|
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Sophus Lie was born on 17 December 1842 in the small town of Nordfjordeid. He was the youngest of six children born to a Lutheran pastor named Johann Herman Lie, and his wife who came from a well-known Trondheim family. He had his primary education in the south-eastern coast of Moss, before attending high school at Oslo (known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to his poor eyesight. It was then that he decided to enrol at the University of Christiania. Sophus Lie's first mathematical work, ''Repräsentation der Imaginären der Plangeometrie'', was published in 1869 by the Academy of Sciences in Christiania and also by ''Crelle's Journal''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nordfjordeid
Nordfjordeid is the administrative centre of the municipality of Stad in Vestland county, western Norway. It is located at the end of the Eidsfjorden, an arm off of the main Nordfjorden, west of the large lake Hornindalsvatnet. The village of Stårheim is located about to the west, the village of Mogrenda is about to the east, and the village of Lote is about to the southeast. The town is a commercial hub for the municipality and greater Nordfjord region. It includes several industries, such as timber, agriculture, electronics and commerce. It is the shopping hub for surrounding villages and towns, due to the shopping center and commerce streets. There are 5 primary schools, one lower secondary school and an upper secondary school connected to the Opera House in the town. Eid Church is also located in the town and it is the main church for the municipality. The European route E39 highway passes by Nordfjordeid on its way from Bergen to Ålesund. Norwegian National Road 15 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'integrable system, complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem (differential topology), Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie–Kolchin Theorem
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if ''G'' is a connected and solvable linear algebraic group defined over an algebraically closed field and :\rho\colon G \to GL(V) a representation on a nonzero finite-dimensional vector space ''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that : \rho(G)(L) = L. That is, ρ(''G'') has an invariant line ''L'', on which ''G'' therefore acts through a one-dimensional representation. This is equivalent to the statement that ''V'' contains a nonzero vector ''v'' that is a common (simultaneous) eigenvector for all \rho(g), \,\, g \in G . It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group ''G'' has dimension one. In fact, this is another way to state the Lie–Kolchin theorem. The result for Lie alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie's Third Theorem
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence. Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a ''local'' Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem. Historical notes The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie's Theorem
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, then there's a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0 of invariant subspaces of \pi(\mathfrak) with \operatorname V_i = i, meaning that \pi(X)(V_i) \subseteq V_i for each X \in \mathfrak and ''i''. Put in another way, the theorem says there is a basis for ''V'' such that all linear transformations in \pi(\mathfrak) are represented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices generate an abelian Lie algebra, which is a fortiori solvable. A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra (see #Consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squeeze Mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , the mapping :(x, y) \mapsto (ax, y/a) is the ''squeeze mapping'' with parameter . Since :\ is a hyperbola, if and , then and the points of the image of the squeeze mapping are on the same hyperbola as is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with ''circular rotations'', which preserve circles. Logarithm and hyperbolic angle The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Sphere Geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius. The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres). To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Product Formula
In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary ''m'' × ''m'' real or complex matrices ''A'' and ''B'', :e^ = \lim_ (e^e^)^n, where ''e''''A'' denotes the matrix exponential of ''A''. The Lie–Trotter product formula and the Trotter–Kato theorem extend this to certain unbounded linear operators ''A'' and ''B''. This formula is an analogue of the classical exponential law :e^ = e^x e^y \, which holds for all real or complex numbers ''x'' and ''y''. If ''x'' and ''y'' are replaced with matrices ''A'' and ''B'', and the exponential replaced with a matrix exponential, it is usually necessary for ''A'' and ''B'' to commute for the law to still hold. However, the Lie product formula holds for all matrices ''A'' and ''B'', even ones which do not commute. The Lie product formula is conceptually related to the Baker–Campbell–Hausdorff formula, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bracket
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] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, 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 [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), 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 space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
