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Maximal Compact Subgroup
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are ''not'' in general unique, but are unique up to conjugation – they are essentially unique. Example An example would be the subgroup O(2), the orthogonal group, inside the general linear group GL(2, R). A related example is the circle group SO(2) inside SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Fixed Point Theorem
Cartan may refer to: * Élie Cartan (1869–1951), French mathematician who worked with Lie groups * Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ... (1904–2008), French mathematician who worked in algebraic topology, son of Élie Cartan * Anna Cartan (1878–1923), French mathematician and teacher, sister of Élie Cartan * Cartan (crater), a lunar crater named for Élie Cartan * Badea Cârțan (1849–1911), Austro-Hungarian Romanian activist {{disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strictly Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its ''epigraph'' (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c as a nonnegative real number). Convex functions play an important role in many areas of mathematics. They are especially important in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras. History and name The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term ''"Killing form"'' first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term ''"Cartan-Killing form"''. At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point Subgroup
In algebra, the fixed-point subgroup G^f of an automorphism ''f'' of a group ''G'' is the subgroup of ''G'': :G^f = \. More generally, if ''S'' is a set of automorphisms of ''G'' (i.e., a subset of the automorphism group of ''G''), then the set of the elements of ''G'' that are left fixed by every automorphism in ''S'' is a subgroup of ''G'', denoted by ''G''''S''. For example, take ''G'' to be the group of invertible ''n''-by-''n'' real matrices and f(g)=(g^T)^ (called the Cartan involution). Then G^f is the group O(n) of ''n''-by-''n'' orthogonal matrices. To give an abstract example, let ''S'' be a subset of a group ''G''. Then each element ''s'' of ''S'' can be associated with the automorphism g \mapsto sgs^, i.e. conjugation by ''s''. Then :G^S = \; that is, the centralizer of ''S''. See also *Ring of invariants In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Involution
In mathematics, the Cartan decomposition is a decomposition of a 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 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 equivalent. Examples * A Cartan involution on \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcenter
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is. Straightedge and compass construction The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. For three non-collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruhat-Tits Fixed Point Theorem
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete metric space such that, given any points x and y, there exists a point m such that for every point z, d(z, m)^2 + \leq . The point m is then the midpoint of x and y: d(x, m) = d(y, m) = d(x, y)/2. In a Hilbert space, the above inequality is equality (with m = (x+y)/2), and in general an Hadamard space is said to be if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space. The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In a Hadamard space, any two points can be joined by a unique geodesic between them; in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Metric Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots of elements from X of a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d(x_m, x_n) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: #Every Cauchy seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Space
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete metric space such that, given any points x and y, there exists a point m such that for every point z, d(z, m)^2 + \leq . The point m is then the midpoint of x and y: d(x, m) = d(y, m) = d(x, y)/2. In a Hilbert space, the above inequality is equality (with m = (x+y)/2), and in general an Hadamard space is said to be if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space. The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In a Hadamard space, any two points can be joined by a unique geodesic between them; in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Symmetric Space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Decomposition
In mathematics, the Cartan decomposition is a decomposition of a 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 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 equivalent. Examples * A Cartan involution on \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
