Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 6

Degenerate quantum general linear groups

Pages: 1375 – 1422



Jin Cheng (School of Mathematics and Statistics, Shandong Normal University, Jinan, China)

Yan Wang (School of Mathematics and Statistics, Shandong Normal University, Jinan, China)

Ruibin Zhang (School of Mathematics and Statistics, University of Sydney, NSW, Australia)


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.

Published 7 July 2021