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Automorphism Of A Lie Algebra
In abstract algebra, an automorphism of a Lie algebra \mathfrak g is an isomorphism from \mathfrak g to itself, that is, a linear map preserving the Lie bracket. The set of automorphisms of \mathfrak are denoted \text(\mathfrak), the automorphism group of \mathfrak. Inner and outer automorphisms The subgroup of \operatorname(\mathfrak g) generated using the adjoint action e^, x \in \mathfrak g is called the inner automorphism group of \mathfrak g. The group is denoted \operatorname^0(\mathfrak). These form a normal subgroup in the group of automorphisms, and the quotient \operatorname(\mathfrak)/\operatorname^0(\mathfrak) is known as the outer automorphism group. Diagram automorphisms It is known that the outer automorphism group for a simple Lie algebra \mathfrak is isomorphic to the group of diagram automorphisms for the corresponding Dynkin diagram in the classification of Lie algebras. The only algebras with non-trivial outer automorphism group are therefore A_n (n \geq 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Subalgebra
In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Borel subalgebra associated to a flag Let \mathfrak g = \mathfrak(V) be the Lie algebra of the endomorphisms of a finite-dimensional vector space ''V'' over the complex numbers. Then to specify a Borel subalgebra of \mathfrak g amounts to specify a flag of ''V''; given a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0, the subspace \mathfrak b = \ is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of ''V''. Borel subalgebra relative to a base of a root system Let \mathfrak g be a complex semisimple Lie algebra, \mathfrak h a Cartan subalgebra and ''R'' the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Lie Algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consists of all linear combinations of Lie brackets of pairs of elements of \mathfrak. The ''derived series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq \mathfrak,\mathfrak mathfrak,\mathfrak \geq [ \mathfrak,\mathfrak mathfrak,\mathfrak, \mathfrak,\mathfrak mathfrak,\mathfrak] \geq ... If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra_is_a_fortiori.html" ;"title="mathfrak,\mathfrak ... is a fortiori">mathfrak,\mathfrak ... is a fortiori solvable but the converse is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group-Lie Algebra Correspondence
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be interpreted as deliberately false statements or misleading statements. Lies may also serve a variety of instrumental, interpersonal, or psychological functions for the individuals who use them. Generally, the term "lie" carries a negative connotation, and depending on the context a person who communicates a lie may be subject to social, legal, religious, or criminal sanctions; for instance, perjury, or the act of lying under oath, can result in criminal and civil charges being pressed against the perjurer. Although people in many cultures believe that deception can be detected by observing nonverbal behaviors (e.g. not making eye contact, fidgeting, stuttering) research indicates that people overestimate both the significance of such cues and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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(h)= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negligi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of "mathematical ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named. Classification of semisimple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph is a permutation of the vertex set , such that the pair of vertices form an edge if and only if the pair also form an edge. That is, it is a graph isomorphism from to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Computational complexity Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
