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Jacobi Identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jacob Jacobi. The cross product a\times b and the Lie bracket operation ,b/math> both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket. Definition Let + and \times be two binary operations, and let 0 be the neutral element for +. The is :x \times (y \times z) \ +\ y \times (z \times x) \ +\ z \times (x \times y)\ =\ 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three Subgroups Lemma
In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity. Notation In what follows, the following notation will be employed: * If ''H'' and ''K'' are subgroups of a group ''G'', the commutator of ''H'' and ''K'', denoted by 'H'', ''K'' is defined as the subgroup of ''G'' generated by commutators between elements in the two subgroups. If ''L'' is a third subgroup, the convention that 'H'',''K'',''L''= ''H'',''K''''L''] will be followed. * If ''x'' and ''y'' are elements of a group ''G'', the Conjugate (group theory), conjugate of ''x'' by ''y'' will be denoted by x^. * If ''H'' is a subgroup of a group ''G'', then the centralizer of ''H'' in ''G'' will be denoted by CG(''H''). Statement Let ''X'', ''Y'' and ''Z'' be subgroups of a group ''G'', and assume : ,Y,Z1 and ,Z,X1. Then ,X,Y1. More generally, for a normal subgroup N of G, if ,Y,Zsubseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Jacobi Identity
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ''even'' elements of the superalgebra correspond to bosons and ''odd'' elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around). Definition Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or ''superalgebra'', over a commutative ring (typically R or C) whose product ·, · called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading): Super skew-symmetry: : ,y-(-1)^ ,x\ The super Jacobi identity: :(-1)^ ,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^ ,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_ ... [...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 used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting product is bilinear 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 algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles. Recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket or commutator. Definition Given a set of basis vectors \ for the underlying vector space of the algebra, the structure constants or structure coefficients c_^ express the multiplication \cdot of pairs of vectors as a linear combination: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Homomorphism
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Algebra
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module ''L'' over a commutative ring ''R'' with a bilinear product _ , _ satisfying the Leibniz identity : a,bc] = ,c.html"_;"title=",,[b,c+__a,cb.html"_;"title=",c">,[b,c+__a,c.html"_;"title=",c.html"_;"title=",[b,c">,[b,c+__a,c">,c.html"_;"title=",[b,c">,[b,c+__a,cb">,c">,[b,c+__a,c.html"_;"title=",c.html"_;"title=",[b,c">,[b,c+__a,c">,c.html"_;"title=",[b,c">,[b,c+__a,cb_\,_ In_other_words,_right_multiplication_by_any_element_''c''_is_a_derivation_(abstract_algebra).html" "title=",c">,[b,c+__a,cb.html" ;"title=",c">,[b,c+__a,c.html" ;"title=",c.html" ;"title=",[b,c">,[b,c+ a,c">,c.html" ;"title=",[b,c">,[b,c+ a,cb">,c">,[b,c+__a,c.html" ;"title=",c.html" ;"title=",[b,c">,[b,c+ a,c">,c.html" ;"title=",[b,c">,[b,c+ a,cb \, In other words, right multiplication by any element ''c'' is a derivation (abstract algebra)">derivation. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation (abstract Algebra)
In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Leibniz's law: : D(ab) = a D(b) + D(a) b. More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by . Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R''n''. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the ... [...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 GL(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 (conjugation) :\Psi_g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Ring
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
