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...

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure...

In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written...

Schubert cell decomposition of flag varieties: see Weyl group for this. More generally, any group with a (B, N) pair has a Bruhat decomposition. G is a connected...

rectangular matrices A. Cartan decomposition Algebraic polar decomposition Polar decomposition of a complex measure Lie group decomposition Hall 2015 Section 2.5...

In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi (1905)...

theorem, decomposition of a measure Lie group decomposition, used to analyse the structure of Lie groups and associated objects Manifold decomposition, decomposition...

semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable...

Tangent Lie group Tate Lie algebra Toral Lie algebra Lie bracket of vector fields Lie derivative Lie group Lie group decomposition Lie groupoid Lie subgroup...

Gauss decomposition is a generalization of the LU decomposition for the general linear group and a specialization of the Bruhat decomposition. For GL(V)...

Local Lie group Formal group law Hilbert's fifth problem Hilbert-Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification...

Jordan–Chevalley decomposition of a matrix Deligne–Lusztig theory, and its Jordan decomposition of a character of a finite group of Lie type The Jordan–Hölder...

before Gram and Schmidt. In the theory of Lie group decompositions, it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process...

analogues in Lie algebras. Analogues of the Jordan–Chevalley decomposition also exist for elements of Linear algebraic groups and Lie groups via a multiplicative...

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...

In mathematics, the term cycle decomposition can mean: Cycle decomposition (graph theory), a partitioning of the vertices of a graph into subsets, such...

complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan. Using the Lie correspondence...

algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent...

a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of...

Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups...

determinant. This is also a Lie group of dimension n2; it has the same Lie algebra as GL(n, R). The polar decomposition, which is unique for invertible...

topological group G (Peter & Weyl 1927). The theorem is a collection of results generalizing the significant facts about the decomposition of the regular...

linear group, this decomposition is the QR decomposition, and the deformation retraction is the Gram-Schmidt process. For a general semisimple Lie group, the...

to lie in B, then we obtain the Bruhat decomposition G = ⋃ w ∈ W B w B {\displaystyle G=\bigcup _{w\in W}BwB} which gives rise to the decomposition of...

group H3(R). It is a nilpotent real Lie group of dimension 3. In addition to the representation as real 3×3 matrices, the continuous Heisenberg group...

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of...

is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for...

whereas U(n) forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition. Further symplectic matrix properties can...

normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism. Two Lie groups are locally isomorphic...

complementary representation). The decomposition of a semisimple representation into simple ones, called a semisimple decomposition, need not be unique; for example...