A quantum group is usually a Hopf algebra whose underlying ring is a noncommutative
ring, whose coproduct is noncocommutative, or both.
Noncommutativity means that the product can satisfy 
. Noncocommutativity means that
, where 
is the coproduct and 
reverses the
tensor factors.
Two standard constructions deform either the algebra of functions on a group or
the universal enveloping algebra of
a Lie algebra. The first is an algebra
of suitable scalar-valued functions on the group,
with pointwise multiplication. The group multiplication induces the coproduct 
. A typical deformation
depends on a parameter 
and reduces to the corresponding classical object in a limit such as 
.
The coproduct therefore encodes a generalized group multiplication. The antipode 
is the Hopf
algebra operation satisfying 
, so it plays
the role of group inversion. Quantum groups arise in integrable systems, knot
invariants, representation theory, and quantum statistical mechanics. Despite
the name, a quantum group is generally an algebraic object rather than a group
in the ordinary sense of set theory.
