In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Lie group decompositions, named after Sophus Lie

Sophus Lie

Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...

, are used to analyse the structure of Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s and associated objects, by showing how they are built up out of subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s. They are essential technical tools in the representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

 of Lie groups and Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

s; they can also be used to study the algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

 of such groups and associated homogeneous space

Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

s. 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 group

Algebraic group

In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

s and p-adic number

P-adic number

In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

 analogues, making it harder to summarise the facts into a unified theory.

List of decompositions

• The Bruhat decomposition
  Bruhat decomposition
  In mathematics, the Bruhat decomposition G = BWB 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...
  
   G = BWB of a semisimple algebraic group
  Semisimple algebraic group
  In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
  
   into double coset
  Coset
  In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
  
  s of a Borel subgroup
  Borel subgroup
  In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
  
   can be regarded as a general expression of the principle of Gauss–Jordan elimination
  Gauss–Jordan elimination
  In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards....
  
  , which generically writes a matrix as the product of an upper triangular matrix with a lower triangular matrix—but with exceptional cases. It is related to the Schubert cell decomposition of Grassmannian
  Grassmannian
  In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
  
  s: see Weyl group
  Weyl group
  In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
  
   for more details.
• The Cartan decomposition
  Cartan decomposition
  The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition of matrices.- Cartan involutions on Lie algebras :...
  
   writes a semisimple real Lie algebra as the sum of eigenspaces of a Cartan involution.
• The Iwasawa decomposition
  Iwasawa decomposition
  In mathematics, the Iwasawa decomposition KAN of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix...
  
   G = KAN of a semisimple group G as the product of compact
  Compact group
  In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
  
  , abelian, and nilpotent
  Nilpotent group
  In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
  
   subgroups generalises the way a square real matrix can be written as a product of an orthogonal matrix
  Orthogonal matrix
  In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
  
   and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization).
• The Langlands decomposition
  Langlands decomposition
  In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product P=MAN of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.- Applications :...
  
   P = MAN writes a parabolic subgroup P of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.
• The Levi decomposition
  Levi decomposition
  In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra....
  
   writes a finite dimensional Lie algebra as a semidirect product
  Semidirect product
  In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
  
   of a normal solvable subalgebra by a semisimple subalgebra.
• The Polar decomposition G = KAK writes a semisimple Lie group G in terms of a maximal compact subgroup K and an abelian subgroup A.