The two notions of the exponential map coincide in the case of Lie groups equipped with bi-invariant metrics (i.e. Riemannian metrics invariant under left and right translation). In this case the geodesics through the identity are precisely the one-parameter subgroups of G. Take the example that gives the "honest" exponential map. Consider the positive real numbers R+, a Lie group under the usual multiplication. Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product <u,v>y< ...

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Exponential map, Exponential map - Lie theory, Exponential map - Definition, Exponential map - Properties, Exponential map - Riemannian geometry, Exponential map - Definition, Exponential map - Properties, Exponential map - Relationships

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When the map f between manifolds is a diffeomorphism, that is, it is both smooth and invertible, then the pullback can be defined for the tangent space as well as for the cotangent space, and thus, by extension, for an arbitrary mixed tensor bundle on the manifold. The matrix can be inverted to define and thus one has, at each point p, that the pushforward is the inverse of the pullback, now acting on the tangent space (instead of the cotangent space): f * (p) = [f ...

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Pullback, Pullback - Pullback on tensors, Pullback - Pullback of cotangent bundles, Pullback - Pullback on tensor bundles, Pullback - Pullback of diffeomorphisms

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Let C be a category and let {Xi | i ∈ I} be an indexed family of objects in C. The product of the set {Xi} is an object X together with a collection of morphisms πi : X → Xi (called projections) which satisfy a universal property: for any object Y and any collection of morphisms fi : Y → Xi, there exists a unique morphism f : Y → XSee also:

Product category theory, Product category theory - Definition, Product category theory - Examples, Product category theory - Discussion

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In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. If V has even dimension then Cℓ(V,Q) is a central simple algebra over K. If V has even dimension then ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

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More formally, a left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent conditions: If Q is a submodule of some other left R-module M, then there exists another submodule K of M such that M is the internal direct sum of Q and K, i.e. Q + K = M and Q ∩ K = {0}. If X is a submodule of the left R-module Y and g : X → Q ...

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Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure of the totality of all homology or cohomology groups of a space. Locally compact quantum groups generalize Hopf algebras and carry a topology. The algebra of all continuous functions on a Lie group is a locally compact quantum group. ...

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Hopf algebra, Hopf algebra - Examples, Hopf algebra - Quantum groups and non-commutative geometry, Hopf algebra - Related concepts

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Trivially, the zero module {0} is injective. Every vector space Q is injective. Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map g in ...

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Injective module, Injective module - Definition, Injective module - Examples, Injective module - Facts, Injective module - Generalization

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If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn. The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where a ...

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Product topology, Product topology - Definition, Product topology - Examples, Product topology - Properties, Product topology - Relation to other topological notions

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The product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, and for every i in I, fi : Y → Xi is a continuous map, then there exists precisely one continuous map f : Y → X such that the following diagram commutes: This shows that the product space is a product in the category of topological spaces. If follows from the above universal pr ...

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Product topology, Product topology - Definition, Product topology - Examples, Product topology - Properties, Product topology - Relation to other topological notions

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In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavele ...

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Hamiltonian quantum mechanics, Hamiltonian quantum mechanics - The quantum Hamiltonian, Hamiltonian quantum mechanics - Energy eigenket degeneracy symmetry and conservation laws, Hamiltonian quantum mechanics - Hamilton's equations

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Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations. The Galilean symmetries (interpreted as active transformations): Spatial translations: Time translations: Boosts: Rotations: ...

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Galilean transformation, Galilean transformation - History, Galilean transformation - Translation one dimension, Galilean transformation - Galilean transformations, Galilean transformation - Central extension of the Galilean group, Galilean transformation - Notes

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A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism φ : MorD(F–, –) → MorC(–, G–) consisting of bijections: φX,Y : MorD(F(X), Y) → Mor< ...

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Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

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Free objects and forgetful functors. If F : Set → Grp is the functor assigning to each set X the free group over X, and if G : Grp → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G. The unit of this adjoint pair is the embedding of a set X into the free group over X. In general, free constructions in mathematics tend to be left adjoints of forgetful functors. Free rings, free abelian groups, ...

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Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

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In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum isn't conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). For a system of point particles without any intrinsic angular momentum, it ...

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Angular momentum, Angular momentum - Angular momentum in classical mechanics, Angular momentum - Definition, Angular momentum - Conservation of angular momentum, Angular momentum - Angular momentum in relativistic mechanics, Angular momentum - Angular momentum in quantum mechanics

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In quantum mechanics, angular momentum is defined like momentum - not as a quantity but as an operator on the wave function: where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as where is the gradient operator. This is a commonly encountered form of the angular momentum operator, though not th ...

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Angular momentum, Angular momentum - Angular momentum in classical mechanics, Angular momentum - Definition, Angular momentum - Conservation of angular momentum, Angular momentum - Angular momentum in relativistic mechanics, Angular momentum - Angular momentum in quantum mechanics

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A morphism from the vector bundle π1 : E1 → X1 to the vector bundle π2 : E2 → X2 is given by a pair of continuous maps f : E1 → E2 and g : X1 → X2 such that gπ1 = π2f for every x in X1, the map π1−1({x}) → π2−1({g(x)}) induced by f ...

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Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

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Given a vector bundle π : E → X and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s : U → E with πs = idU. Essentially, a section assigns to every point of U a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a di ...

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Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

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In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.) The Pin group PinV(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group SpinV(K) is the subgroup of elements of Dickson invariant 0 in PinV(K). When the characteristic is not 2, these are the elements of determinant ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

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In this section we assume that V is finite dimensional and the bilinear form of Q is non-singular. The Clifford group Γ is defined to be the set of invertible elements x of the Clifford algebra such that xvα(x)−1 ∈ V for all v in V. This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group cont ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

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Suppose that p+q=2n is even. Then the Clifford algebra Cℓp,q(C) is a matrix algebra, and so has a complex representation of dimension 2n. By restricting to the group Pinp,q(R) we get a complex representation of the Pin group of the same dimension, called the spinor representation. If we restrict this to the spin group Spinp,q(R) then it splits as the sum of two half spin representations (or Weyl representatio ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

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Clifford algebra - Differential geometry. One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

Read more here: » Clifford algebra: Encyclopedia II - Clifford algebra - Applications

In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism. In preadditive categories it makes sense to add and subtract morphisms (the hom-sets actually form abelian groups). In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference: coeq(f, < ...

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Coequalizer, Coequalizer - Definition, Coequalizer - Examples, Coequalizer - Special cases

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