By M. Kneser, notes taken by P. Jothilingam
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The proof of this is exactly the same as in the case of S p2n ; we have only to use the fact that the centre of S O2n is ±1. Consider the spin group now; let p : Spin2n → S O2n be the covering homomorphism. 2. 1. Remark 2 that T˜ is an algebraic torus. Let V1 be a two dimensional anisotropic quadratic space. Then the √ Clifford algebra C(V1 ) is of dimension 4 and C + (V1 ) is just K( c) where c is the discriminant of V1 and the ∗ automorphism is the non-trivial K√ automorphism of K( c). Hence we have Spin2 (V1 ) = {x ∈ C + /xx∗ = √ 1} is isomorphic to the group of elements of K( c)/K with norm 1 which is a torus.
Bilinear and hermitian forms; discriminants 33 A = Mn (D); for Z = (zi j ) ∈ A denote by Z ∗ the element (zIji ). e. a∗ = −a. Then if X, Y are 1 × n matrices over D such that XaX ∗ = YaY ∗ = C is non-singular in D, there exists a proper a-unitary matrix t ∈ A such that Y = Xt. Proof. If D is a division algebra, this follows immediately from Witt’s theorem and lemma 1. If D = M2 (K), the action of I is as follows: α → S t αS −1 where S is some fixed 2 × 2 skew-symmetric matrix S S.
28 2. e (d−1 a)I = d−1 a. Moreover the equation x J = axI a−1 is unaltered if we replace a by d−1 a since d ∈ L. Hence if I is of the second kind we can assume in the equation x J = axI a−1 that aI = a. 37 Examples of algebras with involution: A quaternion algebra with the standard involution x → x¯, the conjugate of x, is a simple algebra and the standard involution is of the first kind. If D is any division algebra over K then Mn (D) has an involution of the rth kind (r = 1, 2) if and only if D has an involution of the rth kind; for if I in an involution of the rth kind on D then if we define X J for X = (xi j ) ∈ Mn (D) as X J = (xIji ) we get an involution of the rth kind; the converse follows from Theorem 1 below by taking A = Mn (D), B = Mn (K).