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Casimir Operator
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. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. More generally, Casimir elements can be used to refer to ''any'' element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center Of A Ring
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(''R''); 'Z' stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples * The center of a commutative ring ''R'' is ''R'' itself. * The center of a skew-field is a field. * The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. * Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then . * The center of the universal enveloping algebra of a Lie algebra plays an important role in the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure Constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors. Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vectors allows to compute the product of any elements (just like a matrix allows to compute the action of the linear operator on any vector by providing the action of the operator on basis vectors). Therefore, the structure constants can be used to specify the product operation of the algebra (just like a matrix defines a linear operator). Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra. Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used when discussing Lie alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PBW Theorem , a prosopographical database project
* Poincaré-Birkhoff-Witt theorem, a result in mathematics
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PBW may refer to: * Philadelphia-Baltimore-Washington Stock Exchange * Peanut Butter Wolf, American hip hop record producer * Proton beam writing, a lithography process * Play by Web, Play-by-post role-playing game * Prosopography of the Byzantine World The Prosopography of the Byzantine World (PBW) is a project to create a prosopographical database of individuals named in textual sources in the Byzantine Empire and surrounding areas in the period from 642 to 1265. The project is a collaboration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space Basis
In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference. Definition A basis of a vector space over a field (such as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Algebra
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. For example, if ''G'' is \mathrm(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal property: for every linear map from to a commutative algebra , there is a unique algebra homomorphism such that , where is the inclusion map of in . If is a basis of , the symmetric algebra can be identified, through a canonical isomorphism, to the polynomial ring , where the elements of are considered as indeterminates. Therefore, the symmetric algebra over can be viewed as a "coordinate free" polynomial ring over . The symmetric algebra can be built as the quotient of the tensor algebra by the two-sided ideal generated by the elements of the form . All these definitions and properties extend naturally to the case where is a module (not necessarily a free one) over a commutative ring. Construction From tensor algebra It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms ''homogeneous polynomial'' and ''form'' are sometimes considered as synonymous. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Theory
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. It therefore forms a branch of operator theory and functional analysis. The theory is named in honour of Erik Ivar Fredholm. Fredholm equation of the first kind Much of Fredholm theory concerns itself with the following integral equation for ''f'' when ''g'' and ''K'' are given: :g(x)=\int_a^b K(x,y) f(y)\,dy. This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is asked to solve the differential equation :Lg(x)=f(x) where the function is given and is unknown. Here, stands for a linear differential operator. For example, one might take to be an elliptic op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-differential Operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space. History The study of pseudo-differential operators began in the mid 1960s with the work of Kohn, Nirenberg, Hörmander, Unterberger and Bokobza. They played an influential role in the second proof of the Atiyah–Singer index theorem via K-theory. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of pseudo-differential operators. Motivation Linear differential operators with constant coefficients Consider a linear differential operator with constant coefficients, : P(D) := \sum_\alpha a_\alpha \, D^\alpha which acts on smooth functions u with compact support in R''n''. This operator can be writt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian Operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
