Representation theory of Hopf algebras information
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In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H over a field K is a K-vector space V with an action H × V → V usually denoted by juxtaposition ( that is, the image of (h,v) is written hv ). The vector space V is called an H-module.
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abstract algebra, a representationof a Hopfalgebra is a representationof its underlying associative algebra. That is, a representationof a Hopfalgebra H...
satisfying a certain property. The representationtheoryof a Hopfalgebra is particularly nice, since the existence of compatible comultiplication, counit...
abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these...
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representation Principal series representation Borel–Weil–Bott theorem Weyl's character formula AlgebrarepresentationRepresentationtheoryofHopf algebras...
article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification...
to other categories in abstract algebra: associative algebras, rings, Lie algebras, Lie superalgebras, Hopfalgebras to name some. Representations or...
(which are quasitriangular Hopfalgebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum...
complex group ring C[G]. The importance of the group algebra is that it captures the unitary representationtheoryof G as shown in the following Theorem...
in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representationtheory is...
homological algebra, Lie algebras, free algebras, and homology groups. The influence ofalgebra is wide reaching and includes many branches of mathematics...
associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representationtheoryof groups, in which elements of a group...
special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard...
affine algebra (or affine quantum group) is a Hopfalgebra that is a q-deformation of the universal enveloping algebraof an affine Lie algebra. They were...
Coalgebras occur naturally in a number of contexts (for example, representationtheory, universal enveloping algebras and group schemes). There are also F-coalgebras...
of graded Hopfalgebras with a single character. Hence any such Hopfalgebra has a morphism to the ring of quasisymmetric functions. One example of this...
Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers. These algebras, called geometric algebras, form a...
enveloping algebras are used in the representationtheoryof Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal...
2010 Aguiar, M. and Mahajan, S.2010. "Monoidal Functors, Species, and HopfAlgebras". Meseguer, J., Montanari, U.: 1990 Petri nets are monoids. Information...
results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic...
p} , the Steenrod algebra A p {\displaystyle A_{p}} is the graded Hopfalgebra over the field F p {\displaystyle \mathbb {F} _{p}} of order p {\displaystyle...
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