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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
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
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
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
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
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
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
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
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
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)
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
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
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
-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
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
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)
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
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
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)
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)
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
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)
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
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
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
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
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
In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P4 introduced by Geoffrey
Horrocks–Mumford_bundle
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
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
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
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
In algebraic geometry, Procesi bundles are vector bundles of rank n ! {\displaystyle n!} on certain symplectic resolutions of quotient singularities, particularly
Procesi_bundle
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
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
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
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
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
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
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
Concept in differential topology
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
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)
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
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
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
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
Concept in mathematics
for a vector bundle (i.e., a variety corresponding to a locally free sheaf of constant rank). We say a vector bundle E on an algebraic variety X acted
Equivariant_sheaf
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
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
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
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
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
construction Vector bundle Integral monoid ring construction Integral group ring construction Category of Eilenberg–Moore algebras Kleisli category Adjunction
List of algebraic constructions
List_of_algebraic_constructions
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
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
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)
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
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
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
Algebra in algebraic topology
In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {\displaystyle
Steenrod_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
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
) {\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
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
Type of Kac–Moody algebras
affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given
Affine_Lie_algebra
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
Matrix operation which flips a matrix over its diagonal
In linear algebra, transposition is an operation that flips a matrix over its diagonal; that is, transposition switches the row and column indices of
Transpose
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
algebra is an algebraic structure introduced by Beilinson and Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in
Chiral_algebra
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
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
Group of unitary complex matrices with determinant of 1
Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} of SU(n) consists of n × n skew-Hermitian matrices with trace zero. This (real) Lie algebra has
Special_unitary_group
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
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
Mathematical theorem
In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by Herbet Lange [de] and proved by Montserrat
Lange's_conjecture
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
group whose underlying space is the tangent bundle TG of a Lie group G. As a Lie group, the tangent bundle is a semidirect product of a normal abelian
Tangent_Lie_group
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
Algebraic geometry scheme
In algebraic geometry, a Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for
Gorenstein_scheme
Long exact sequence
of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa
Gysin_homomorphism
\wedge ,[\cdot ,\cdot ])} into a Gerstenhaber algebra. Since the tangent bundle is dual to the cotangent bundle, multivector fields of degree k {\displaystyle
Polyvector_field
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
theory (algebraic or topological) are characterized by the following properties. ψk are ring homomorphisms. ψk(l)= lk if l is the class of a line bundle. ψk
Adams_operation
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
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
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
Projective variety that is also an algebraic group
particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is
Abelian_variety
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
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)
Hopf_fibration
Right inverse of a fiber bundle map
mathematical field of topology, a section (or cross section) of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection function
Section_(fiber_bundle)
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)
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
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
Scheme in algebraic geometry
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
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
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