|
Pareigis Hopf Algebra
In algebra, the Pareigis Hopf algebra is the Hopf algebra over a field ''k'' whose left comodules are essentially the same as complexes over ''k'', in the sense that the corresponding monoidal categories are isomorphic. It was introduced by as a natural example of a Hopf algebra that is neither commutative nor cocommutative. Construction As an algebra over ''k'', the Pareigis algebra is generated by elements ''x'',''y'', 1/''y'', with the relations ''xy'' + ''yx'' = ''x''2 = 0. The coproduct takes ''x'' to ''x''⊗1 + (1/''y'')⊗''x'' and ''y'' to ''y''⊗''y'', and the counit takes ''x'' to 0 and ''y'' to 1. The antipode takes ''x'' to ''xy'' and ''y'' to its inverse and has order 4. Relation to complexes If ''M'' = ⊕''M''''n'' is a complex with differential ''d'' of degree –1, then ''M'' can be made into a comodule over ''H'' by letting the coproduct take ''m'' to Σ ''y''''n''⊗''m''''n'' + ''y''''n''+1''x''⊗''dm''''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variable (mathematics), variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in mathematical education, education, to the study of algebraic structures such as group (mathematics), groups, ring (mathematics), rings, and field (mathematics), fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, ... [...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 (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comodule
In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. Formal definition Let ''K'' be a field, and ''C'' be a coalgebra over ''K''. A (right) comodule over ''C'' is a ''K''-vector space ''M'' together with a linear map :\rho\colon M \to M \otimes C such that # (\mathrm \otimes \Delta) \circ \rho = (\rho \otimes \mathrm) \circ \rho # (\mathrm \otimes \varepsilon) \circ \rho = \mathrm, where Δ is the comultiplication for ''C'', and ε is the counit. Note that in the second rule we have identified M \otimes K with M\,. Examples * A coalgebra is a comodule over itself. * If ''M'' is a finite-dimensional module over a finite-dimensional ''K''-algebra ''A'', then the set of linear functions from ''A'' to ''K'' forms a coalgebra, and the set of linear functions from ''M'' to ''K'' forms a comodule over that coalgebra. * A graded vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every ( small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cocommutative
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] |
Counit
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 angul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sweedler's Hopf Algebra
In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra ''H''4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative. Definition The following infinite dimensional Hopf algebra was introduced by . The Hopf algebra is generated as an algebra by three elements ''x'', ''g'' and ''g''-1. The coproduct Δ is given by :Δ(g) = ''g'' ⊗''g'', Δ(''x'') = 1⊗''x'' + ''x'' ⊗''g'' The antipode ''S'' is given by :''S''(''x'') = –''x'' ''g''−1, ''S''(''g'') = ''g''−1 The counit ε is given by :ε(''x'')=0, ε(''g'') = 1 Sweedler's 4-dimensional Hopf algebra ''H''4 is the quotient of this by the relations :''x''2 = 0, ''g''2 = 1, ''gx'' = –''xg'' so it has a basis 1, ''x'', ''g'', ''xg'' . Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on ''H''4⊗''H''4. This Hopf algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
