spaces Suppose V is a finite-dimensional vector space over K. For any bilinear form B : V × V → K, define a linear map B : V → V ∗ , v → B(v, ·). The bilinear form B is called symmetric if it satisfies B(v1 , v2 ) = B(v2 , v1 ) for all v1 , v2 ∈ V . Since dim V < ∞ this is equivalent to (B )∗ = B . The symmetric bilinear form B is uniquely determined by the associated quadratic form QB (v) = B(v, v), using the polarization identity 1 (1.1) B(v, w) = QB (v + w) − QB (v) − QB (w) . 2 The kernel (also called radical) of B is the subspace ker(B) = {v ∈ V | B(v, w) = 0 for all w ∈ V }, i.e., the kernel of the linear map B . The bilinear form B is called non-degenerate if ker(B) = 0, i.e., if and only if B is an isomorphism. A vector space V together with a non-degenerate symmetric bilinear form B will be referred to as a quadratic E. Meinrenken, Clifford Algebras and Lie Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 58, DOI 10.1007/978-3-642-36216-3_1, © Springer-Verlag Berlin Heidelberg 2013
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Symmetric bilinear forms
vector space. Assume for the rest of this chapter that (V , B) is a quadratic vector space. Definition 1.1 A vector v ∈ V is called isotropic if B(v, v) = 0. n B(z, w) = nFor instance, if V = C over K = C, with the standard bilinear form 2 over K = R, z w , then v = (1, i, 0, . . . , 0) is an isotropic vector. If V = R i=1 i i with bilinear form B(x, y) = x1 y1 − x2 y2 , then the set of isotropic vectors x = (x1 , x2 ) is given by the “light cone” x1 = ±x2 . The orthogonal group O(V ) is the group
O(V ) = {A ∈ GL(V )| B(Av, Aw) = B(v, w) for all v, w ∈ V }.
(1.2)
The subgroup of orthogonal transformations of determinant 1 is denoted by SO(V ), and is called the special orthogonal group. For any subspace F ⊆ V , the orthogonal or perpendicular subspace is defined as F ⊥ = {v ∈ V | B(v, v1 ) = 0 for all v1 ∈ F }. The image of B (F ⊥ ) ⊆ V ∗ is the annihilator of F . From this one deduces the dimension formula dim F + dim F ⊥ = dim V
(1.3)
and the identities (F ⊥ )⊥ = F, (F1 ∩ F2 )⊥ = F1⊥ + F2⊥ , (F1 + F2 )⊥ = F1⊥ ∩ F2⊥ for all F, F1 , F2 ⊆ V . For any subspace F ⊆ V the restriction of B to F has kernel ker(B|F ×F ) = F ∩ F ⊥ . Defi
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