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In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that
Algebra_bundle
Concept in topology (mathematics)
In mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,} over
Lie_algebra_bundle
Generalization of vector bundles
calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle O ( 1 ) {\displaystyle
Coherent_sheaf
Continuous surjection satisfying a local triviality condition
Affine bundle Algebra bundle Characteristic class Covering map Equivariant bundle Fibered manifold Fibration Gauge theory Hopf bundle I-bundle Natural
Fiber_bundle
Universal construction in multilinear algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any order) with multiplication being
Tensor_algebra
adjoint bundle is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure
Adjoint_bundle
Concept in algebraic geometry
canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle
Canonical_bundle
Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure
Clifford_bundle
Mathematical parametrization of vector spaces by another space
{\displaystyle X} (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X {\displaystyle X} . The simplest example
Vector_bundle
Concept in algebraic geometry
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others
Ample_line_bundle
Generalization of a vector bundle
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec C = Spec X R {\displaystyle
Cone_(algebraic_geometry)
the theory of connections on a principal bundle as well as in the theory of Cartan connections. A Lie-algebra-valued differential k {\displaystyle k} -form
Lie algebra–valued differential form
Lie_algebra–valued_differential_form
Study of vector bundles, principal bundles, and fibre bundles
alternative descriptions of important structures in algebraic geometry such as moduli spaces of vector bundles and coherent sheaves. Gauge theory has its origins
Gauge_theory_(mathematics)
In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach
Vector bundles on algebraic curves
Vector_bundles_on_algebraic_curves
Infinitesimal version of Lie groupoid
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
Lie_algebroid
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Vector bundle of rank 1
tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of
Line_bundle
and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The
Stable_principal_bundle
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
Vector bundle of cotangent spaces at every point in a manifold
sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and
Cotangent_bundle
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Relates the geometric vector bundles to algebraic projective modules
topology and algebraic geometry, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept
Serre–Swan_theorem
Concept in mathematics
transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P {\displaystyle
Connection_(principal_bundle)
are Lie algebra-valued forms (a connection form is an example of such a form.) Let M be a smooth manifold and E → M be a smooth vector bundle over M.
Vector-valued differential form
Vector-valued_differential_form
Subject area in mathematics
about Euler characteristics: The Euler characteristic of a vector bundle on an algebraic variety (which is the alternating sum of the dimensions of its cohomology
Algebraic_K-theory
Vector bundle existing over a Grassmannian
Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible
Tautological_bundle
Tangent spaces of a manifold
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself.
Tangent_bundle
Term in differential geometry
principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Let G be a Lie group with Lie algebra g {\displaystyle
Curvature_form
Carathéodory–Jacobi–Lie theorem Lie algebra Lie-* algebra Lie algebra bundle Lie algebra cohomology Lie algebra representation Lie algebroid Lie bialgebra
List of things named after Sophus Lie
List_of_things_named_after_Sophus_Lie
Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves
Hodge_bundle
Clifford algebras. The canonical example is a spinor bundle. In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle. The
Clifford_module_bundle
In homological algebra, a monad is a 3-term complex A → B → C of objects in some abelian category whose middle term B is projective, whose first map A → B
Monad_(homological_algebra)
Concept in mathematics
a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or
Normal_bundle
Non-tensorial representation of the spin group
spinor Spin-1/2 Spinor bundle Supercharge Twistor theory Spacetime algebra Spinors in three dimensions are points of a line bundle over a conic in the projective
Spinor
Concept in algebraic geometry
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line
Nef_line_bundle
Algebraic geometry analog of a principal bundle in algebraic topology
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski
Torsor_(algebraic_geometry)
Complex vector bundle on a complex manifold
vector bundles on a smooth complex projective variety X (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e.
