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81 relations: Addison-Wesley, ArXiv, Élie Cartan, Bilinear form, Canonical commutation relation, Canonical coordinates, Canonical transformation, Center (group theory), Characteristic (algebra), Classical group, Compact group, Compact space, Complex number, Complexification, Conjugate transpose, Connected space, Determinant, Diagonal matrix, Diffeomorphism, Encyclopedia of Mathematics, Exponential map (Lie theory), Field (mathematics), Fundamental group, Graduate Texts in Mathematics, Group (mathematics), Group action, Group isomorphism, Hamiltonian (quantum mechanics), Hamiltonian matrix, Hamiltonian mechanics, Heisenberg picture, Hermann Weyl, Hilbert space, If and only if, Integer, Isomorphism, Θ10, Lie algebra, Lie group, Linear map, Manifold, Mathematics, MathWorld, Matrix (mathematics), Matrix multiplication, Metaplectic group, Metric signature, Nondegenerate form, Orthogonal group, Phase space, ..., Positive-definite matrix, Projective unitary group, Quantum state, Quaternion, Real form (Lie theory), Real number, Semisimple Lie algebra, Sesquilinear form, Simple Lie group, Simply connected space, Skew-Hermitian matrix, Skew-symmetric matrix, Special linear group, Special unitary group, Split Lie algebra, Springer Science+Business Media, Stack Exchange, Subgroup, Surjective function, Symmetric matrix, Symplectic geometry, Symplectic manifold, Symplectic matrix, Symplectic representation, Symplectic vector space, Symplectomorphism, Tangent space, Transpose, Unitary group, Vector space, 3-sphere. Expand index (31 more) »
Addison-Wesley
Addison-Wesley is a publisher of textbooks and computer literature.
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ArXiv
arXiv (pronounced "archive") is a repository of electronic preprints (known as e-prints) approved for publication after moderation, that consists of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance, which can be accessed online.
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Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Bilinear form
In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.
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Canonical commutation relation
In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).
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Canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time.
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Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations.
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Center (group theory)
In abstract algebra, the center of a group,, is the set of elements that commute with every element of.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Classical group
In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.
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Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Complexification
In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers.
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Conjugate transpose
In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry.
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Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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Encyclopedia of Mathematics
The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.
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Exponential map (Lie theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Fundamental group
In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Hamiltonian (quantum mechanics)
In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases.
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Hamiltonian matrix
In mathematics, a Hamiltonian matrix is a -by- matrix such that is symmetric, where is the skew-symmetric matrix \begin 0 & I_n \\ -I_n & 0 \\ \end and is the -by- identity matrix.
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Hamiltonian mechanics
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
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Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.
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Hermann Weyl
Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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MathWorld
MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix multiplication
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
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Metaplectic group
In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n.
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Metric signature
The signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis.
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Nondegenerate form
In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if.
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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.
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Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
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Positive-definite matrix
In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.
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Projective unitary group
In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center,, embedded as scalars.
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Quantum state
In quantum physics, quantum state refers to the state of an isolated quantum system.
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Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers.
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Real form (Lie theory)
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.
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Sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space.
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Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.
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Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
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Skew-Hermitian matrix
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or antihermitian if its conjugate transpose is equal to the original matrix, with all the entries being of opposite sign.
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Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.
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Special linear group
In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.
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Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
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Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak is a splitting Cartan subalgebra, where "splitting" means that for all x \in \mathfrak, \operatorname_ x is triangularizable.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stack Exchange
Stack Exchange is a network of question-and-answer (Q&A) websites on topics in varied fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process.
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
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Symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
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Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.
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Symplectic matrix
In mathematics, a symplectic matrix is a 2n×2n matrix M with real entries that satisfies the condition where MT denotes the transpose of M and Ω is a fixed 2n×2n nonsingular, skew-symmetric matrix.
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Symplectic representation
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form where F is the field of scalars.
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Symplectic vector space
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form.
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Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.
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Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.
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Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).
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Unitary group
In mathematics, the unitary group of degree n, denoted U(n), is the group of unitary matrices, with the group operation of matrix multiplication.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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3-sphere
In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.
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References
[1] https://en.wikipedia.org/wiki/Symplectic_group
