By Jacques Tits, Richard M. Weiss (auth.)

Spherical constructions are definite combinatorial simplicial complexes intro duced, before everything within the language of "incidence geometries," to supply a sys tematic geometric interpretation of the phenomenal advanced Lie teams. (The definition of a construction by way of chamber structures and definitions of many of the comparable notions utilized in this creation equivalent to "thick," "residue," "rank," "spherical," and so forth. are given in bankruptcy 39. ) through the inspiration of a BN-pair, the idea became out to use to basic algebraic teams over an arbitrary box. extra accurately, to any completely basic algebraic team of optimistic rela tive rank £ is linked a thick irreducible round development of a similar rank (these are the algebraic round constructions) and the most results of structures of round variety and Finite BN-Pairs [101] is that the speak, for £ ::::: three, is nearly real: (1. 1) Theorem. each thick irreducible round development of rank at the least 3 is classical, algebraic' or combined. Classical constructions are these outlined by way of the geometry of a classical team (e. g. unitary, orthogonal, and so on. of finite Witt index or linear of finite size) over an arbitrary box or skew-field. (These should not algebraic if, for example, the skew-field is of endless measurement over its heart. ) combined constructions are extra unique; they're on the topic of teams that are in a few experience algebraic teams outlined over a couple of fields ok and okay of attribute p, the place KP eke ok and p is 2 or (in one case) three.

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Ub - ub . a = [u,a,b] + [a,u,b]- [u,b,a] +uv = [a,b,u] +uv. Thus [a, b, u] = -uv. 27) Let a, b, c, d E A and u [a,dju = o. = [a, b, c]. If [a, d, c] = [a, d, u] = 0, then Proof. 26) leaves u and [a, b, u] unchanged, since [a, d, c] = [a, d, u] = 0, but transforms v into v + [a, d]. Thus [a, d]u = O. 28) Suppose that A is not associative and that [a, d, A] = 0 for some a,d E A. Then [a,d] = o. Proof. 27), [a, d] = 0 would follow if [a, A, A] =f. o. By symmetry, [a, d] = also follow if [d, A, A] =f.

For (i, j) E h, we must make U[i,j] into the direct product of Ui and Uj since U[i+l,j-l] = {(I, ... , I)}. We now suppose that k E [2, n - 1] and impose the following conditions inductively: A k . For each (i, j) E h \h-l and all ai, bi E U i and aj E U j , the equation ~ij(aibi' aj) = ~ij(ai' aj)bi~ij(bi' aj) holds in the group U[i,j-l]. B k . For each (i,j) E h \h-l and all ai E U i and aj, bj E U j , the equation ~ij(ai' ajbj ) = ~ij(ai' bj)~ij(ai' aj)b j holds in the group U[i+l,j]' Ck . 13) Theorem.

Then Uo lies in the root group of \II associated with the root (0,1, ... ,n) and acts transitively on \Ilo\{l}. 3), Uo is the root group of \II associated with (0,1, ... , n). Its pre-image under 1f is therefore Uo. The action of Uo on U[l,n-l] is uniquely determined by n and its apartment ~. , Un), it follows that Uo is unique. 1), therefore, /1 is unique. • We now take one step further back. Suppose groups Ul, ... , Un for some Let W = U1 X '" X Un. For i, j E [1, n], let n:::: 3 are given but not the larger group U[l,n]' U[i,j] For each i E = {(al, ...