Holomorphic_vector_bundle
Manifold upon which it is possible to perform calculus
example, the tangent bundle to M can be defined as the derivations of the algebra of smooth functions on M. This "algebraization" of a manifold (replacing
Differentiable_manifold
Group of unitary complex matrices with determinant of 1
structure of this Lie algebra can be found below in § Lie algebra structure. In the physics literature, it is common to identify the Lie algebra with the space
Special_unitary_group
Branch of mathematics
manifolds. It uses the techniques of vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry
Differential_geometry
Defines a notion of parallel transport on a bundle
bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic
Connection_(vector_bundle)
Branch of mathematics
bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry
K-theory
vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may
Stable_vector_bundle
Algebraic structure in linear algebra
exterior algebra. A vector bundle is a family of vector spaces parametrized continuously by a topological space X. More precisely, a vector bundle over X
Vector_space
Mathematical concept
group algebra of (the dual to) a vector space is the symmetric algebra, and the group algebra of the Heisenberg group (of the dual) is the Weyl algebra: one
Symplectic_vector_space
In algebraic geometry, Procesi bundles are vector bundles of rank n ! {\displaystyle n!} on certain symplectic resolutions of quotient singularities, particularly
Procesi_bundle
Characteristic classes of vector bundles
algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles
Chern_class
Set of vectors used to define coordinates
program Coordinate system Change of basis – Coordinate change in linear algebra Frame of a vector space – Similar to the basis of a vector space, but not
Basis_(linear_algebra)
Fiber bundle whose fibers are projective spaces
projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally
Projective_bundle
4-manifold invariants
K=c_{1}(W^{+})=c_{1}(W^{-})} . The spinor bundle W {\displaystyle W} comes with a graded Clifford algebra bundle representation i.e. a map γ : C l i f f
Seiberg–Witten_invariants
Module over a sheaf of differential operators
symbols, which in the good case is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems). The techniques were taken up
D-module
Right inverse of a morphism
homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a
Section_(category_theory)
Algebraic topology uses abstract algebra to study topological spaces
This is a list of algebraic topology topics. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces
List of algebraic topology topics
List_of_algebraic_topology_topics
-bundle P → M {\displaystyle P\to M} is always: Transitive (so its unique orbit is the entire M {\displaystyle M} and its isotropy Lie algebra bundle is
Atiyah_algebroid
to the space of smooth sections of f if we work with the algebra bundle with the graded algebra of V-tensors as fibers. Assume also that under this Poisson
First-class_constraint
Type of vector bundle
projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over this variety
Higgs_bundle
Type of derivative in differential geometry
T {\displaystyle T\mapsto {\mathcal {L}}_{X}T} is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes
Lie_derivative
Differential form of degree one or section of a cotangent bundle
cotangent bundle. Equivalently, a one-form on a manifold M {\displaystyle M} is a smooth mapping of the total space of the tangent bundle of M {\displaystyle
One-form
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product C × C × ... × C or Cn by the
Symmetric product of an algebraic curve
Symmetric_product_of_an_algebraic_curve
Various mathematical dualites
of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototypical
Koszul_duality
filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and
Filtered_algebra
Branch of mathematics
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument
Multilinear_algebra
Assignment of a tensor continuously varying across a region of space
language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional)
Tensor_field
Fiber bundle
theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the
Associated_bundle
space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Set of topological invariants
particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe
Stiefel–Whitney_class
Exterior algebraic map taking tensors from p forms to n-p forms
Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate
Hodge_star_operator
Isomorphism between the tangent and cotangent bundles of a manifold
isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of
Musical_isomorphism
Relation between genus, degree, and dimension of function spaces over surfaces
important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic
Riemann–Roch_theorem
In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution to a Cartesian equation of the form: X 2 + a X Y + b Y 2 = P (
Conic_bundle
Generalization of an orientation of a vector space
orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of
