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Simple Lie Group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, \mathbb, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "building blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(''n'', \mathbb) of ''n'' by ''n'' matrices with determinant equal to 1 is simple for all odd ''n'' > 1, when it is isomorphic to the projective special linear group. The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Lie Group
In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb 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. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph '' The Classical Groups''. The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
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. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Group
In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, \mathbb, and that of the unit-magnitude complex numbers, Circle group, U(1) (the unit circle), simple Lie groups give the atomic "building blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(''n'', \mathbb) of ''n'' by ''n'' matrices with determinant equal to 1 is simple for all odd ''n'' > 1, when it is isomorphic to the projective special linear group. The first classification of simple Lie groups was by Wilhelm Killing, and this work was la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Linear Group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over \R (the set of real numbers) is the group of n\times n invertible matrices of real numbers, and is denoted by \operatorname_n(\R) or \operatorname(n,\R). More generally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Lie Algebra
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra \mathfrak, its conjugate \overline is a complex Lie algebra with the same underlying real vector space but with i = \sqrt acting as -i instead. As a real Lie algebra, a complex Lie algebra \mathfrak is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers). Real form Given a complex Lie algebra \mathfrak, a real Lie algebra \mathfrak_0 is said to be a real form of \mathfrak if the complexification \mathfrak_0 \otimes_\mathbb is isomorphic to \mathfrak. A real form \mathfrak_0 is abelian (resp. nilpotent, solvable, semisimple) if and only if \mathfrak is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form \mathfrak_0 is simple if and only if either \mathfrak is simple or \mathfrak is of the form \mathfrak \times \overli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are Holomorphic function, holomorphic. The term "complex manifold" is variously used to mean a complex manifold in the sense above (which can be specified as an ''integrable'' complex manifold) or an almost complex manifold, ''almost'' complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth manifold, smooth and complex manifolds have very different flavors: compact space, compact complex manifolds are much closer to algebraic variety, algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be Embedding, embedded as a smooth subma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Lie Group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mathbb), the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group \mathbb C^*). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie algebra. Examples *A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way. *A connected compact complex Lie group ''A'' of dimension ''g'' is of the form \mathbb^g/L, a complex torus, where ''L'' is a discrete subgroup of rank 2g. Indeed, its Lie algebra \mathfrak can be shown to be abelian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exceptional Object
Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series. These are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values. For example, spin groups in low dimensions are isomorphic to other classical Lie groups. Regular polytopes The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular ''n''-gons for ''n'' ≥ 3. In every dimension above 2, one can find a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Simple Lie Groups
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, \mathbb, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "building blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(''n'', \mathbb) of ''n'' by ''n'' matrices with determinant equal to 1 is simple for all odd ''n'' > 1, when it is isomorphic to the projective special linear group. The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erlangen Program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is named after the University Erlangen-Nürnberg, where Klein worked. By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: :* Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more restrictive than affine geometry, which in turn is more restrictive than projective geometry. :* Klein proposed that group theory, a branch of mathematics that uses algebraic methods to abstract the idea of symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the theory of equations in the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
