Degenerate quantum general linear groups

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 $\mathfrak{gl}_{m+n}$. 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 $\mathbb{C}(q)^{m+n}$. 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 $\mathbb{C}(q)^{m+n}$ 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.

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Published 7 July 2021