|
Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. He became a professor at the seminary college Collegium Hosianum in Braunsberg (now Braniewo). He took holy orders in order to take his teaching position. He became rector of the college and chair of the town council. As a professor and administrator Killing was widely liked and respected. Finally, in 1892 he became professor at the University of Münster. In 1886, Killing and his spouse entered the Third Order of Franciscans. Work In 1878 Killing wrote on space forms in terms of non-Euclidean geometry in Crelle's Journal, which he further developed in 1880 as well as in 1885. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burbach, North Rhine-Westphalia
Burbach is a municipality in the Siegen-Wittgenstein district, in North Rhine-Westphalia, Germany. Geography Burbach is located in Siegen-Wittgenstein district on the river Heller, about 15 km south of Siegen. Burbach also is a community in the southern part of the state Northrhine Westphalia. Constituent divisions The community of Burbach consists of the following subdivisions: Burbach, Gilsbach, Holzhausen, Lippe, Lützeln, Niederdresselndorf, Oberdresselndorf, Wahlbach, Würgendorf. Politics Municipal council The council's 32 seats are apportioned thus, in accordance with municipal elections held on 25 May 2014: * CDU 19 seats *SPD 10 seats * Greens 3 seats Coat of arms Burbach's civic coat of arms might heraldically be described thus: Party per pale, dexter in azure spangled with billets Or a two-tailed lion rampant Or armed and langued gules, sinister in Or three lozenges sable arranged vertically. The lion stands for the princely House of Nassau-Siegen. The thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braniewo
Braniewo () (german: Braunsberg in Ostpreußen, la, Brunsberga, Old Prussian: ''Brus'', lt, Prūsa), is a town in northern Poland, in Warmia, in the Warmian-Masurian Voivodeship, with a population of 16,907 as of June 2021. It is the capital of Braniewo County. Braniewo is the second biggest city of Warmia after Olsztyn and one of the historical centers of the region. Location Braniewo lies on the Pasłęka River about 5 km from the Vistula Lagoon, about 35 km northeast of Elbląg and southwest of Kaliningrad ( pl, Królewiec). The Polish border with Russia's Kaliningrad Oblast lies 6 km north, and may be reached from Braniewo via National road 54. History Middle Ages According to the German geographer Johann Friedrich Goldbeck (1748-1812), the town originally was named Brunsberg after Bruno von Schauenburg (1205–1281), bishop of Olomouc in Moravia, who accompanied King Ottokar II of Bohemia in 1254 and 1267 when the latter participated in the crusade of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exceptional Lie Algebra
In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: \mathfrak_2, \mathfrak_4, \mathfrak_6, \mathfrak_7, \mathfrak_8; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are: * G2 : * F4 : * E6 : * E7 : * E8 : In contrast, simple Lie algebras that are not exceptional are called classical Lie algebra The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A_n , B_n , C_n and D_n , where for \mathfrak(n) the general linear Lie algebra and I_n the n \times n identity matrix: ...s (there are infinitely many of them). Construction There is no simple universally accepted way to construct exceptional Lie algebras; in fact, they were discovered only in the process of the classification program. Here are some constructions: *§ 22.1-2 of give a detailed construction of \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Linear span, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century. His son Henri Cartan was an influential mathematician working in algebraic topology. Life Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel". Élie had an elder sister Je ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Matrix
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. Lie algebras A (symmetrizable) generalized Cartan matrix is a square matrix A = (a_) with integral entries such that # For diagonal entries, a_ = 2 . # For non-diagonal entries, a_ \leq 0 . # a_ = 0 if and only if a_ = 0 # A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix. For example, the Cartan matrix for ''G''2 can be decomposed as such: : \begin 2 & -3 \\ -1 & 2 \end = \begin 3&0\\ 0&1 \end\begin \frac & -1 \\ -1 & 2 \end. The third condition is not independent but is really a consequence of the first and fourth conditions. We can always choose a ''D'' with positive diagonal entries. In that case, if ''S'' in the ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra \mathfrak over a field of characteristic 0 . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements ''x'' such that the adjoint endomorphism \operatorname(x) : \mathfrak \to \mathfrak is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231 In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Lie Algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of simple Lie algebras is called a semisimple Lie algebra. A simple Lie group is a connected Lie group whose Lie algebra is simple. Complex simple Lie algebras A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: \mathfrak_n \mathbb, \mathfrak_n \mathbb, \mathfrak_ \mathbb (classical Lie algebras) or one of the five exceptional Lie algebras. To each finite-dimensional complex semisimple Lie algebra \mathfrak, there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed (or not jointed) by a number of lines depending on the angles between the simple roots and the arrows are put to indicate whether the roots are longer or shorter. The Dynk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Lorentz Transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v, representing a velocity confined to the -direction, is expressed as \begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end where and are the coordinates of an event in two frames with the origins coinciding at 0, where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and \gamma = \left ( \sqrt\right )^ is the Lorentz factor. When speed is much smaller than , the Lorentz factor is negligibly different from 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid Model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloid in (''n''+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and ''m''-planes are represented by the intersections of (''m''+1)-planes passing through the origin in Minkowski space with ''S''+ or by wedge products of ''m'' vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the ''n''-sphere is embedded in (''n''+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of ''S''+: the Beltrami–Klein model is the projection of ''S''+ through the origin onto a plane perpendic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
