|
Equivariant Differential Form
In differential geometry, an equivariant differential form on a manifold ''M'' acted upon by a Lie group ''G'' is a polynomial map :\alpha: \mathfrak \to \Omega^*(M) from the Lie algebra \mathfrak = \operatorname(G) to the space of differential forms on ''M'' that are equivariant; i.e., :\alpha(\operatorname(g)X) = g\alpha(X). In other words, an equivariant differential form is an invariant element ofProof: with V = \Omega^*(M), we have: \operatorname_G(\mathfrak, V) = \operatorname(\mathfrak, V)^G = (\operatorname(\mathfrak, \mathbb)\otimes V)^G. Note \mathbbmathfrak/math> is the ring of polynomials in linear functionals of \mathfrak; see ring of polynomial functions. See also https://math.stackexchange.com/q/101453 for M. Emerton's comment. :\mathbbmathfrak\otimes \Omega^*(M) = \operatorname(\mathfrak^*) \otimes \Omega^*(M). For an equivariant differential form \alpha, the equivariant exterior derivative d_\mathfrak \alpha of \alpha is defined by :(d_\mathfrak \alpha)(X) = d(\al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group Action
In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. __TOC__ Definition Let \sigma: G \times M \to M, (g, x) \mapsto g \cdot x be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map \sigma is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism G \to \mathrm(M). A smooth manifold endowed with a Lie group action is also called a ''G''-manifold. Properties The fact that the action map \sigma is smooth has a couple of immediate consequences: * the stabilizers G_x \subseteq G of the group action are closed, thus are Lie subgroups of ''G'' * the orbits G \cdot x \subseteq M of the group action are immersed submanifolds. Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action. Examples For every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. 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 (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, 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 circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Map
In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as :P(v) = \sum_ \lambda_(v) \cdots \lambda_(v) w_ where the \lambda_: V \to k are linear functionals and the w_ are vectors in ''W''. For example, if W = k^m, then a polynomial mapping can be expressed as P(v) = (P_1(v), \dots, P_m(v)) where the P_i are (scalar-valued) polynomial functions on ''V''. (The abstract definition has an advantage that the map is manifestly free of a choice of basis.) When ''V'', ''W'' are finite-dimensional vector spaces and are viewed as algebraic varieties, then a polynomial mapping is precisely a morphism of algebraic varieties. One fundamental outstanding question regarding polynomial mappings is the Jacobian conjecture, which concerns the sufficiency of a polynomial mapping to be invertible. See also * Polynomial functor References *Claudio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) \, dx is an example of a -form, and can be integrated over an interval ,b/math> contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. In general, a -form is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Polynomial Functions
In mathematics, the ring of polynomial functions on a vector space ''V'' over a field ''k'' gives a coordinate-free analog of a polynomial ring. It is denoted by ''k'' 'V'' If ''V'' is finite dimensional and is viewed as an algebraic variety, then ''k'' 'V''is precisely the coordinate ring of ''V''. The explicit definition of the ring can be given as follows. Given a polynomial ring k _1, \dots, t_n/math>, we can view t_i as a coordinate function on k^n; i.e., t_i(x) = x_i where x = (x_1, \dots, x_n). This suggests the following: given a vector space ''V'', let ''k'' 'V''be the commutative ''k''-algebra generated by the dual space V^*, which is a subring of the ring of all functions V \to k. If we fix a basis for ''V'' and write t_i for its dual basis, then ''k'' 'V''consists of polynomials in t_i. If ''k'' is infinite, then ''k'' 'V''is the symmetric algebra of the dual space V^*. In applications, one also defines ''k'' 'V''when ''V'' is defined over some subfield of ''k' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product \iota_X \omega is sometimes written as X \mathbin \omega. Definition The interior product is defined to be the contraction of a differential form with a vector field. Thus if X is a vector field on the manifold M, then \iota_X : \Omega^p(M) \to \Omega^(M) is the map which sends a p-form \omega to the (p - 1)-form \iota_X \omega defined by the property that (\iota_X\omega)\left(X_1, \ldots, X_\right) = \omega\left(X, X_1, \ldots, X_\right) for any vector fields X_1, \ldots, X_. When \omega is a scalar field (0-form), \iota_X \omega = 0 by convention ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Vector Field
In the study of mathematics, and especially of differential geometry, fundamental vector fields are instruments that describe the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions. Motivation Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if M is a smooth manifold and X is a smooth vector field, one is interested in finding integral curves to X . More precisely, given p \in M one is interested in curves \gamma_p: \mathbb R \to M such that: : \gamma_p'(t) = X_, \qquad \gamma_p(0) = p, for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If X is furthermore a complete vector field, then the flow of X , defined as the collection of all integral curves for X , is a diffeomorphism of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Cohomology
In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^*(X; \Lambda) = H^*(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^*(X;A) o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Construction
In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^*(X; \Lambda) = H^*(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^*(X;A) of X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Index Theory
In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed point (mathematics), fixed points of the element. If the element is neutral, then the theorem reduces to the usual index theorem. The classical formula such as the Atiyah–Bott formula is a special case of the theorem. Statement Let \pi: E \to M be a clifford module bundle. Assume a compact Lie group ''G'' acts on both ''E'' and ''M'' so that \pi is equivariant bundle, equivariant. Let ''E'' be given a connection that is compatible with the action of ''G''. Finally, let ''D'' be a Dirac operator on ''E'' associated to the given data. In particular, ''D'' commutes with ''G'' and thus the kernel of ''D'' is a finite-dimensional representation of ''G''. The equivariant index of ''E'' is a Brauer's theorem on induced characters, virtual character given by taking t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Formula For Equivariant Cohomology
In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form \alpha on an orbifold ''M'' with a torus action and for a sufficient small \xi in the Lie algebra of the torus ''T'', we have : \int_M \alpha(\xi) = \sum_F \int_F where the sum runs over all connected components ''F'' of the set M^T of fixed points, d_M is the orbifold multiplicity of M (which equals 1 if M is a manifold), and e_T(F) is the equivariant Euler form of the normal bundle of F. The formula allows one to compute the equivariant cohomology ring of the orbifold ''M'' (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms. No analog of such results holds in the non-equivariant cohomology. One important consequence of the formula is the Duistermaat–Heckman theorem, which states: supposing there is a Hamiltonian circle action (for simplicity) on a compact symp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
