Degenerate quantum general linear groups

Abstract

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.