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Braided Vector Space
In mathematics, a braided vector space\;V is a vector space together with an additional structure map \tau symbolizing interchanging of two vector tensor copies: :\tau:\; V\otimes V\longrightarrow V\otimes V such that the Yang–Baxter equation is fulfilled. Hence drawing tensor diagrams with \tau an overcrossing the corresponding composed morphism is unchanged when a Reidemeister move is applied to the tensor diagram and thus they present a representation of the braid group. As first example, every vector space is braided via the trivial braiding (simply flipping). A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a V-base x_i we have :\tau(x_i\otimes x_j)=q_(x_j\otimes x_i) A good source for braided vector spaces entire braided monoidal categories with braidings between any objects \tau_, most importantly the modules over quasitriangular Hopf algebras and Yetter–Drinfeld modules over finite g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, ideals.) Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of Characteristic (algebra), characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form \kappa(x, y) = \operatorname(\operatorname(x)\operatorname(y)) is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable Lie algebra, solvable ideals; * the Radical of a Lie algebra, radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré–Birkhoff–Witt Theorem
In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms ''PBW type theorem'' and ''PBW theorem'' may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups. Statement of the theorem Recall that any vector space ''V'' over a field has a basis; this is a set ''S'' such that any element of ''V'' is a unique (finite) linear combination of elements of ''S''. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases of which the elements are totally ordered by some relation which we denote ≤. If ''L'' is a Lie algebra over a field K, let ''h'' denote the canonical K-linear map from ''L'' into the universal envelop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), 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 graph, directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected graph, undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System
In mathematics, a root system is a configuration of vector space, 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 Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nichols Algebra
Nichols may refer to: People *Nichols (surname) * Nichol, a surname Places Canada * Nichols Islands, Nunavut United States * Nichols, California, an unincorporated community * Nichols Canyon, Los Angeles, California * Nichols, Connecticut * Nichols Farms Historic District, a village within Trumbull, Connecticut. * Nichols, Iowa * Nichols (village), New York * Nichols (town), New York * Nichols, South Carolina, a town * Nichols, Wisconsin, a village * Nichols Shore Acres, Wisconsin, an unincorporated community Military * Nichols Field, a former U.S. air base in the Philippines * Nichols' Regiment of Militia, a U.S. Revolutionary War unit * Camp Nichols, a historic fortification in Cimarron County, Oklahoma Organisations Education * Nichols College, in Dudley, Massachusetts * Nichols School, in Buffalo, New York * Nichols Hall, Kansas State University * Nichols House (Baltimore, Maryland), home of the president of Johns Hopkins University * Nichols Arboretum, Ann Arb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (that is, if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the '' commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many group theorists define the commutator as : . Using the first definition, this can be expressed as . Identities (group theory) Commutator identities are an important tool in group th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braided Hopf Algebra
In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra ''H'', particularly the Nichols algebra of a braided vector space in that category. ''The notion should not be confused with quasitriangular Hopf algebra.'' Definition Let ''H'' be a Hopf algebra over a field ''k'', and assume that the antipode of ''H'' is bijective. A Yetter–Drinfeld module ''R'' over ''H'' is called a braided bialgebra in the Yetter–Drinfeld category ^H_H\mathcal if * (R,\cdot ,\eta ) is a unital associative algebra, where the multiplication map \cdot :R\times R\to R and the unit \eta :k\to R are maps of Yetter–Drinfeld modules, * (R,\Delta ,\varepsilon ) is a coassociative coalgebra with counit \varepsilon , and both \Delta and \varepsilon are maps of Yetter–Drinfeld modules, * the maps \Delta :R\to R\otimes R and \varepsilon :R\to k are algebra maps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yetter–Drinfeld Category
In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms. Definition Let ''H'' be a Hopf algebra over a field ''k''. Let \Delta denote the coproduct and ''S'' the antipode of ''H''. Let ''V'' be a vector space over ''k''. Then ''V'' is called a (left left) Yetter–Drinfeld module over ''H'' if * (V,\boldsymbol) is a left ''H''- module, where \boldsymbol: H\otimes V\to V denotes the left action of ''H'' on ''V'', * (V,\delta\;) is a left ''H''- comodule, where \delta : V\to H\otimes V denotes the left coaction of ''H'' on ''V'', * the maps \boldsymbol and \delta satisfy the compatibility condition :: \delta (h\boldsymbolv)=h_v_S(h_) \otimes h_\boldsymbolv_ for all h\in H,v\in V, :where, using Sweedler notation, (\Delta \otimes \mathrm)\Delta (h)=h_\otimes h_ \otimes h_ \in H\otimes H\otimes H denotes the twofold coproduct of h\in H , and \delta (v)= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
