Clifford Algebras and Orthogonal Groups over Commutative Rings
The Clifford algebra of a quadratic module M over a form ring (in the orthogonal case with form parameter {0}) is an algebra which is compatible with the structure of M in a universal way. We will study this algebra in this chapter and see that it has imp
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The Clifford algebra of a quadratic module M over a form ring (in the orthogonal case with form parameter {O}) is an algebra which is compatible with the structure of M in a universal way. We will study this algebra in this chapter and see that it has important impact on the structure ofthe orthogonal groups. It will be instrumental not only in clearing up earlier complications that were special to the orthogonal groups but also in generalizing previous results from fields to commutative rings. For example, the Clifford algebra provides a generalized spinor norm which adds insight into the structure of the K0 1 groups. When applied in the special case of the orthogonal group OZn(R) with n ;:::: 2 and R a commutative Euclidean domain, the spinor norm gives rise to the isomorphism OZn(R)/EOzn(R) c::::: R*/(R*)2 x Zz(R)
which was pointed out without proof in the discussion that followed proposition 5.3.5. In addition, certain subgroups of the multiplicative group of the Clifford algebra and associated projection maps turn out to be non-trivial central extensions ofthe orthogonal groups. As such they provide information about the group KO z . It is this connection which leads to the orthogonal versions of Theorems 6.5.14, 6.5.15 and 6.5.16, and thus provides, respectively, a presentation and universal central extension of the finite elementary orthogonal group. In addition, the internal structure of the Clifford algebra will allow us to construct a number of isomorphisms between orthogonal groups and other classical groups of small ranks. There is, for example, the isomorphism PO~(M) c:::::
PSLz(R) x PSLz(R),
where R is any commutative ring, M is a hyperbolic quadratic module of rank 4, and O~(M) is the kernel of the spinor norm. This isomorphism shows that the 4dimensional orthogonal groups excluded from the hypothesis of Theorem 6.3.16 are never simple. The first section introduces the Clifford algebra and studies its basic properties. The second examines certain subgroups of the group of units of the Clifford algebra and the relationships of these groups with the orthogonal groups. This section also discusses Bass' generalized spinor norm. The third section shows how the Clifford algebra leads to isomorphisms between classical groups of small ranks.
A. J. Hahn et al., The Classical Groups and K-Theory © Springer-Verlag Berlin Heidelberg 1989
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7. Clifford Algebras and Orthogonal Groups over Commutative Rings
In this chapter (R, A) is anyform ring with A = 0 and M = (M, [f]A) = (M, h, q) is a (right) quadratic module over (R, A). So R is an arbitrary commutative ring, E = 1, and J = id R . We identify R/A = R/{O} = R without further ado. So q(x) = f(x, x) and h(x, y) = f(x, y) + f(y, x) for all .x and y in M. In view of Example (C) of §5.2B, q:V--+R
is an ordinary quadratic form on M, and h is the associated symmetric bilinear form determined by the equation h(x,y) = q(x + y) - q(x) - q(y),
for all x and y in M. Clearly h(x, x) = 2q(x) for all x in M. Since h is determined by q, we will abbreviate the not
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