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Highest-weight Category
In the mathematical field of representation theory, a highest-weight category is a ''k''-linear category C (here ''k'' is a field) that *is ''locally artinian'' *has enough injectives *satisfies ::B\cap\left(\bigcup_\alpha A_\alpha\right)=\bigcup_\alpha\left(B\cap A_\alpha\right) :for all subobjects ''B'' and each family of subobjects of each object ''X'' and such that there is a locally finite poset Λ (whose elements are called the weights of C) that satisfies the following conditions: * The poset Λ indexes an exhaustive set of non-isomorphic simple objects in C. * Λ also indexes a collection of objects of objects of C such that there exist embeddings ''S''(''λ'') → ''A''(''λ'') such that all composition factors ''S''(''μ'') of ''A''(''λ'')/''S''(''λ'') satisfy ''μ'' :is also finite. *Each ''S''(''λ'') has an injective envelope ''I''(''λ'') in C equipped with an increasing filtration ::0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\la ... [...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] |
Filtration (mathematics)
In mathematics, a filtration \mathcal is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S_j. If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S_i gaining in complexity with time. Hence, a process that is adapted to a filtration \mathcal is also called non-anticipating, because it cannot "see into the future". Sometimes, as in a filtered algebra, there is instead the requirement that the S_i be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication) that satisfy only S_i \cdot S_j \subseteq S_, where the index set is the natural numbers; this is by analogy with a graded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter De Gruyter
Walter de Gruyter GmbH, known as De Gruyter (), is a German scholarly publishing house specializing in academic literature. History The roots of the company go back to 1749 when Frederick the Great granted the Königliche Realschule in Berlin the royal privilege to open a bookstore and "to publish good and useful books". In 1800, the store was taken over by Georg Reimer (1776–1842), operating as the ''Reimer'sche Buchhandlung'' from 1817, while the school’s press eventually became the ''Georg Reimer Verlag''. From 1816, Reimer used the representative Sacken'sche Palace on Berlin's Wilhelmstraße for his family and the publishing house, whereby the wings contained his print shop and press. The building became a meeting point for Berlin salon life and later served as the official residence of the president of Germany. Born in Ruhrort in 1862, Walter de Gruyter took a position with Reimer Verlag in 1894. By 1897, at the age of 35, he had become sole proprietor of the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin, Germany
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constituent states, Berlin is surrounded by the State of Brandenburg and contiguous with Potsdam, Brandenburg's capital. Berlin's urban area, which has a population of around 4.5 million, is the second most populous urban area in Germany after the Ruhr. The Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions. Berlin straddles the banks of the Spree, which flows into the Havel (a tributary of the Elbe) in the western borough of Spandau. Among the city's main topographical features are the many lakes in the western and southeastern boroughs formed by the Spree, Havel and Dahme, the largest of which is Lake Müggelsee. Due to its locat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–1928 Kurt Hens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Algebra
In abstract algebra, a cellular algebra is a finite-dimensional associative algebra ''A'' with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of ''A''. History The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer. However, the terminology had previously been used by Weisfeiler and Lehman in the Soviet Union in the 1960s, to describe what are also known as coherent algebras. Definitions Let R be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also A be an R-algebra. The concrete definition A ''cell datum'' for A is a tuple (\Lambda,i,M,C) consisting of :* A finite partially ordered set \Lambda. :* A R-linear anti-automorphism i:A\to A with i^2 = \operatorname_A. :* For every \lambda\in\Lambda a non-empty finite set M(\lambda) of indices. :* An injective map :::C: \dot_ M(\lambda)\times M(\lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditary Ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring ''R'' is called hereditary if all submodules of projective modules over ''R'' are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring ''R'', the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective ''left'' ''R''-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated) submodules of projective ''right'' ''R''-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary and vice versa. Equivalent definitions * The ring ''R'' is left (semi-)hereditary if and only if all ( finitely generated) left ideals of ''R'' are projective modules. * The ring ''R'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see ''Semisimple representation''. Definition A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module ''M'', the following are equivalent: # ''M'' is semisimple; i.e., a dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highest-weight Module
In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Envelope
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . Definition A module ''E'' is called the injective hull of a module ''M'', if ''E'' is an essential extension of ''M'', and ''E'' is injective. Here, the base ring is a ring with unity, though possibly non-commutative. Examples * An injective module is its own injective hull. * The injective hull of an integral domain is its field of fractions . * The injective hull of a cyclic ''p''-group (as Z-module) is a Prüfer group . * The injective hull of ''R''/rad(''R'') is Hom''k''(''R'',''k''), where ''R'' is a finite-dimensional ''k''-algebra with Jacobson radical rad(''R'') . * A simple module is necessarily the socle of its injective hull. * The injective hull of the residue field of a discrete valuation ring (R,\mathfrak,k) where \mathfrak = x\c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
