* Iwasawa decomposition ( KAN ) a mathematical process dealing with Lie groups

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra ( the Iwasawa decomposition ), and this provides a canonical bicrossproduct quantum group associated to g. For su ( 2 ) one obtains a quantum group deformation of the Euclidean group E ( 3 ) of motions in 3 dimensions.

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 ( a consequence of Gram-Schmidt orthogonalization ).

Then the Iwasawa decomposition of is

and the Iwasawa decomposition of G is

For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.

Before that he worked on Lie groups and Lie algebras, introducing the general Iwasawa decomposition.

Ribet's methods were pushed further by Barry Mazur and Andrew Wiles in order to prove the Main Conjecture of Iwasawa theory ,< ref > a corollary of which is a strengthening of the Herbrand-Ribet theorem: the power of p dividing B < sub > p − n </ sub > is exactly the power of p dividing the order of G < sub > n </ sub >.

Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a ( disconnected ) maximal compact subgroup provided the center of G is finite.

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

In order to get an interesting Galois module here, Iwasawa took the ideal class group of, and let be its p-torsion part.

In fact, is a module over the Iwasawa algebra ( i. e. the completed group ring of over ).

Ralph Greenberg has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motives in the context of Iwasawa theory.

In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties.

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Kenkichi Iwasawa ( Iwasawa Kenkichi, September 11, 1917 – October 26, 1998 ) was a Japanese mathematician who is known for his influence on algebraic number theory.

* Kenkichi Iwasawa ( 1917 – 1998 ) Mathematician

Sometimes Iwasawa tells this story ).

The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur.

The second part of the theorem gives the existence of a decomposition of a unitary representation of G into finite-dimensional representations.

* The Bruhat decomposition G = BWB of a semisimple algebraic group into double cosets of a Borel subgroup can be regarded as a general expression of the principle of Gauss – Jordan elimination, which generically writes a matrix as the product of an upper triangular matrix with a lower triangular matrix — but with exceptional cases.

* The Polar decomposition G = KAK writes a semisimple Lie group G in terms of a maximal compact subgroup K and an abelian subgroup A.

More strongly, either G has a decomposition of this type, or for every vertex v of G there is a k-coloring in which v is the only vertex of its color and every other color class has at least two vertices.

A representation of a chordal graph as an intersection of subtrees forms a tree decomposition of the graph, with treewidth equal to one less than the size of the largest clique in the graph ; the tree decomposition of any graph G can be viewed in this way as a representation of G as a subgraph of a chordal graph.

More generally the Cartan-Helgason theorem gives the decomposition when G / H is a compact symmetric space, in which case all multiplicities are one ; a generalization to arbitrary σ has since been obtained by.

In mathematics, the Bruhat decomposition ( named after François Bruhat ) 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.

The Bruhat decomposition of G is the decomposition

: where by definition D, the decomposition group of p, is the subgroup of elements of G sending a given P < sub > j </ sub > to itself.

That is, since the degree of L / K and the order of G are equal by basic Galois theory, the order of the decomposition group D is the degree of the residue field extension F ′/ F.

Each decomposition group D, for a given P < sub > j </ sub >, contains an inertia group I consisting of the g in G that send P < sub > j </ sub > to itself, but induce the identity automorphism on

The decomposition group must be equal to all of G, since there is only one prime of Z above 2.

Consider a group G and subgroups H and K, with K contained in H. Then the left cosets of H in G are each the union of left cosets of K. Not only that, but translation ( on one side ) by any element g of G respects this decomposition.

It can be produced by thermal decomposition of only certain compounds, especially those with an-N = N-linkage.

Conversely, any Hermitian positive semidefinite matrix M can be written as M = A < sup >*</ sup > A ; this is the Cholesky decomposition.

b / a, then z = a ( 1 + m ε ) is the polar decomposition of the dual number z, and the slope m is its angular part.

) The singular value decomposition of X is X = WΣV < sup > T </ sup >, where the m × m matrix W is the matrix of eigenvectors of the covariance matrix XX < sup > T </ sup >, the matrix Σ is an m × n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and the n × n matrix V is the matrix of eigenvectors of X < sup > T </ sup > X.

:; QR decomposition: M = QR, Q orthogonal, R upper triangular.

:; Singular value decomposition: M = UΣV < sup > T </ sup >, U and V orthogonal, Σ non-negative diagonal.

:; Polar decomposition: M = QS, Q orthogonal, S symmetric non-negative definite.

The Cholesky decomposition is mainly used for the numerical solution of linear equations Ax =

# U and P commute, where we have the polar decomposition A = UP with a unitary matrix U and some positive semidefinite matrix P.

On an almost Kähler manifold, one can write this decomposition as h = g + iω, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and ω is the almost symplectic structure.

If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.

In particular, any prime larger than can enter at most once in this decomposition ( that is, with an exponent r = 1 ).

In linear algebra, a QR decomposition ( also called a QR factorization ) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.

* The Langlands decomposition P = MAN writes a parabolic subgroup P of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.

The decomposition for X = S < sup > 2 </ sup >

The decomposition above further shows, that γ behaves near z = 0 asymptotically like:

An abstract definition of ( real ) Hodge structure is now given: for a real vector space W, a Hodge structure of integer weight k on W is a direct sum decomposition of W < sup > C </ sup > = W ⊗ C, the complexification of W, into graded pieces W < sup > p, q </ sup > where k = p + q, and the complex conjugation of W < sup > C </ sup > interchanges this subspace with W < sup > q, p </ sup >.

In other words, each Ω < sup > r </ sup >( M )< sup > C </ sup > admits a decomposition into a sum of Ω < sup >( p, q )</ sup >( M ), with r = p + q.

-- ( x = v | rec ) -- record value decomposition, pattern fields must be non empty