The Classical Groups and K-Theory by Alexander J. Hahn, O.Timothy O'Meara, J. Dieudonne

By Alexander J. Hahn, O.Timothy O'Meara, J. Dieudonne

It is a smart pride for a mathematician to witness the expansion and enlargement of a concept during which he has taken a few half in the course of its early years. while H. Weyl coined the phrases "classical groups", greatest in his brain have been their connections with invariant thought, which his well-known ebook helped to restore. even supposing his procedure in that publication was once intentionally algebraic, his curiosity in those teams without delay derived from his pioneering examine of the exact case during which the scalars are actual or advanced numbers, the place for the 1st time he injected Topology into Lie idea. yet ever because the definition of Lie teams, the analogy among basic classical teams over finite fields and easy classical teams over IR or C have been saw, whether the concept that of "simplicity" used to be no longer rather an analogous in either circumstances. With the invention of the outstanding easy advanced Lie algebras by way of Killing and E. Cartan, it was once usual to seem for corresponding teams over finite fields, and already round 1900 this used to be performed by means of Dickson for the outstanding Lie algebras G and E • notwithstanding, a deep explanation for this 2 6 parallelism was once lacking, and it is just Chevalley who, in 1955 and 1961, found that to every complicated uncomplicated Lie algebra corresponds, via a uniform strategy, a bunch scheme (fj over the hoop Z of integers, from which, for any box okay, might be derived a bunch (fj(K).

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2. The groups Ak and Bk together generate Stn(R) for any fixed k. Proof. Let G be the subgroup generated by Ak and Bk and let xij(r) be any generator of Stn(R). If j = k then xij(r)EA k, and if i = k then xij(r)EBk. If neither i nor j equals k, then by (E3), xij(r) = [xik(r), xki1)] and xij(r)EG. 3. The restrictions of ¢: Stn(R) -+ En(R) to Ak and Bk are injective. Proof. r(Mn). Since the other case is analogous, we consider only A k. Let xEA k and put x = nixik(r;). If i > j, then by (E2), xik(rJ and xjk(r j ) commute.

Let sER* with s"# 1. Consider the element O"EGL(M) defined by O"V = vs and O"IN = IN' Define p EM* by putting pv = s - 1 and p IN = O. It is clear that 0" = 'v,p, so that 0" is a dilation. 9. Let 'v,p be a transvection or dilation. If ann R v = 0, then 'v,p = 1M O. If p is surjective, then 'v,p = 1M if and only if v = O. if p = if and only The elements 'v,p satisfy the following properties, all easily checked. p,,-l for O"EGL(M), 'v,p 'u,,,, = 'u,,,, 'v,P' if pu = ({JV = O. Let 'v,p be a transvection.

Let a be in the centralizer. For any i =I j and rER, consider the element TXi,rpjEE,{(M). 4 that ax, = XiSi with SiER*. 9, SirsJ:-l = r. rs:-lp. "J J that Si = sjE(CenR)*. So aERL(M), D I} 28 1. General Linear Groups, Steinberg Groups, and K-Groups So far we have dealt primarily with the interrelationship between En(R) and GLn(R). We now turn to the study of the group En(R). Consider the generators eJr) of En(R). It is easy to see that they satisfy the relations (El) (E2) (E3) eij(r)eij(s) = eij(r + s) [eij(r)Az(s)J = I if j 1= k and i 1= I, and [eij(r), ejk(s)J = eik(rs) if i,j, and k are distinct.

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