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Adjoint Representation Of A Lie Algebra
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 \mathrm(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 (conjugatio ... [...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] |
Lie Algebra Automorphism
In abstract algebra, an automorphism of a Lie algebra \mathfrak g is an isomorphism from \mathfrak g to itself, that is, a bijective linear map preserving the Lie bracket. The automorphisms of \mathfrak form a group denoted \operatorname \mathfrak, the automorphism group of \mathfrak. Inner and outer automorphisms The subgroup of \operatorname \mathfrak generated using the adjoint action e^, x \in \mathfrak g is called the inner automorphism group of \mathfrak g. The group is denoted \operatorname^0(\mathfrak). These form a normal subgroup in the group of automorphisms, and the quotient \operatorname(\mathfrak)/\operatorname^0(\mathfrak) is known as the outer automorphism group. Diagram automorphisms It is known that the outer automorphism group for a simple Lie algebra \mathfrak is isomorphic to the group of diagram automorphisms for the corresponding Dynkin diagram in the classification of Lie algebras. The only algebras with non-trivial outer automorphism group are therefo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative Of The Exponential Map
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into . In case is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted , is analytic and has as such a derivative , where is a path in the Lie algebra, and a closely related differential . Appendix on analytic functions. The formula for was first proved by Friedrich Schur (1891). It was later elaborated by Henri Poincaré (1899) in the context of the problem of expressing Lie group multiplication using Lie algebraic terms. It is also sometimes known as Duhamel's formula. The formula is important both in pure and applied mathematics. It enters into proofs of theorems such as the Baker–Campbell–Hausdorff formula, and it is used frequently in physics for example in quantum field theory, as in the Magnus expansion in perturbation theory, and in lattice gauge theory. Throughout, the notations and will be used inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X T. The differential operator T \mapsto \mathcal_X T is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow. Formal definition A flow on a set is a group action of the additive group of real numbers on . More explicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bracket Of Vector Fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted ,Y/math>. Conceptually, the Lie bracket ,Y/math> is the derivative of Y along the flow generated by X, and is sometimes denoted ''\mathcal_X Y'' ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X. The Lie bracket is an R- bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra. The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems. V. I. Arnold refers to this as the "fisherman derivative", ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Algebra
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra. More specifically, ''differential algebra'' refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \mathbb(t), where the derivation is differentiation with respect to t. More generally, every differential e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Of A Lie Algebra
Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a group of people * Representation in contract law, a pre-contractual statement that may (if untrue) result in liability for misrepresentation * Labor representation, or worker representation, the work of a union representative who represents and defends the interests of fellow labor union members * Legal representation, provided by a barrister, lawyer, or other advocate * Lobbying or interest representation, attempts to influence the actions, policies, or decisions of officials * " No taxation without representation", a 1700s slogan that summarized one of the American colonists' 27 colonial grievances in the Thirteen Colonies, which was one of the major causes of the American Revolution * Permanent representation, a type of diplomatic missio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropy Representation
In differential geometry, the isotropy representation is a natural linear representation of a Lie group, that is acting on a manifold, on the tangent space to a fixed point. Construction Given a Lie group action (G, \sigma) on a manifold ''M'', if ''G''''o'' is the stabilizer of a point ''o'' (isotropy subgroup at ''o''), then, for each ''g'' in ''G''''o'', \sigma_g: M \to M fixes ''o'' and thus taking the derivative at ''o'' gives the map (d\sigma_g)_o: T_o M \to T_o M. By the chain rule, :(d \sigma_)_o = d (\sigma_g \circ \sigma_h)_o = (d \sigma_g)_o \circ (d \sigma_h)_o and thus there is a representation: :\rho: G_o \to \operatorname(T_o M) given by :\rho(g) = (d \sigma_g)_o. It is called the isotropy representation at ''o''. For example, if \sigma is a conjugation action of ''G'' on itself, then the isotropy representation \rho at the identity element ''e'' is the adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Map (Lie Theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects. Definitions Let G be a Lie group and \mathfrak g be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map :\exp\colon \mathfrak g \to G which can be defined in several different ways. The typical modern definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
