In mathematics, a quantum or quantized enveloping algebra is a
''q''-analog of a
universal enveloping algebra
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 representati ...
.
Given a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, the quantum enveloping algebra is typically denoted as
. The notation was introduced by Drinfeld and independently by Jimbo.
Among the applications, studying the
limit led to the discovery of
crystal base
A crystal base for a representation of a quantum group on a \Q(v)-vector space
is not a base of that vector space but rather a \Q-base of L/vL where L is a \Q(v)-lattice in that vector spaces. Crystal bases appeared in the work of and also in the ...
s.
The case of
Michio Jimbo
is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist .
Career
After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at t ...
considered the algebras with three generators related by the three commutators
:
When
, these reduce to the commutators that define the
special linear Lie algebra
In mathematics, the special linear Lie algebra of order n (denoted \mathfrak_n(F) or \mathfrak(n, F)) is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket ,Y=XY-YX. This algebra is well studied and understood, and ...
. In contrast, for nonzero
, the algebra defined by these relations is not a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
but instead an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
that can be regarded as a deformation of the universal enveloping algebra of
.
See also
*
quantum group
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) ...
References
*
*
External links
Quantized enveloping algebraat the
nLab
Quantized enveloping algebras at at
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a ...
U_q(g)?">Does there exist any "quantum Lie algebra" imbedded into the quantum enveloping algebra ?at MathOverflow
{{algebra-stub
Quantum groups
Representation theory
Mathematical quantization