* 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.

* The Iwasawa decomposition G = KAN of a semisimple group G as the product of compact, abelian, and nilpotent subgroups 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 ).

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

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.

Via the theory of zeta integrals initiated by Kenkichi Iwasawa and by John Tate in Tate's thesis it is related to the study of the zeta function of global fields.

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields.

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.

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

This idea is much used in Iwasawa theory.

It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

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

In 1950, Iwasawa was invited to Cambridge, Massachusetts to give a lecture at the International Congress of Mathematicians on his method to study Dedekind zeta functions using integration over ideles and duality of adeles ; this method was also independently obtained by John Tate and it is sometimes called Tate's thesis or the Iwasawa-Tate theory.

Iwasawa is perhaps best known for introducing what is now called Iwasawa theory, which developed from researches on cyclotomic fields from the later parts of the 1950s.

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.

This latter reaction is in accord with the reported decomposition of Af.

The small reaction occurring at 337-degrees-C is probably caused by decomposition of occluded nitrates, and perhaps by a small amount of some hydrous material other than Af.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

If the direct-sum decomposition ( A ) is valid, how can we get hold of the projections Af associated with the decomposition??

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

# As exploratory data analysis, an ANOVA is an organization of an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition ( or, equivalently, each set of terms of a linear model ).

For example, the Banach – Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition.

In the formation of an ordinary agate, it is probable that waters containing silica in solution — derived, perhaps, from the decomposition of some of the silicates in the lava itself — percolated through the rock and deposited a siliceous coating on the interior of the vapour-vesicles.

The first deposit on the wall of a cavity, forming the " skin " of the agate, is generally a dark greenish mineral substance, like celadonite, delessite or " green earth ", which are rich in iron probably derived from the decomposition of the augite in the enclosing volcanic rock.

The volatile hydride generated by the reaction that occurs is swept into the atomization chamber by an inert gas, where it undergoes decomposition.

If sugars are heated so that all water of crystallisation is driven off, then caramelization starts, with the sugar undergoing thermal decomposition with the formation of carbon, and other breakdown products producing caramel.

In fact, the decomposition of hydrogen peroxide is so slow that hydrogen peroxide solutions are commercially available.

The decomposition process is aided by shredding the plant matter, adding water and ensuring proper aeration by regularly turning the mixture.

A decomposition reaction is the opposite of a synthesis reaction, where a more complex substance breaks down into its more simple parts.

Pyrolysis gas chromatography mass spectrometry is a method of chemical analysis in which the sample is heated to decomposition to produce smaller molecules that are separated by gas chromatography and detected using mass spectrometry.