晶体和量子群在$q=0$上的范畴重建
A categorical reconstruction of crystals and quantum groups at $q=0$
摘要：
与Kac-Moody李代数$\mathfrak{g}$相关联的量子坐标代数$A_{q}(\mathfrak{g})$形成一个Hopf代数，其余模正是BGG范畴$\mathcal{O}_{\mathfrak{g}}$中的$U_{q}(\mathfrak{g})$模。在本文中，我们调查是否类似的结果是正确的，当$q=0$。我们将晶体基分类为点集范畴上的余单元函子上的余代数，并将晶体的monoidal结构编码为双单元结构。在这样做，我们证明了，有没有余代数的范畴内的点集的余模是等价的晶体基地。然后，我们构造了一个双代数在$\mathbb{Z}$的基础余模是等价的晶体，我们推测这是联系到Lusztig的量子群在$v = \infty$。
英文摘要：
The quantum co-ordinate algebra $A_{q}(\mathfrak{g})$ associated to a Kac-Moody Lie algebra $\mathfrak{g}$ forms a Hopf algebra whose comodules are precisely the $U_{q}(\mathfrak{g})$ modules in the BGG category $\mathcal{O}_{\mathfrak{g}}$. In this paper we investigate whether an analogous result is true when $q=0$. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over $\mathbb{Z}$ whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig's quantum group at $v = \infty$.
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