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Spinc Group
In spin geometry, a spinᶜ group (or complex spin group) is a Lie group obtained by the spin group through twisting with the first unitary group. C stands for the complex numbers, which are denoted \mathbb. An important application of spinᶜ groups is for spinᶜ structures, which are central for Seiberg–Witten theory. Definition The spin group \operatorname(n) is a double cover of the special orthogonal group \operatorname(n), hence \mathbb_2 acts on it with \operatorname(n)/\Z_2\cong\operatorname(n). Furthermore, \mathbb_2 also acts on the first unitary group \operatorname(1) through the antipodal identification y\sim -y. The ''spinᶜ group'' is then: : \operatorname^\mathrm(n) :=\left( \operatorname(n)\times\operatorname(1) \right)/\mathbb_2 with (x,y)\sim(-x,-y). It is also denoted \operatorname^\mathbb(n). Using the exceptional isomorphism \operatorname(2) \cong\operatorname(1) , one also has \operatorname^\mathrm(n) =\operatorname^2(n) with: : \operatorname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Geometry
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics. An important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields of mathematical research. See also * Contact geometry * Symplectic topology * Spinor * Spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\co ... * Spin manifold Books * * Differential topology Differential geometry {{physics-stub ... [...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] |
Spin Group
In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \operatorname(n) \to \operatorname(n) \to 1. The group multiplication law on the double cover is given by lifting the multiplication on \operatorname(n). As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group. For , Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n''). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −. Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(''n''). A distinct article discusses the spin representations. Use for physics models The spin group is use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Group
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Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigroup Politics * Unitary authority * Unitary state See also * Unital (other) * Unitarianism Unitarianism () is a Nontrinitarianism, nontrinitarian sect of Christianity. Unitarian Christians affirm the wikt:unitary, unitary God in Christianity, nature of God as the singular and unique Creator deity, creator of the universe, believe that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinc Structure
In spin geometry, a spinᶜ structure (or complex spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinᶜ manifolds. C stands for the Complex number, complex numbers, which are denoted \mathbb and appear in the definition of the underlying Spinc group, spinᶜ group. In four dimensions, a spinᶜ structure defines two complex plane bundles, which can be used to describe negative and positive Chirality (physics), chirality of Spinor, spinors, for example in the Dirac equation of relativistic quantum field theory. Another central application is Seiberg–Witten theory, which uses them to study 4-manifold, 4-manifolds. Definition Let M be a n-dimensional orientable manifold. Its tangent bundle TM is described by a classifying map M\rightarrow\operatorname(n) into the Classifying space for SO(n), classifying space \operatorname(n) of the special orthogonal group \operatorname(n). It can factor over the map \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seiberg–Witten Theory
In theoretical physics, Seiberg–Witten theory is an \mathcal = 2 supersymmetric gauge theory with an exact low-energy effective action (for massless degrees of freedom), of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low-energy effective action, the theory is known as \mathcal = 2 supersymmetric Yang–Mills theory, as the field content is a single \mathcal = 2 vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field (in particle theory language) or connection (in geometric language). The theory was studied in detail by Nathan Seiberg and Edward Witten . Seiberg–Witten curves In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic (really, meromorphic) properties and their behavior near the singularities. In gauge theory with \mathcal = 2 extended supersymmetry, the moduli space of vacua is a spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Cover (topology)
Double, The Double or Dubble may refer to: Mathematics and computing * Multiplication by 2 * Double precision, a floating-point representation of numbers that is typically 64 bits in length * A double number of the form x+yj, where j^2=+1 * A 2-tuple, or ordered list of two elements, commonly called an ordered pair, denoted (a,b) * Double (manifold), in topology Food and drink * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doubling cube, in backgammon ** Double, doubling a blackjack bet in a favorable situation ** Double, a bet offered by UK bookmakers which combines two selections * Double, villain in the video game '' Mega Man X4'' * A kart racing game '' Mario Kart: Double Dash'' * An arcade action game ''Double Dragon'' Sports * Double (association football), the act of a winning a division and pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antipodal Point
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center. Given any point on a sphere, its antipodal point is the unique point at greatest distance, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically ( chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exceptional Isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a''''i'' and ''b''''j'' of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general pattern of such isomorphisms.Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions), but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity). These coincidences are at times considered a matter of trivia, but in other respects they can give rise to consequential phenomena, such as exceptional objects. In the following, coincidences are organized according to the structures where they occur. Groups Finite simple groups The exceptional isomorphisms between the series of finite simple groups mostly involve projective special linear groups and alternating groups, and are: * PSL(4) ≅ PSL(5) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinh Group
In spin geometry, a spinʰ group (or quaternionic spin group) is a Lie group obtained by the spin group through twisting with the first symplectic group. H stands for the quaternions, which are denoted \mathbb. An important application of spinʰ groups is for spinʰ structures. Definition The spin group \operatorname(n) is a double cover of the special orthogonal group \operatorname(n), hence \mathbb_2 acts on it with \operatorname(n)/\Z_2\cong\operatorname(n). Furthermore, \mathbb_2 also acts on the first symplectic group \operatorname(1) through the antipodal identification y\sim -y. The ''spinʰ group'' is then:Bär 1999, page 16 : \operatorname^\mathrm(n) :=\left( \operatorname(n)\times\operatorname(1) \right)/\mathbb_2 mit (x,y)\sim(-x,-y). It is also denoted \operatorname^\mathbb(n). Using the exceptional isomorphism \operatorname(3) \cong\operatorname(1) , one also has \operatorname^\mathrm(n) =\operatorname^3(n) with: : \operatorname^k(n) :=\left( \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
