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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 map ... [...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] |
Hopf Algebra
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braided Monoidal Category
In mathematics, a ''commutativity constraint'' \gamma on a monoidal category ''\mathcal'' is a choice of isomorphism \gamma_ : A\otimes B \rightarrow B\otimes A for each pair of objects ''A'' and ''B'' which form a "natural family." In particular, to have a commutativity constraint, one must have A \otimes B \cong B \otimes A for all pairs of objects A,B \in \mathcal. A braided monoidal category is a monoidal category \mathcal equipped with a braiding—that is, a commutativity constraint \gamma that satisfies axioms including the hexagon identities defined below. The term ''braided'' references the fact that the braid group plays an important role in the theory of braided monoidal categories. Partly for this reason, braided monoidal categories and other topics are related in the theory of knot invariants. Alternatively, a braided monoidal category can be seen as a tricategory with one 0-cell and one 1-cell. Braided monoidal categories were introduced by André Joyal an ... [...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)=v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nichols Algebra
In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by \mathfrak(V) and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional truncations. Nichols algebras can immediately be used to write down new such quantum groups by using the Radford biproduct. The classification of all such Nichols algebras and even all associated quantum groups (see Application) has been progressing rapidly, although still much is open: The case of an abelian group was solved in 2005, but otherwise this phenomenon seems to be very rare, with a handful examples known and powerful negation criteria established (see below). See also this List of finite-dimensional Nichols algebras. The finite-dimensional theory is greatly governed by a theory of root systems and Dynkin diagrams, strikingl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasitriangular Hopf Algebra
In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the coproduct on ''H'', and the linear map T : H \otimes H \to H \otimes H is given by T(x \otimes y) = y \otimes x, :*(\Delta \otimes 1)(R) = R_ \ R_, :*(1 \otimes \Delta)(R) = R_ \ R_, where R_ = \phi_(R), R_ = \phi_(R), and R_ = \phi_(R), where \phi_ : H \otimes H \to H \otimes H \otimes H, \phi_ : H \otimes H \to H \otimes H \otimes H, and \phi_ : H \otimes H \to H \otimes H \otimes H, are algebra morphisms determined by :\phi_(a \otimes b) = a \otimes b \otimes 1, :\phi_(a \otimes b) = a \otimes 1 \otimes b, :\phi_(a \otimes b) = 1 \otimes a \otimes b. ''R'' is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the Yang–Baxter equation (and so a module ''V'' of ''H'' c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. ''Note'': In this article, all a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
