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.

中文翻译：

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简并量子广义线性群

给定任意一对正整数 m 和 n，我们构造了一个新的 Hopf 代数，它可以看作是 $\mathfrak{gl}_{m+n}$ 量子群的简并版本。我们研究了它的结构并开发了一个最高权重表示理论。有限维简单模块根据最高权重进行分类，其主要特征是 $m + n - 2$ 非负整数和两个任意非零标量。在 $m = 2$ 和 $n = 1$ 的特殊情况下，为每个有限维简单模块构造显式基。对于所有 $m$ 和 $n$，简并量子群具有作用于 $\mathbb{C}(q)^{m+n}$ 的自然不可约表示。它承认一个 $R$-矩阵，它满足 Yang-Baxter 方程并且将乘法和它的对立交织在一起。这尤其会导致 $\mathbb{C}(q)^{m+n}$ 的任何张量幂的两个模块结构之间的同构，该张量幂分别相对于乘法及其相反定义。节点的拓扑不变量是从这个 $R$-matrix 构造的，它再现了著名的 HOMFLY 多项式。简要讨论了其他经典类型的简并量子群。