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 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 triangular and lower triangular matrices - but with exceptional cases. It is related to the Schubert cell decomposition of Grassmannians.
A Cartan subalgebra of a semisimple Lie algebra L given the canonical decomposition, is a maximal Abelian subalgebra of M with which the symmetric space G/K is homeomorphic.
The Cartan subalgebra of u(n) is a Lie algebra of diagonal Hermitean matrices, that of su(n) is a Lie algebra of Hermitean matrices of trace zero.
The Cartan metric metric provides a map between L and its dual space L^+, the space of linear functionals acting on L. The adjoint action of the group on L is then mapped by the Cartan metric to an action of the group on L^+, the coadjoint representation.
Feeds |
Contact