Lie group decompositions

In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces. Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back to decompositions.

The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues, making it harder to summarise the facts into a unified theory.

Some of the standard techniques are these:

• The Levi decomposition is a general fact of Lie algebra theory, giving rise to decompositions of semidirect product type for groups, as extensions of a solvable group by a semisimple group.

• The Iwasawa decomposition KAN of a semisimple group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of orthogonalization).

• The Bruhat decomposition into cells can be regarded as a general expression of the principle of Gauss-Jordan elimination, which generically writes a matrix as a product of an upper triamgular and lower triangular matrices - but with exceptional cases. It is related to the Schubert cell decomposition of Grassmannians.

• Double coset decompositions have a natural connection with induced representations.