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...

satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit...

SU(2) Representation theory of the Lorentz group Representation theory of Hopf algebras Adjoint representation of a Lie group List of Lie group topics Symmetry...

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...

polynomial of a matrix in one variable. Representation theory Intertwiner Representation theory of Hopf algebras Lie algebra representation Schur’s lemma...

hence avoids the use of auxiliary Grassmann numbers. Graded vector space Lie algebra representation Representation theory of Hopf algebras v t e v t e...

article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification...

complex group ring C[G]. The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following Theorem...

discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential...

in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is...

(mathematics) Representation theory of Hopf algebras General Associative property, Associator Heap (mathematics) Magma (algebra) Loop (algebra), Quasigroup...

enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal...

to other categories in abstract algebra: associative algebras, rings, Lie algebras, Lie superalgebras, Hopf algebras to name some. Representations or...

results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic...

in the representation theory of semisimple Lie algebras (or the closely related representation theory of compact Lie groups), the weights of the dual...

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...

Discrete series representation Principal series representation Borel–Weil–Bott theorem Algebra representation Representation theory of Hopf algebras Quiver (mathematics)...

that every Lie algebra is contained in the Lie algebra associated to some associative algebra. The representation theory of Lie algebras plays an important...

Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.[citation...

(which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum...

→ A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped...

special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard...

which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point...

p} , the Steenrod algebra A p {\displaystyle A_{p}} is the graded Hopf algebra over the field F p {\displaystyle \mathbb {F} _{p}} of order p {\displaystyle...

selfadjoint graded Hopf algebra that is both commutative and cocommutative. The study of symmetric functions is based on that of symmetric polynomials...

related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely...

fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational...

enveloping algebras) Shirshov–Cohn theorem (Jordan algebras) Shirshov–Witt theorem (Lie algebras) Beck's monadicity theorem (category theory) Bruguières...

of the compact Lie group, a reductive algebraic group. Let G be connected compact Lie group with maximal torus T. Hopf showed that the centralizer of...

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...