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...

rectangular matrices A. Cartan decomposition Algebraic polar decomposition Polar decomposition of a complex measure Lie group decomposition Hall 2015, Section 2...

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...

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

Helmholtz decomposition, decomposition of a vector field Indecomposable continuum Lebesgue's decomposition theorem, decomposition of a measure Lie group decomposition...

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 {\displaystyle...

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...

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

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...

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...

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...

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

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

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...

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

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

discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary...

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...

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...

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...

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...

equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components...

as Frobenius himself. The Mackey decomposition was defined and explored by George Mackey in the context of Lie groups, but is a powerful tool in the character...

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

In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states...

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

In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra...

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...