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Lie Group Decomposition
{{unreferenced, date=September 2009 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 subgroups. They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces. Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back to decompositions. The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues, making it harder to summarise the facts into a unified theory. List of decompositions * The Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements * The Bruhat decomposition ''G'' = ''BWB'' of a semisimple algebraic group into double cosets of a Borel subgroup c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Decomposition
In mathematics, the Cartan decomposition is a decomposition of a Semisimple Lie algebra, semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing. Cartan involutions on Lie algebras Let \mathfrak be a real semisimple Lie algebra and let B(\cdot,\cdot) be its Killing form. An Involution (mathematics), involution on \mathfrak is a Lie algebra automorphism \theta of \mathfrak whose square is equal to the identity. Such an involution is called a ''Cartan involution'' on \mathfrak if B_\theta(X,Y) := -B(X,\theta Y) is a positive definite bilinear form. Two involutions \theta_1 and \theta_2 are considered equivalent if they differ only by an inner automorphism. Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LU Decomposition
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish mathematician Tadeusz Banachiewicz in 1938. Definitions Let ''A'' be a square matrix. An LU factorization refers to the factorization of ''A'', with proper row and/or column orderings or permutations, into two factors – a lower triangular matrix ''L'' and an upper triangular matrix ''U'': : A = LU. In the lower triangular matrix all elements above the diagonal are zero, in the upper triangular matrix, all the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Lie Algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consists of all linear combinations of Lie brackets of pairs of elements of \mathfrak. The ''derived series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq \mathfrak,\mathfrak mathfrak,\mathfrak \geq [ \mathfrak,\mathfrak mathfrak,\mathfrak, \mathfrak,\mathfrak mathfrak,\mathfrak] \geq ... If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra_is_a_fortiori.html" ;"title="mathfrak,\mathfrak ... is a fortiori">mathfrak,\mathfrak ... is a fortiori solvable but the converse is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group with identity element , a subgroup , and a normal subgroup , the following statements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple. Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form :\exp(\mathrm(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Decomposition
In mathematics, the Langlands decomposition writes a parabolic subgroup ''P'' of a semisimple Lie group as a product P=MAN of a reductive subgroup ''M'', an abelian subgroup ''A'', and a nilpotent subgroup ''N''. Applications A key application is in parabolic induction, which leads to the Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...: if G is a reductive algebraic group and P=MAN is the Langlands decomposition of a parabolic subgroup ''P'', then parabolic induction consists of taking a representation of MA, extending it to P by letting N act trivially, and inducing the result from P to G. See also * Lie group decompositions References Sources * A. W. Knapp, Structure theory of semisimple Lie groups. . Lie groups Algebraic groups {{Mathan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity matrix. This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse: Q^\mathrm=Q^, where is the inverse of . An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint (conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of orthogonal matrices, under multiplication, forms the group , known as the orthogonal gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. Definition The definition uses the idea of a central series for a group. The followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
