  Given any pair of positive integers m and n, we construct a new Hopf algebra,
which may be regarded as a degenerate version of the quantum group of gl(m+n).
We study its structure and develop a highest weight representation theory. The
finite dimensional simple modules are classified in terms of highest weights,
which are essentially characterised by m+n-2 nonnegative integers and two
arbitrary nonzero scalars. In the special case with m=2 and n=1, an explicit
basis is constructed for each finite dimensional simple module. For all m and
n, the degenerate quantum group has a natural irreducible representation acting
on C(q)^(m+n). It admits an R-matrix that satisfies the Yang-Baxter equation
and intertwines the co-multiplication and its opposite. This in particular
gives rise to isomorphisms between the two module structures of any tensor
power of C(q)^(m+n) defined relative to the co-multiplication and its opposite
respectively. A topological invariant of knots is constructed from this
R-matrix, which reproduces the celebrated HOMFLY polynomial. Degenerate quantum
groups of other classical types are briefly discussed.
