This week Sükrü led the discussion which focussed on root systems, algebraic groups and the Curtis-Tits Theorem.
Recall from an earlier week that given and a positive definite bilinear form on , we can define the reflection
Then a root system is a subset of such that
- and ,
- If then the only scalar multiples of in are ,
- for all ,
- for all .
A root system is called irreducible if it cannot be written as a disjoint union of and such that .
A fundamental system is then a subset of which is a basis for and such that each element of can be written as a linear combination of the elements of such that the coefficients are either all nonnegative integers or all nonpositive integers.
The Dynkin diagram is the graph with vertices such that the vertex is joined to the vertex with strength
Note that . The table below shows what happens for each value of .
|angle between and||name of restriction of diagram to|
Now let , where is an algebraically closed field. A maximal torus is a subgroup maximal subject to being isomorphic to a subgroup of and hence can be represented by all diagonal matrices. For each we can define
Let . The elements lie in the dual space of and we can think of as a subset of , a vector space over . In this setting is a root system with fundamental system .
An alternative way to see these roots is as follows. Let be a Borel subgroup of , that is, a maximal connected solvable subgroup. Then where is unipotent and normal in , while is a torus. We can then choose a minimal -invariant subgroup of . Then . Since normalises we obtain a map .
We can take to be the group of all upper triangular matrices and then is the subgroup of all upper triangular matrices such that the entries on the diagonal are all equal to 1, and is the subgroup of all diagonal matrices. Then there exists such that . Now has a unique conjugate such that and then . The group is the group of all lower triangular matrices.
Note that for and we have .
Now . These generators are known as the Steinberg generators.
We can now state the Curtis-Tits Theorem.
Theorem Let be an irreducible root system of rank at least 3 with fundamental system and Dynkin diagram . Assume that is a group which satisfies the following conditions.
- There exist subgroups for with for some field and such that .
- if and are not adjacent in the Dynkin diagram.
- if and are joined by a single bond.
- if and are joined by a double bond.
- for all .
Then there exists a group of Lie type with corresponding root system , fundamental system and Dynkin diagram and a surjective homomorphism from onto .
We note it is possible to have isomorphic to or . For example, for groups of Lie type with root systems of type , the long roots all have while the short root has .
Sükrü will continue next week. He is also giving a Groups and Combinatorics seminar on Tuesday on related material.