Unitary Groups over Division Rings
Let R be a division ring. Let (R, Λ), with underlying J and ∈, be a form ring over R and let V = (V, q, h) be an n-dimensional, non-degenerate quadratic module over (R,Λ). The unitary group U(V) of V will now be denoted U n (V). While R is restricted to b
- PDF / 8,149,682 Bytes
- 89 Pages / 439.37 x 666.142 pts Page_size
- 61 Downloads / 210 Views
Let R be a division ring. Let (R, A), with underlying J and E, be a form ring over R and let V = (V, q, h) be an n-dimensional, non-degenerate quadratic module over (R,A). The unitary group U(V) of V will now be denoted Un (V). While R is restricted to be a division ring, V will be general in this chapter. This contrasts sharply to the previous chapter which restricted itself for the most part to hyperbolic modules. An outline of the chapter follows. An initial section introduces the necessary material about forms over division rings. Basic properties of unitary transformations, such as the Witt extension theorem, the generation by transvections of the symplectic groups, and the generation by symmetries of the unitary groups which are not symplectic, are established next. A subsequent section considers, for isotropic V, the group EUn(V) generated by the Eichler transformations, and proves that EU n(V)/Cen EU n(V) is simple aside from certain exceptions. Then the spinor norm, a generalization of the classical spinor norm, is introduced. Its kernel U~(V) is (with some exceptions) shown to be equal to EUn(V) for isotropic V. It follows that in most situations Un(V)/EUn(V) is isomorphic to a certain Abelian quotient group of Ii which depends only on (R,A)-and not on V. The kernel U: (V) of the Dieudonne determinant, as well as the unitary norm one group SUn (V), play an important role in a more careful analysis of the structure of this quotient. A final section studies the unitary K-groups and proves presentation theorems for the hyperbolic unitary groups. It concludes with a discussion of a number of important results about KU 2(R, A)-in particular the remarkable fact that the symplectic case of Sharpe's sequence weaves the theorems (the cases m = 2) of Merkurjev-Suslin of §2.3B into an exact diagram. We know from §5.2B that the following is the case. If V is normalized, an operation which keeps V non-degenerate and does not change the group Un(V), then (R, A), h, q, and Un (V) specialize to one of the rows in the table below:
A. J. Hahn et al., The Classical Groups and K-Theory © Springer-Verlag Berlin Heidelberg 1989
6. Unitary Groups over Division Rings
J
id R
Group: notation
E
A
-1
R
Commu- Alternating tative
Trivial
Symplectic: Spn(V)
0
Commu- Symmetric tative bilinear; alternating, if X(R) = 2
Ordinary quadratic
Orthogonal: On (V)
OcAcR
Commu- Alternating tative, X(R) = 2
id R
id R
-1
J2 = id R J#id R
-1
j2 = id R
-1
J #id R
293
Amax
A c
Amax
R
h
q
Defective orthogonal: On (V)
Commu- Tracevalued, tative or not skewhermitian
Classical unitary: Un (V)
not Tracecommu- valued, tative, skewX(R) = 2 hermitian
Restricted classical unitary: Un(V)
The theory developed in this chapter is, therefore, a theory of the groups in the table; however for the most part case by case considerations will be necessary only when the conclusions of a particular discussion differ from group to group. Accordingly: Throughout this chapter, R will be a division ring, J will be an anti-automorphism of
Data Loading...
