Mark Ronan's website
Buildings
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Buildings are geometric
structures created by Jacques Tits
(1930– ), and my book Lectures on Buildings gives the
basics of the subject. Tits' original aim was to find analogues of the simple
Lie groups over any field, and buildings are closely connected with Lie
theory. The simplest ones — those of type An — are obtained as follows. The An buildings There is one An building for each field or division ring k — for example k might be the rational numbers, the real numbers,
complex numbers, quaternions, or indeed a finite field. The building is
obtained from an (n+1)‑dimensional
vector space V whose field of
scalars is k. The proper
subspaces of V are the vertices
of the building, and when one subspace contains another, the corresponding
vertices are joined by an edge. Each nested sequence of proper subspaces
forms a face, or simplex, of this building; a sequence of length two gives an
edge, a sequence of length three gives a triangular face, and so on. A
maximal sequence has length n, and
a maximal face is called a chamber.
For example if n = 3,
then each chamber has three vertices, corresponding to nested subspaces of V having dimensions 1, 2 and 3 — in this case
a chamber is a triangle (or in other words a 2-dimensional simplex). Apartments Every building
contains important substructures called 'apartments'. In the An example above, take a basis for the vector space V — the proper subspaces spanned by subsets of
this basis are the vertices of an apartment. For example, when dim V = 4 a basis has four elements and the
vertices of an apartment comprise: four 1‑spaces, six 2‑spaces,
and four 3‑spaces. We can represent this geometrically by taking the
four 1‑spaces as the vertices of a tetrahedron, the six 2‑spaces
as midpoints of its edges, and the four 3‑spaces as the midpoints of
its triangular faces. There are 24 chambers — six on each triangular
face of the tetrahedron. When the
apartments of a building are tilings of a sphere, as they are in this case,
the building is called spherical.
Every building is a direct product of irreducible buildings, and the
irreducible spherical buildings of rank at least 3 (the rank is
1+dimension, in the spherical case) come in one of the types labelled
classically as follows: An, Bn = Cn, Dn,
E6, E7, E8, F4, H3, H4. The subscript denotes the rank. Spherical Buildings Tits determined
all irreducible spherical buildings of rank at least 3, assuming a mild
non-degeneracy condition. For type An there is one building for each field (possibly
non-commutative), as mentioned above. For types E and Dn (n at
least 4) there is one for each commutative field. The classification for
types Bn=Cn
and F4 is more
complex. For types H3 and
H4 there are only
the apartments themselves. Tits achieves his
classification by first proving that if two spherical buildings have the same
local structure, then there is an isomorphism from one to the other. In
particular the local structure determines the global structure. When the two
buildings are the same, Tits' results yield automorphisms that impose
restrictions on the local structure, and this is a vital feature of the
eventual classification. Affine Buildings A building is
called affine if its apartments can be
realised as tilings of Euclidean space — for example a tiling of the
Euclidean plane by equilateral triangles. A building having such apartments
arises from the group SL3 over the p-adic numbers Qp. This group also yields a spherical building (of
type A2), in the way
described earlier, and there is a relationship between the two: the affine
building yields the spherical building as a structure "at
infinity". This is a general feature of affine buildings. Each affine
building of rank n has a
spherical building of rank n-1
at infinity, and Tits used this in classifying affine buildings that are
irreducible and have rank at least 4. Unlike the spherical case, the
local structure of an affine building does not determine its global
structure. For example, in the p-adic
building mentioned above, the local structure is coordinatized by the residue
field of Qp, which
is the finite field of p
elements. There are infinitely many fields having this same residue field, so
in general there are infinitely many affine buildings having the same local
structure. Other buildings, and
(B,N)-pairs Buildings that
are neither spherical nor affine arise from Kac-Moody groups, which are
infinite dimensional analogues of simple Lie groups. Such buildings can be
constructed directly from the group by using subgroups B and N,
forming what is called a (B,N)-pair,
where B is a chamber
stabilizer, and N stabilizes an
apartment. The method of (B,N)-pairs
works for all types of buildings, and in the spherical building for the group
SLn, mentioned
earlier, B is conjugate to the
subgroup of upper triangular matrices. This conjugacy class of subgroups is
in bijective correspondence with the set of chambers in the building. In a Kac-Moody group there are
two conjugacy classes of subgroups B.
These yield two buildings — both of the same type — forming a twin
building. The theory of twin buildings is
analogous to the theory of spherical buildings, but is less well-developed. |
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