Everything about Quantum Group totally explained
In
mathematics and
theoretical physics,
quantum groups are certain
noncommutative algebras that first appeared in the theory of
quantum integrable systems, and which were then formalized by
Vladimir Drinfel'd and
Michio Jimbo. There is no single, all-encompassing definition of quantum group.
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. Distinct but related objects, also called quantum groups, are deformations of the algebra of functions on a semisimple
algebraic group or a
compact Lie group.
Since the discovery of quantum groups, it has become fashionable to introduce the attribute
quantum into the names of many other mathematical objects, such as
quantum plane or
quantum grassmanian. They may also be loosely referred to as aspects of "quantum groups".
Intuitive meaning
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, can't be deformed. One of the ideas behind quantum groups is that if we consider in some sense equivalent but larger structure, namely a group algebra or a universal enveloping algebra, then it
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. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of
Alain Connes'
noncommutative geometry. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum
Yang-Baxter equation and
quantum inverse scattering method developed by the Leningrad School (
Ludwig Faddeev,
Leon Takhtajan,
Evgenii Sklyanin,
Nicolai Reshetikhin and others) and related work by the Japanese School.
Drinfel'd-Jimbo type quantum groups
One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfel'd and Michio Jimbo as a deformation of the
universal enveloping algebra of a
semisimple Lie algebra or, more generally, a
Kac-Moody algebra, in the category of
Hopf algebras. The resulting algebra has additional structure, making it into a
quasitriangular Hopf algebra.
Let
is equal to the concrete compact group
.
Further Information
Get more info on 'Quantum Group'.
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