Quantum Groups, q-Oscillators and q-Deformed Quantum Mechanics
93
J 0 , J ±
= ±J ±
J + , J −
= 2J 0
(4)
A prototype q-deformation of this algebra is
J 0 , J ±
= ±J ±
J + , J −
= [2J 0 ] q
(5)
[2J 0 ] q is the q-deformation of 2J 0 , which in general depends on a parameter q. There
are more than one definition of q-deformation in q-analysis all of which reproduces
the original structure in the limit q → 1. Thus different su q (2) are possible and all
of them reduces to the su(2) algebra in the limit q → 1.
1.3 Formal Definition of a Quantum Group
Quantum groups are not groups; they are deformed Lie algebras. Formally, quantum groups are defined to be Hopf algebras, which are in general non-commutative
[4–11] algebra is a bialgebra with an antipode. Bialgebra is a vector space which
is an algebra as well as coalgebra. As an algebra is a way of multiplying things, a
coalgebra is a way of unmultiplying things. Analogous to the notion of product in an
algebra, there is the notion of co-product in a coalgebra. For a bialgebra A defined
over a field k, product is defined as the mapping
m : A ⊗ A → A
whereas co-product is defined by
: A → A ⊗ A
unit is defined by
η : k → A
co-unit is defined by
c : A → k
The antipode is a linear map
: A → A
so that the following conditions are satisfied.
93
J 0 , J ±
= ±J ±
J + , J −
= 2J 0
(4)
A prototype q-deformation of this algebra is
J 0 , J ±
= ±J ±
J + , J −
= [2J 0 ] q
(5)
[2J 0 ] q is the q-deformation of 2J 0 , which in general depends on a parameter q. There
are more than one definition of q-deformation in q-analysis all of which reproduces
the original structure in the limit q → 1. Thus different su q (2) are possible and all
of them reduces to the su(2) algebra in the limit q → 1.
1.3 Formal Definition of a Quantum Group
Quantum groups are not groups; they are deformed Lie algebras. Formally, quantum groups are defined to be Hopf algebras, which are in general non-commutative
[4–11] algebra is a bialgebra with an antipode. Bialgebra is a vector space which
is an algebra as well as coalgebra. As an algebra is a way of multiplying things, a
coalgebra is a way of unmultiplying things. Analogous to the notion of product in an
algebra, there is the notion of co-product in a coalgebra. For a bialgebra A defined
over a field k, product is defined as the mapping
m : A ⊗ A → A
whereas co-product is defined by
: A → A ⊗ A
unit is defined by
η : k → A
co-unit is defined by
c : A → k
The antipode is a linear map
: A → A
so that the following conditions are satisfied.