Quantum Groups, q-Oscillators and q-Deformed Quantum Mechanics
97
If T 1 and T 2 are two 2 × 2 matrices with non-commuting elements, whose elements
satisfy the relations (21), but the elements of T 1 commuting with those of T 2 , then the
elements of the matrix product T 1 T 2 also satisfy the relations (21). But conditions (21)
can be obtained by proposing the idea of a quantum plane, in which the coordinates
do not commute:
x y = qyx
(23)
If x and y commute with the matrix elements a, b, c, d satisfying the relations (21),
then x, y defined by
x
y
=
a b
c d
x
y
(24)
will satisfy
xy = qyx
(25)
In other words, if A is the algebra generated by x and y with relations (23), and H that
generated by a, b, c, d with relations (21), then the map δ : A → H ⊗ A defined by
δ
x
y
=
a b
c d
⊗
x
y
(26)
is a homomorphism. Thus the relations (21) constitute a sufficient condition on the
elements of the matrix T for the action x → Tx on a column vector x to preserve
the relations (23) between the components of x. The same is true for a row vector ˜
x
and the action ˜
x → ˜
xT.
Conversely, (21) are the necessary conditions for the quantum plane condition to
be preserved for both column and row vectors: i.e., if
x y = qyx ⇒ xy
(27)
then a, b, c, d satisfy the relations (21). Thus the relations (21) are consequences of
the non-commutativity of space.
Relations (21) define the quantum general linear group C q G L(2). If we further
assume
ad − qbc = 1
(28)
we get the quantum unimodular group C q SL(2).
If we consider the quantization of G = G L(2), taking the algebra of functions on
G to be the algebra H generated by the non-commuting matrix elements a, b, c, d
defined by (21). Further, if we define
97
If T 1 and T 2 are two 2 × 2 matrices with non-commuting elements, whose elements
satisfy the relations (21), but the elements of T 1 commuting with those of T 2 , then the
elements of the matrix product T 1 T 2 also satisfy the relations (21). But conditions (21)
can be obtained by proposing the idea of a quantum plane, in which the coordinates
do not commute:
x y = qyx
(23)
If x and y commute with the matrix elements a, b, c, d satisfying the relations (21),
then x, y defined by
x
y
=
a b
c d
x
y
(24)
will satisfy
xy = qyx
(25)
In other words, if A is the algebra generated by x and y with relations (23), and H that
generated by a, b, c, d with relations (21), then the map δ : A → H ⊗ A defined by
δ
x
y
=
a b
c d
⊗
x
y
(26)
is a homomorphism. Thus the relations (21) constitute a sufficient condition on the
elements of the matrix T for the action x → Tx on a column vector x to preserve
the relations (23) between the components of x. The same is true for a row vector ˜
x
and the action ˜
x → ˜
xT.
Conversely, (21) are the necessary conditions for the quantum plane condition to
be preserved for both column and row vectors: i.e., if
x y = qyx ⇒ xy
(27)
then a, b, c, d satisfy the relations (21). Thus the relations (21) are consequences of
the non-commutativity of space.
Relations (21) define the quantum general linear group C q G L(2). If we further
assume
ad − qbc = 1
(28)
we get the quantum unimodular group C q SL(2).
If we consider the quantization of G = G L(2), taking the algebra of functions on
G to be the algebra H generated by the non-commuting matrix elements a, b, c, d
defined by (21). Further, if we define