This is an example of the unipotent radical of a Borel subgroup ( of the M�bius group, or of SL ( 2, "'C "') for the matrix group; the notion is defined for any reductive Lie group ).
32.
Ratner's theorems provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form ? \ " G ", where " G " is a Lie group and ? is a lattice in " G ".
33.
An arbitrary irreducible character has a " Jordan decomposition " : to it one can associate a semisimple character ( corresponding to some semisimple element " s " of the dual group ), and a unipotent representation of the centralizer of " s ".
34.
By using these holomorphic affine structures we can prove that the period map is affine map into the unipotent orbit of the period domain, and proves a global Torelli theorem for Calabi-Yau type manifolds which asserts that the period maps from the Teichm�ller spaces to the period domains are injective.
35.
If an epigroup " S " has a partition in unipotent subepigroups ( i . e . each containing a single idempotent ), then this partition is unique, and its components are precisely the unipotency classes defined above; such an epigroup is called " unipotently partionable ".
36.
More precisely, extensions by algebraic functions correspond to finite differential Galois groups, extensions by integrals correspond to subquotients of the differential Galois group that are 1-dimensional and unipotent, and extensions by exponentials of integrals correspond to subquotients of the differential Galois group that are 1-dimensional and reductive ( tori ).
37.
The unipotent representations are the trivial representation and the Steinberg representation, and the semisimple representations are all the representations other than the Steinberg representation . ( In this case the semisimple representations do not correspond exactly to geometric conjugacy classes of the dual group, as the center of " G " is not connected .)
38.
For the special linear group " SL " " n ", the unipotent conjugacy classes are parametrized by partitions of " n " : if " u " is a unipotent element, the corresponding partition is given by the sizes of the Jordan blocks of " u ".
39.
For the special linear group " SL " " n ", the unipotent conjugacy classes are parametrized by partitions of " n " : if " u " is a unipotent element, the corresponding partition is given by the sizes of the Jordan blocks of " u ".
40.
As irreducible representations are always "'indecomposable "'( i . e . cannot be decomposed further into a direct sum of representations ), these terms are often confused; however, in general there are many reducible but indecomposable representations, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices.
