Кіріспе
Математикада Черн-Вейль гомоморфизмі — Черн-Вейль теориясының негізгі құрылымы болып табылады, ол M тегіс көптұқымдысындағы векторлық бундельдер мен негізгі бундельдердің топологиялық инварианттарын M-нің де Рэм кохомология сақиналарындағы сыныптарды білдіретін байланыстар мен қисықтықтар арқылы есептейді. Яғни, бұл теория алгебралық топология және дифференциалдық геометрия салалары арасындағы байланыс құрайды. Ол 1940 жылдардың соңында Шиинг Шен Черн мен Андре Вейль тарапынан жалпыланған Гаусс-Бонне теоремасының дәлелденуінен кейін жасалды. Бұл теория сипаттамалық кластар теориясындағы маңызды қадам болды. G-ді Lie алгебрасымен бірге нақты немесе кешенді Lie тобы деп есептейік, ал — алгебраны үстіндегі бағаланған полиномдардың алгебрасы деп белгілейік (біз орнына қолданғанда да дәл осы аргумент жұмыс істейді). — G-дің жалғау әрекеті бойынша ішіндегі тұрақты нүктелердің субальгебрасы болсын; яғни, — G-дегі барлық g және ішіндегі барлық x үшін тең болатын барлық полиномдардан тұратын субальгебра.
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra , and let denote the algebra of valued polynomials on (exactly the same argument works if we used instead of ). Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
M-дегі негізгі G бунделі P берілген кезде, алгебралардың гомоморфизмі бар:
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra , and let denote the algebra of valued polynomials on (exactly the same argument works if we used instead of ). Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
,
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra , and let denote the algebra of valued polynomials on (exactly the same argument works if we used instead of ). Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
осы гомоморфизм Черн-Вейль гомоморфизмі деп аталады, мұнда оң жақтағы кохомология — де Рэм кохомологиясы. Бұл гомоморфизм берілген бунделдегі кез келген байланыстың қисықтығындағы инвариантты полиномдарды алу арқылы алынады. Егер G компакт немесе жартылай жайлы болса, онда G бундельдерінің жіктеу кеңістігінің кохомология сақинасы , инвариантты полиномдар алгебрасына изоморфты:
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra , and let denote the algebra of valued polynomials on (exactly the same argument works if we used instead of ). Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
(BG кохомология сақинасын де Рэм мағынасында да беруге болады:
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra , and let denote the algebra of valued polynomials on (exactly the same argument works if we used instead of ). Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
қайда және — көптүрліліктер.)
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra , and let denote the algebra of valued polynomials on (exactly the same argument works if we used instead of ). Let be the subalgebra of fixed points in under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that , for all g in G and x in ,
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Given a principal G bundle P on M, there is an associated homomorphism of algebras,
,
called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi simple, then the cohomology ring of the classifying space for G bundles, , is isomorphic to the algebra of invariant polynomials:
(The cohomology ring of BG can still be given in the de Rham sense:
when and are manifolds.)
Голоморфты векторлы бундельдер үшін гомоморфизм
E-нің күрделі көптік M-де голоморфты (кешенді) векторлық бунделі болсын. E-нің қисықтық формасы, белгілі бір гермиттік метрикаға қатысты, жай ғана 2-форма емес, шын мәнінде (1, 1)-форма болып табылады (голоморфты векторлық бундель#Голоморфты векторлық бундельдегі гермиттік метрикалар қараңыз). Сондықтан Черн-Вейль гомоморфизмі мына түрде келеді: ,
Let E be a holomorphic (complex )vector bundle on a complex manifold M. The curvature form of E, with respect to some hermitian metric, is not just a 2 form, but is in fact a (1, 1) form (see holomorphic vector bundle#Hermitian metrics on a holomorphic vector bundle). Hence, the Chern–Weil homomorphism assumes the form: with ,