Orientation of a vector bundle
Orientation_of_a_vector_bundle
space, Zariski tangent space Function field of an algebraic variety Ample line bundle Ample vector bundle Linear system of divisors Birational geometry Blowing
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
Canadian-American mathematician
analysis and representation theory. He is known for Fell bundles (i.e. Banach *-algebraic bundles). He was an accomplished linguist who knew Sanskrit, Icelandic
James_Michael_Gardner_Fell
Function in mathematics
Connection (principal bundle) Connection (vector bundle) Connection (affine bundle) Connection (composite bundle) Connection (algebraic framework) Gauge theory
Connection_(mathematics)
Principal bundle associated to a vector bundle
In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber
Frame_bundle
Local ring in commutative algebra
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many
Gorenstein_ring
System of partial differential equations used in Higgs field theory
{\text{ad}}P^{\mathbb {C} }} and gives this Lie algebra bundle the structure of a holomorphic vector bundle. Therefore, the condition ∂ ¯ A Φ = 0 {\displaystyle
Hitchin's_equations
scholar and polymath Hermann Grassmann: Grassmann's laws Grassmann algebra Grassmann bundle Grassmann dimensions Grassmann graph Grassmann integral Grassmann
List of things named after Hermann Grassmann
List_of_things_named_after_Hermann_Grassmann
Objects extending the notion of functions
and some contemporary developments are closely related to Mikio Sato's algebraic analysis. In the mathematics of the nineteenth century, aspects of generalized
Generalized_function
Expression that may be integrated over a region
geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential
Differential_form
Branch of mathematics
operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative rings and graded algebras;
Noncommutative_geometry
Mathematical operation on vector spaces
tensor algebra can be constructed as quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the
Tensor_product
American mathematician
representation theory, C*-algebra characterizations, the notion of an approximate identity in a Banach algebra, and Banach bundle theory. Doran taught at
Robert_S._Doran
) {\displaystyle \operatorname {Pic} ^{0}(A)} is called a degree 0 line bundle on A. To A one then associates a dual abelian variety Av (over the same
Dual_abelian_variety
In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: p : G d ( E ) → X {\displaystyle
Grassmann_bundle
Concept in algebraic geometry
of vector bundles. There is a notion of a coherent analytic sheaf on a complex analytic space, and an analogous notion of a coherent algebraic sheaf on
Coherent_sheaf_cohomology
Algebraic geometry term
In algebraic geometry, Horrocks bundles are certain indecomposable rank 3 vector bundles (locally free sheaves) on 5-dimensional projective space, found
Horrocks_bundle
Construct allowing differentiation of tangent vector fields of manifolds
a bundle isomorphism of T(FM) with the trivial bundle FM × aff(n), where aff(n) is the Cartesian product of Rn and gl(n) (viewed as the Lie algebra of
Affine_connection
Classical field theories on fiber bundles
Lie algebra through the Lie group–Lie algebra correspondence is denoted g {\displaystyle {\mathfrak {g}}} . A principal G {\displaystyle G} -bundle P {\displaystyle
Covariant classical field theory
Covariant_classical_field_theory
Branch of mathematics
determined by the Weyl algebra. This deformation is related to the symbol of a differential operator and that A2 is the cotangent bundle of the affine line
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Geometric space whose points represent algebro-geometric objects of some fixed kind
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent
Moduli_space
Topology in mathematics
inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation. Let E → M {\displaystyle E\rightarrow M} be a fibre bundle. A bundle
Inverse_bundle
Generalization of the Dirac equation
vierbein is equivalent to a section of the frame bundle, and so defines a local trivialization of the frame bundle. To write down the equation we also need the
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Riemannian manifold with SU(n) holonomy
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties
Calabi–Yau_manifold
Proof that every structure with certain properties is isomorphic to another structure
functor that sends a bundle to its sheaf of (local) sections. The Gelfand–Naimark–Segal construction embeds any C*-algebra in an algebra of bounded operators
Representation_theorem
a sphere bundle is a fiber bundle in which the fibers are spheres S n {\displaystyle S^{n}} of some dimension n. Similarly, in a disk bundle, the fibers
Sphere_bundle
on the corresponding compact Lie algebra. More precisely, there is a metric tensor k defined on the vertical bundle E = VP such that k is invariant under
Bundle_metric
ALGEBRA BUNDLE
ALGEBRA BUNDLE
Girl/Female
Arabic
Aristocratic Lady
Girl/Female
Italian
Joyful.
Female
Italian
Italian name ALLEGRA means "cheerful and lively."
