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Lie's Third Theorem
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence. Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a ''local'' Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem. Historical notes The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second h ... [...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] |
Maurer–Cartan Form
In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer. As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the Lie algebra associated to the Lie group . The Lie algebra is identified with the tangent space of at the identity, denoted . The Maurer–Cartan form is thus a one-form defined globally on which is a linear mapping of the tangent space at each into . It is given as the pushforward of a vector in along the left-translation in the group: :\omega(v) = (L_)_* v,\quad v\in T_gG. Motivation and interpretation A Lie group acts on itself by multiplication under the mapping :G\times G \ni (g,h) \mapsto gh \in G. A question of importance to Cartan and his contempora ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebroid
In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra. Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid. Lie algebroids were introduced in 1967 by Jean Pradines. Definition and basic concepts A Lie algebroid is a triple (A, cdot,\cdot \rho) consisting of * a vector bundle A over a manifold M * a Lie bracket cdot,\cdot/math> on its space of sections \Gamma (A) * a morphism of vector bundles \rho: A\rightarrow TM, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groupoid
In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations :s,t : \operatorname \to \operatorname are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by Charles Ehresmann under the name ''differentiable groupoids''. Definition and basic concepts A Lie groupoid consists of * two smooth ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path (topology)
In mathematics, a path in a topological space X is a continuous function from the closed unit interval , 1/math> into X. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space X is often denoted \pi_0(X). One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x_0, then a path in X is one whose initial point is x_0. Likewise, a loop in X is one that is based at x_0. Definition A ''curve'' in a topological space X is a continuous function f : J \to X from a non-empty and non-degenerate interval J \subseteq \R. A in X is a curve f : , b\to X whose domain , b/math> is a compact non-degenerate interval (meaning a is homeomorphic to , 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Proof
In mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or existence theorem, ''pure existence theorem''), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics. Constructivism (mathematics), Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Duistermaat
Johannes Jisse (Hans) Duistermaat (The Hague, December 20, 1942 – Utrecht, March 19, 2010) was a Dutch mathematician. Biography Duistermaat attended primary school in Jakarta, at the time capital of the Dutch East Indies, where his family moved after the end of World War II. In 1957, a few years after the Indonesian independence, they came back to the Netherlands and Duistermaat completed his high school studies in Vlaardingen. From 1959 to 1965 he studied studied mathematics at Utrecht University, and he obtained his PhD degree at the same institution in 1968, with a thesis on the mathematical structures of thermodynamics entitled "''Energy and Entropy as Real Morphisms for Addition and Order''". His original supervisor was the applied mathematician Günther K. Braun, who passed away one year before the thesis defense, so the official supervision was taken over by geometer Hans Freudenthal. After a postdoctoral stay in Lund (1969–70), Duistermaat returned to the Netherlan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chevalley-Eilenberg Complex
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] |
Center Of A Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
