96
G. Vinod
This is the standard form of U q sl(2). Further if we assume that H
†
= H ; X
†
± =
X ∓ , U q sl(2) → U q su(2). If we represent the generators by J 0 , J + , J − , U q su(2),
or simply SU q (2) takes the form
J 0 , J ±
= ±J ±
J + , J −
= [2J 0 ] q
(16)
where
[N ] q =
q
N
− q
−N
q − q −1
(17)
which approaches N in the limit q → 1. There are two distinct expressions for the
Casimir operator corresponding to integer and half-integer values of j in the representation of the algebra. They are J − J + + [J 0 ] q [J 0 + 1] q and J − J + + [J 0 ] q [J 0 +
1
2
] q
respectively. Quantization of other Lie algebras have also appeared in the literature.
Callegini et al. [12] have constructed H q (1) and E q (2) by contracting SU q (2). The
existence of a Jacobi identity
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
(18)
is an essential requirement of any Lie algebra. In order to construct the q-analogue,
Chaichian et al. [13] defined a q-deformed commutator
[A, B] q = AB − q B A
(19)
They have observed that the following identity holds for arbitrary values of p and q:
A, [B, C] p
q
+ q
B, [C, A] p
q
+
C, [A, B] pq
= 0
( 2 0 )
1.5 Non-commutative Differential Geometric Approach
Manin [7] showed that a system of 2 × 2 matrices T =
a b
c d
with non-commuting
matrix elements could be a quantum group provided the bialgebra A generated by
a, b, c, d satisfy the following rules for multiplication:
ab = qba; ac = qca;
bd = qdb; cd = qdc;
bc = cb;
ad − da =
q − q
−1
bc
(21)
T = T ⊗ T
(22)