q-Oscillators and a q-deformed Debye Model for Lattice Heat Capacity
75
The invariance of Eq. (5) implies
(ax + cy)(bx + dy) = q(bx + dy)(ax + cy)
(9)
Equations (8) and (9) give a complete set of conditions to be obeyed by the noncommuting objects a, b, c, d to preserve the structure of the quantum plane:
ab = qba; cd = qdc; ac = qca; bd = qdb; bc = cb;
ad − da = (q − q
−1
)bc
(10)
These are commutation relations obeyed by a, b, c, d. T is called a quantum matrix.
It is shown that [25] the matrices T satisfy all the axioms of a non-commutative Hopf
algebra and thus constitute a quantum group. It is denoted by GLq(2), the quantum
linear general group in two dimensions. It is the group of linear transformations in
two-dimensional non-commutative space that preserves the commutation relation
(5). The additional relation
ad − qbc = 1
(11)
yields the quantum unimodular group SLq(2). The object defined by
det q (T ) = ad − qbc
(12)
is called the quantum determinant or q-determinant. Relation (5) can be generalised
to the case of two or more copies of quantum planes [26].
1.2 q-Deformed Numbers and q-Differential Calculus
The q-deformation of numbers was introduced by Heine [27] in 1878. The q-number
[n] q corresponding to the ordinary number n is defined as
[n] q =
q
n
− q
−n
q − q −1
(13)
This definition of q-deformation possesses q ↔ q
−1 symmetry
lim q→1 [n] q = n
(14)
The q-functions are also defined [28]. For example, the q-exponential function
e q (x) =
∞
n=0
x
n
[n] q !
(15)
75
The invariance of Eq. (5) implies
(ax + cy)(bx + dy) = q(bx + dy)(ax + cy)
(9)
Equations (8) and (9) give a complete set of conditions to be obeyed by the noncommuting objects a, b, c, d to preserve the structure of the quantum plane:
ab = qba; cd = qdc; ac = qca; bd = qdb; bc = cb;
ad − da = (q − q
−1
)bc
(10)
These are commutation relations obeyed by a, b, c, d. T is called a quantum matrix.
It is shown that [25] the matrices T satisfy all the axioms of a non-commutative Hopf
algebra and thus constitute a quantum group. It is denoted by GLq(2), the quantum
linear general group in two dimensions. It is the group of linear transformations in
two-dimensional non-commutative space that preserves the commutation relation
(5). The additional relation
ad − qbc = 1
(11)
yields the quantum unimodular group SLq(2). The object defined by
det q (T ) = ad − qbc
(12)
is called the quantum determinant or q-determinant. Relation (5) can be generalised
to the case of two or more copies of quantum planes [26].
1.2 q-Deformed Numbers and q-Differential Calculus
The q-deformation of numbers was introduced by Heine [27] in 1878. The q-number
[n] q corresponding to the ordinary number n is defined as
[n] q =
q
n
− q
−n
q − q −1
(13)
This definition of q-deformation possesses q ↔ q
−1 symmetry
lim q→1 [n] q = n
(14)
The q-functions are also defined [28]. For example, the q-exponential function
e q (x) =
∞
n=0
x
n
[n] q !
(15)