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Representation Up To Homotopy
A representation up to homotopy has several meanings. One of the earliest appeared in physics, in constrained Hamiltonian systems. The essential idea is lifting a non-representation on a quotient to a representation up to strong homotopy on a resolution of the quotient. As a concept in differential geometry, it generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. As such, it was introduced by Abad and Crainic. As a motivation consider a regular Lie algebroid (''A'',''ρ'', ,. (regular meaning that the anchor ''ρ'' has constant rank) where we have two natural ''A''- connections on ''g''(''A'') = ker ''ρ'' and ''ν''(''A'')= ''TM''/im ''ρ'' respectively: :\nabla\colon \Gamma(A)\times\Gamma(\mathfrak(A))\to\Gamma(\mathfrak(A)): \nabla_\psi:= phi,\psi :\nabla\colon \Gamma(A)\times\Gamma(\nu(A))\to\Gamma(\nu(A)): \nabla_\overline:=\overline. In the deformation theory of the Lie algebroid ''A'' there is a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian System
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory. Overview Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\boldsymbol,\boldsymbol,t), also known as the Hamiltonian. The state of the system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Of A Lie Algebra
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The univers ... [...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] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marius Crainic
Marius Nicolae Crainic (February 3, 1973, Aiud) is a Romanian mathematician working in the Netherlands. Education and career Born in Aiud, Romania, Crainic obtained a bachelor's degree at Babeș-Bolyai University (Cluj-Napoca) in 1995. He then moved to the Netherlands and obtained a master's degree in 1996 at Nijmegen University. He received his Ph.D. in 2000 from Utrecht University under the supervision of Ieke Moerdijk. His Ph.D. thesis is titled "''Cyclic cohomology and characteristic classes for foliations''". He was a Miller Research Fellow at the University of California, Berkeley from 2001 to 2002. He then returned to Utrecht University as a Fellow of the Royal Netherlands Academy of Arts and Sciences (KNAW). In 2007 he became an associate professor at Utrecht University, and since 2012 he is a full professor. In 2016 he was elected member of KNAW. In 2008 Crainic was awarded the André Lichnerowicz Prize in Poisson Geometry and in 2016 he received the De Bruijn Priz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The European Mathematical Society
'' Journal of the European Mathematical Society'' is a monthly peer review, peer-reviewed mathematical journal. Founded in 1999, the journal publishes articles on all areas of Mathematics, pure and applied mathematics. Most published articles are original research articles but the journal also publishes survey articles.Summary of the journal The journal has been published by Springer_Science+Business_Media, Springer until 2003. Since 2004, it is published by the European Mathematical Society. The first editor-in-chief was Jürgen Jost, followed in 2004 by Haïm Brezis. The journal was founded in order to promote interdisciplinary work within the mathematical community and to preserve unity across pure and applied mathematics. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Group
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matched Pair
* ''Matched'' (book), the first book in the trilogy
{{disambiguation ...
Matched may refer to: * Matched filter, a filter used in signal processing * Matched betting, a betting technique * ''Matched'' trilogy, a dystopian fiction trilogy of books, by Ally Condie Allyson Braithwaite Condie is an author of young adult and middle grade fiction."Ally Condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
