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G. Vinod
be reproduced by a suitable algebraic quantisation of Poisson Lie algebras. The same
relations were obtained by Jimbo [5] through a somewhat different scheme. Drinfeld
coined the term quantum group during the international Conference on Mathematics
in Berkeley, in 1986. Drinfeld has been awarded the field medal in that ICM for
his contributions to mathematics including quantum groups. Quantum groups or
quantum universal enveloping algebras arise topologically in the theory of knot and
link invariants [6] and geometrically in the study of non-commuting geometries [7].
1.2 Quantum Group as a Key to New Physics
From the viewpoint of physics, quantum group includes two basic ideas, namely,
the deformation of an algebraic structure and the notion of a non-commutative comultiplication [8]. The idea of deformation is familiar in physics: the Poincare group
is a deformation of the Galilie group, which is regained in the limit c → ∞. Also
quantum mechanics can be considered as a deformation of classical mechanics which
is regained in the limit → 0. In the deformation of algebraic structure usually
deformation parameter q is introduced and in the limit q → 1, the original structure
is regained. As a result of q-deformation, commutative algebra becomes a noncommuting one. This is the origin of the term quantum in quantum groups since
quantization is in effect replacement of commuting things by non-commuting things.
The concept of co-multiplication is also inherent in quantum physics. For instance,
consider the action of angular momentum operator J in quantum mechanics: Angular
momentum is additive in both classical and quantum mechanics,
J total = J 1 + J 2
(1)
In the ketspace formed from the product of the ket spaces spanned by the eigen kets
of J 1 and J 2 , the action of J total can be expressed as
J total = J 1 ⊗ 1 + 1 ⊗ J 2
(2)
This is actually a co-multiplication:
(J) = J ⊗ 1 + 1 ⊗ J
(3)
Thus the vector addition of angular momentum in quantum mechanics defines a comultiplication in a bialgebra. This is an example of commutative co-multiplication.
The raising and lowering operators in angular momentum theory obey such a comultiplication.
The z-component of total angular momentum J 0 and the ladder operators J ±
constitute an su(2) algebra.
G. Vinod
be reproduced by a suitable algebraic quantisation of Poisson Lie algebras. The same
relations were obtained by Jimbo [5] through a somewhat different scheme. Drinfeld
coined the term quantum group during the international Conference on Mathematics
in Berkeley, in 1986. Drinfeld has been awarded the field medal in that ICM for
his contributions to mathematics including quantum groups. Quantum groups or
quantum universal enveloping algebras arise topologically in the theory of knot and
link invariants [6] and geometrically in the study of non-commuting geometries [7].
1.2 Quantum Group as a Key to New Physics
From the viewpoint of physics, quantum group includes two basic ideas, namely,
the deformation of an algebraic structure and the notion of a non-commutative comultiplication [8]. The idea of deformation is familiar in physics: the Poincare group
is a deformation of the Galilie group, which is regained in the limit c → ∞. Also
quantum mechanics can be considered as a deformation of classical mechanics which
is regained in the limit → 0. In the deformation of algebraic structure usually
deformation parameter q is introduced and in the limit q → 1, the original structure
is regained. As a result of q-deformation, commutative algebra becomes a noncommuting one. This is the origin of the term quantum in quantum groups since
quantization is in effect replacement of commuting things by non-commuting things.
The concept of co-multiplication is also inherent in quantum physics. For instance,
consider the action of angular momentum operator J in quantum mechanics: Angular
momentum is additive in both classical and quantum mechanics,
J total = J 1 + J 2
(1)
In the ketspace formed from the product of the ket spaces spanned by the eigen kets
of J 1 and J 2 , the action of J total can be expressed as
J total = J 1 ⊗ 1 + 1 ⊗ J 2
(2)
This is actually a co-multiplication:
(J) = J ⊗ 1 + 1 ⊗ J
(3)
Thus the vector addition of angular momentum in quantum mechanics defines a comultiplication in a bialgebra. This is an example of commutative co-multiplication.
The raising and lowering operators in angular momentum theory obey such a comultiplication.
The z-component of total angular momentum J 0 and the ladder operators J ±
constitute an su(2) algebra.