94
G. Vinod
m ( ⊗ id) = m (id ⊗ ) = η c
Quantum group is a bialgebra with an antipode, but with the conditions of commutativity and co-commutativity relaxed.
The concept of duality is a key point in defining the Lie algebra of a quantum group
[12]. Given a Hopf algebra A(m, ,, η, c, ), the dual of A, A
∗ will be endowed with
the mappings m
∗
, ,
∗
, η
∗
, c
∗
,
∗ such that
m
∗
( f × g) , x = =( f × g) , ,(x)
∗ f, x × y = = f, x y
η
∗
(α) , x = αα (x)
∗
( f ) = = f, 1
S
∗ f, x = = f, S(x)
(6)
, denotes the pairing between a vector space and its dual.
There are mainly two different approaches to quantum groups:
• Lie algebraic approach developed by Drinfeld and Jumbo
• Non-commutative differential geometric approach developed by Manin.
1.4 Lie Algebraic Approach
Simple Lie algebras do not admit non-trivial deformations in the category of Lie
algebras. Hence Drinfeld [4] and Jimbo [5] independently introduced the idea of
deforming them in the category of Hopf algebras [10]. The resulting algebraic structure is called quantum universal enveloping algebra or popularly, quantum group,
even though they are not at all groups.
There is no general prescription for defining the mappings for a given algebraic
structure so as to make it a Hopf algebra. Consider the sl(2) algebra formed by the
generators X + , X − and H :
H, X ±
= ±X ±
X + , X −
= 2H
(7)
The co-product can be defined as
(H ) = H ⊗ 1 + 1 ⊗ H
(X ± ) = X ± ⊗ 1 + 1 ⊗ X ±
(8)
This is a co-commutative co-product. Now assume that the deformed algebra has the
form