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Cartan Pair
In the mathematical fields of Lie theory and algebraic topology, the notion of Cartan pair is a technical condition on the relationship between a reductive Lie algebra \mathfrak and a subalgebra \mathfrak reductive in \mathfrak. A reductive pair (\mathfrak,\mathfrak) is said to be Cartan if the relative Lie algebra cohomology :H^*(\mathfrak,\mathfrak) is isomorphic to the tensor product of the characteristic subalgebra :\mathrm\big(S(\mathfrak^*) \to H^*(\mathfrak,\mathfrak)\big) and an exterior subalgebra \bigwedge \hat P of H^*(\mathfrak), where *\hat P, the ''Samelson subspace'', are those primitive elements in the kernel of the composition P \overset\tau\to S(\mathfrak^*) \to S(\mathfrak^*), *P is the primitive subspace of H^*(\mathfrak), *\tau is the transgression, *and the map S(\mathfrak^*) \to S(\mathfrak^*) of symmetric algebras is induced by the restriction map of dual vector spaces \mathfrak^* \to \mathfrak^*. On the level of Lie groups, if ''G'' is a compact, ... [...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] |
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] |
Lie Theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Lie Algebra
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: \mathfrak = \mathfrak \oplus \mathfrak; there are alternative characterizations, given below. Examples The most basic example is the Lie algebra \mathfrak_n of n \times n matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an ''n''-dimensional vector space, \mathfrak(V). This is the Lie algebra of the general linear group GL(''n''), and is reductive as it decomposes as \mathfrak_n = \mathfrak_n \oplus \mathfrak, corresponding to traceless matrices and scalar matrices. Any semisimple Lie algebra or abelian Lie algebra is ''a fortiori'' reductive. Over the real numbers, compact Lie algebras are reductive. Definitions A Lie algebra \mathfrak over a field of characteristic 0 is called reductive if any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module. Motivation If G is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on G. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential. The construction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transgression Map
In algebraic topology, a transgression map is a way to transfer cohomology classes. It occurs, for example in the inflation-restriction exact sequence in group cohomology, and in integration in fibers. It also naturally arises in many spectral sequences; see spectral sequence#Edge maps and transgressions. Inflation-restriction exact sequence The transgression map appears in the inflation-restriction exact sequence, an exact sequence occurring in group cohomology. Let ''G'' be a group, ''N'' a normal subgroup, and ''A'' an abelian group which is equipped with an action of ''G'', i.e., a homomorphism from ''G'' to the automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ... of ''A''. The quotient group G/N acts on ::A^N = \. Then the inflation-restriction exact s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal property: for every linear map from to a commutative algebra , there is a unique algebra homomorphism such that , where is the inclusion map of in . If is a basis of , the symmetric algebra can be identified, through a canonical isomorphism, to the polynomial ring , where the elements of are considered as indeterminates. Therefore, the symmetric algebra over can be viewed as a "coordinate free" polynomial ring over . The symmetric algebra can be built as the quotient of the tensor algebra by the two-sided ideal generated by the elements of the form . All these definitions and properties extend naturally to the case where is a module (not necessarily a free one) over a commutative ring. Construction From tensor algebra It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Bundle
In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ... , such that every bundle with the given structure group over is a pullback bundle, pullback by means of a continuous map . Existence of a universal bundle In the CW complex category When the definition of the classifying space takes place within the homotopy category (mathematics), category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem. For compact Lie groups We will first prove: :Proposition. Let be a compact Lie group. There exists a contractible space on which acts freely. The projection is a -principal fibr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Map
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. This was still not t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
