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

of bald spots, is a general property of symmetric spaces in the Cartan decomposition. At each point on the manifold, the algebra of the Killing fields...

}} For the complexification GC, the Cartan decomposition is described above. Derived from the polar decomposition in the complex general linear group...

Q(a) with a = Σ αi ei with αi > 0. In fact the Cartan decomposition for GT follows from the decomposition for GD. Given z in D, there is an element u in...

it follows that H/K = G/P. The polar decomposition in the complex linear group G implies the Cartan decomposition H* = K ⋅ exp i m {\displaystyle i{\mathfrak...

related to the Schubert cell decomposition of Grassmannians: see Weyl group for more details. The Cartan decomposition writes a semisimple real Lie algebra...

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

{g}}\to {\mathfrak {gl}}(V)} there is a decomposition related to the decomposition of the Lie algebra from its Cartan subalgebra. If we set V λ = { v ∈ V...

mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the...

{X}^{t}.} Then g0 decomposes into the direct sum of its skew-symmetric part k0 and its symmetric part p0. This is the Cartan decomposition: g 0 = k 0 ⊕ p...

{g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {p}}_{0}} is the corresponding Cartan decomposition a 0 {\displaystyle {\mathfrak {a}}_{0}} is a maximal abelian subalgebra...

criterion Cartan decomposition Cartan's equivalence method Cartan formalism (physics) Cartan involution Cartan's magic formula Cartan relations Cartan map Cartan...

that context, the Cartan decomposition given above is called the "Z–Y decomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar...

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

form of the noncompact semisimple Lie group and the corresponding Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding...

Q (the superscript t means transpose). The Lie algebra admits a Cartan decomposition g = g − 1 ⊕ g 0 ⊕ g 1 {\displaystyle \mathbf {g} =\mathbf {g} _{-1}\oplus...

the matrix are non-negative integers. The decomposition matrix, multiplied by its transpose, forms the Cartan matrix, listing the composition factors of...

algebras. For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not...

seen directly using the Cartan decomposition associated to the symmetric space G/K, a generalisation of the polar decomposition of invertible matrices...

{\displaystyle {\mathfrak {g}}={\mathfrak {u}}+{\mathfrak {p}}} is a Cartan decomposition, then ( g , s ) {\displaystyle ({\mathfrak {g}},s)} is of noncompact...

nilpotency criterion based on the Jordan–Chevalley decomposition, as explained there. Applying Cartan's criterion to the adjoint representation gives: A...

nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded...

Riemannian holonomy), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of...

the universal covering space G/K is actually contractible (by the Cartan decomposition for Lie groups). As an example take G = SL(2, R), K = SO(2) and Γ...

corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into...

is a semisimple Lie algebra of rank 1 whose compact part (in the Cartan decomposition) is not semisimple. A notable sufficient condition for linearisability...

46928.0c, S2CID 14530010 Kobayashi, Toshiyuki (2007), "A generalized Cartan decomposition for the double coset space (U(n1)×U(n2)×U(n3))\U(n)/(U(p)×U(q))"...

the complexification of K0. Let G0= K0·P0 be the corresponding Cartan decomposition of G0 and let A be a maximal Abelian subgroup of P0. Flensted-Jensen...

{\displaystyle \sigma (x)=\|X\|} for x=k exp X with k in K and X in p for a Cartan decomposition G = K exp p of the Lie group G, where ||X|| is a K-invariant Euclidean...

respectively. The product of the transpose of D with D itself results in the Cartan matrix, usually denoted C; this is a symmetric matrix such that the entries...