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110 relations: Abelian category, Adjoint functors, Adjoint representation, Albert algebra, Algebra over a field, Alternative algebra, American Mathematical Society, Angular momentum operator, Antisymmetric tensor, Associated graded ring, Associative algebra, Baker–Campbell–Hausdorff formula, Basis (linear algebra), Bol loop, C*-algebra, Casimir element, Category (mathematics), Center (ring theory), Change of variables, Characteristic polynomial, Chevalley basis, Coalgebra, Coleman–Mandula theorem, Compact group, Convolution, Coordinate-free, Determinant, Differential algebra, Differential operator, Direct sum, Distribution (mathematics), Dual space, Elliptic operator, Equivalence class, Exterior algebra, Filtered algebra, Filtration (mathematics), Fredholm theory, Free algebra, Free Lie algebra, Free module, Functor, Gelfand–Naimark theorem, Gerstenhaber algebra, Graduate Studies in Mathematics, Group (mathematics), Group algebra, Harish-Chandra homomorphism, Harish-Chandra isomorphism, Heisenberg group, ..., Homogeneous polynomial, Hopf algebra, Ideal (ring theory), Identity element, Indefinite orthogonal group, Indicator function, Infinitesimal transformation, Isometry group, Isomorphism of categories, Israel Gelfand, Jordan algebra, Killing form, Kronecker coefficient, Laplace operator, Lie algebra, Lie algebra representation, Lie group, Lie superalgebra, Limit (category theory), Malcev algebra, Mathematics, Milnor–Moore theorem, Module (mathematics), Moyal product, Natural transformation, Noncommutative geometry, Orthogonal group, Partial derivative, Poincaré–Birkhoff–Witt theorem, Poisson superalgebra, Polynomial, Pseudo-differential operator, Pseudo-Riemannian manifold, Quadratic algebra, Quantum group, Quasitriangular Hopf algebra, Representation theory, Riemannian manifold, Root system, Rotation group SO(3), Semisimple Lie algebra, Shlomo Sternberg, Shuffle algebra, Stress–energy tensor, Structure constants, Symmetric algebra, Symmetric polynomial, Tannaka–Krein duality, Tensor algebra, Tensor product, Total order, Universal property, Vector field, Vector space, Verma module, Weight (representation theory), Weyl algebra, Yang–Mills theory, 6-j symbol, 9-j symbol. Expand index (60 more) »
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
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Adjoint functors
In mathematics, specifically category theory, adjunction is a possible relationship between two functors.
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Adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
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Albert algebra
In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra.
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Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
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Alternative algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum.
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Antisymmetric tensor
In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.
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Associated graded ring
In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring: Similarly, if M is a left R-module, then the associated graded module is the graded module over \operatorname_I R.
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Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
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Baker–Campbell–Hausdorff formula
In mathematics, the Baker–Campbell–Hausdorff formula is the solution to the equation for possibly noncommutative and in the Lie algebra of a Lie group.
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Basis (linear algebra)
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.
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Bol loop
In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group.
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C*-algebra
C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.
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Casimir element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra.
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Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.
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Center (ring theory)
In algebra, the center of a ring R is the subring consisting of the elements x such that xy.
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Change of variables
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables.
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Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.
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Chevalley basis
In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers.
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Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras.
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Coleman–Mandula theorem
The Coleman–Mandula theorem (named after Sidney Coleman and Jeffrey Mandula) is a no-go theorem in theoretical physics.
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Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
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Convolution
In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.
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Coordinate-free
A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system.
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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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Differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule.
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Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
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Direct sum
The direct sum is an operation from abstract algebra, a branch of mathematics.
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Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.
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Dual space
In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.
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Elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator.
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Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra.
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Filtration (mathematics)
In mathematics, a filtration \mathcal is an indexed set S_i of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic object S_i gaining in complexity with time.
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Fredholm theory
In mathematics, Fredholm theory is a theory of integral equations.
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Free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables.
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Free Lie algebra
In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations.
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Free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.
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Functor
In mathematics, a functor is a map between categories.
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Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space.
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Gerstenhaber algebra
In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra.
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Graduate Studies in Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).
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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 algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group.
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Harish-Chandra homomorphism
In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra.
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Harish-Chandra isomorphism
In mathematics, the Harish-Chandra isomorphism, introduced by, is an isomorphism of commutative rings constructed in the theory of Lie algebras.
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Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.
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Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.
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Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.
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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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Indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature, where.
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Indicator function
In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.
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Infinitesimal transformation
In mathematics, an infinitesimal transformation is a limiting form of small transformation.
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Isometry group
In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.
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Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F: C → D and G: D → C which are mutually inverse to each other, i.e. FG.
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Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (ישראל געלפֿאַנד, Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent Soviet mathematician.
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Jordan algebra
In abstract algebra, a Jordan algebra is an nonassociative algebra over a field whose multiplication satisfies the following axioms.
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Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
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Kronecker coefficient
In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (.
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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
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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 algebra representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.
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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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Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading.
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Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.
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Malcev algebra
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that and satisfies the Malcev identity They were first defined by Anatoly Maltsev (1955).
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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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Milnor–Moore theorem
In algebra, the Milnor–Moore theorem, introduced in, states: given a connected graded cocommutative Hopf algebra A over a field of characteristic zero with \dim A_n, the natural Hopf algebra homomorphism from the universal enveloping algebra of the "graded" Lie algebra P(A) of primitive elements of A to A is an isomorphism.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Moyal product
In mathematics, the Moyal product (also called the star product or Weyl–Groenewold product) is perhaps the best-known example of a phase-space star product.
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Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.
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Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).
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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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Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
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Poincaré–Birkhoff–Witt theorem
In mathematics, more specifically in abstract algebra, in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra.
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Poisson superalgebra
In mathematics, a Poisson superalgebra is a Z2-graded generalization of a Poisson algebra.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator.
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Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.
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Quadratic algebra
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2.
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Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebras with additional structure.
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Quasitriangular Hopf algebra
In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H \otimes H such that where R_.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
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Rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.
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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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Shlomo Sternberg
Shlomo Zvi Sternberg (born 1936), is an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.
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Shuffle algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of interlacing them.
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Stress–energy tensor
The stress–energy tensor (sometimes stress–energy–momentum tensor or energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics.
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Structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination.
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Symmetric algebra
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative algebra over K containing V. It corresponds to polynomials with indeterminates in V, without choosing coordinates.
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Symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial.
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Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations.
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Tensor 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 rank) with multiplication being the tensor product.
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.
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Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
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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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Verma module
Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
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Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.
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Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.
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Yang–Mills theory
Yang–Mills theory is a gauge theory based on the SU(''N'') group, or more generally any compact, reductive Lie algebra.
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6-j symbol
Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965.
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9-j symbol
In physics, Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937.
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References
[1] https://en.wikipedia.org/wiki/Universal_enveloping_algebra