Girl/Female
Muslim/Islamic
A star in the constellation Leo
Girl/Female
Teutonic American Spanish
Dearly loved.
Girl/Female
American, Australian, Chinese, Spanish, Teutonic
Speaker of Truth; Feminine of Alvaro
Female
Italian
Variant spelling of Italian Allegra, ALLEGRIA means "cheerful and lively."
Boy/Male
Anglo, British, Christian, Danish, English, French, German, Teutonic
Wearing a Moustache; Noble Spear Man; Elf Spear
Male
English
Variant spelling of Middle English Algar, ALGER means elf spear."Â
Girl/Female
Arabic
Aristocratic Lady
Girl/Female
Arabic, Muslim
Truthful
Girl/Female
Muslim
A star in the constellation Leo
Girl/Female
Arabic, French
A Star in the Constellation Leo
Girl/Female
Indian
Speaker of truth
Girl/Female
Latin
Eagle.
Boy/Male
Anglo Saxon German English Teutonic
Noble spearman.
Boy/Male
Bengali, Hindu, Indian
Algea; Lord; Raper
Girl/Female
Italian
Meaning cheerful or lively, related to the musical term allegro. Allegra was the name given by...
Girl/Female
Italian
Lively. Happy.
Surname or Lastname
English
English : from one or more Middle English personal names variously written Alger, Algar, Alcher, Aucher, etc. These represent a falling together of at least three different Continental Germanic and Old English names: Adalgar ‘noble spear’ (Old English Æ{dh}elgÄr), Albgar ‘elf spear’ (Old English ÆlfgÄr), and Aldgar ‘old spear’ (Old English (E)aldgÄr). The Continental Germanic forms were brought to England from France by the Normans. Compare the French cognate Auger. In Norfolk and northern England, the source is probably the Old Norse name Ãlfgeirr ‘elf spear’. The modern English surname is found mainly in East Anglia.German : from a reduced form of the Germanic personal name Adalgar (see 1 above).Abiezer Alger was a merchant in Easton, MA, in the 18th century, who had many prominent descendants.
ALGEBRA BUNDLE
ALGEBRA BUNDLE
Girl/Female
Hindu, Indian, Marathi
Glory; Beauty; Splendour; Fame
Boy/Male
Indian, Sanskrit
Follower of the Guru
Surname or Lastname
English, Scottish, and northern Irish
English, Scottish, and northern Irish : probably a variant of Jelley.German and Frisian : from a Germanic personal name composed with gelt-, cognate with the verb gelten ‘sacrifice’, ‘repay’.Norwegian : unexplained.
Boy/Male
Muslim
Male
English
English name derived from Latin Rudolphus, RUDOLPH means "famous wolf."
Boy/Male
English
Birch Island
Girl/Female
Australian, Chinese, Danish, French, Latin, Swedish
The Sixth Month of the Year
Girl/Female
Arabic
Nice
Boy/Male
British, English
From the Flax Settlement
Boy/Male
Indian, Telugu
Lord Shiva
ALGEBRA BUNDLE
ALGEBRA BUNDLE
ALGEBRA BUNDLE
ALGEBRA BUNDLE
ALGEBRA BUNDLE
a.
Alt. of Algebraical
a.
Originated or taught by Diophantus, the Greek writer on algebra.
a.
Of or pertaining to Algiers or Algeria.
n.
One versed in algebra.
n.
Anything which is required to be done; as, in geometry, to bisect a line, to draw a perpendicular; or, in algebra, to find an unknown quantity.
n.
An algebraic curve, so called from its resemblance to a heart.
a.
A branch of algebra which relates to the direct search for unknown quantities.
adv.
By algebraic process.
n.
A native of Algeria.
n.
One of the terms in an algebraic expression.
v. t.
To perform by algebra; to reduce to algebraic form.
a.
Of or relating to algebra; as, cossic numbers, or the cossic art.
a.
Of or pertaining to algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings.
a.
Of or pertaining to Algeria.
n.
That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude.
n.
That branch of algebra which treats of quadratic equations.
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
pl.
of Palpebra
n.
A treatise on this science.
n.
The eyelid.