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 depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.
The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative.
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In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal 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 enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators.
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.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
We will establish the major results in the representation theory of semisimple Lie algebras over the field of complex numbers, and that of the related algebraic groups.
La théorie des représentations des groupes étudie les actions linéaires d'un groupe G sur un espace vectoriel V. On peut alors utiliser l'algèbre linéaire pour résoudre certaines questions de théorie
Study the basics of representation theory of groups and associative algebras.
Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-module
Let u(q)(g) be the small quantum group associated with a complex semisimple Lie algebra g and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three d