论文标题
$ \ mathfrak {sl}（n）$和Coinvariants的代数
Quantized nilradicals of parabolic subalgebras of $\mathfrak{sl}(n)$ and algebras of coinvariants
论文作者
论文摘要
令$ p_j $为$ sl_n $的标准抛物线子组，通过删除负面简单根部的子集$ j $获得，让$ p_j = l_ju_j $是标准的Levi分解。在第一作者的工作之后，我们研究量子模拟$θ：{\ Mathcal O} _q（p_j）\ to {\ Mathcal O} _Q（l_j）\ otimes {\ Mathcal O} o} _q（p_j）^{\ operatoTorname {co}θ} \ subseteq {\ Mathcal o} _q（p_j）$ of CoinVariants。结果表明，Smash产品代数$ {\ MATHCAL O} _Q（l_j）\＃{\ MATHCAL O} _Q（p_j）^{\ operatoRatorName {co}θ} $ isomorphic isomorphic to isomorphic至$ {\ nathcal o} _qal o} _q（p_j JJ）$。鉴于此，$ {\ Mathcal O} _q（p_j）^{\ propatoTorname {co}θ} $ - 虽然不是Hopf代数 - 可以看作是坐标环$ {\ Mathcal O}（U_J）$的量子类似物。 在本文中，我们证明，当$ q \ in \ mathbb {k} $是非零的，不是一个统一的根，$ {\ Mathcal o} _q（p_j）^{\ propatatorname {co}θ} $与量子相关的量子$ \ mathfrak {sl}（n）$的Weyl Group中的元素$ W $。对于这些量子舒伯特细胞，就发现了发电机和关系的明确表示。
Let $P_J$ be the standard parabolic subgroup of $SL_n$ obtained by deleting a subset $J$ of negative simple roots, and let $P_J = L_JU_J$ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue $θ: {\mathcal O}_q(P_J) \to{\mathcal O}_q(L_J) \otimes {\mathcal O}_q(P_J)$ of an induced coaction and the corresponding subalgebra ${\mathcal O}_q(P_J)^{\operatorname{co} θ} \subseteq {\mathcal O}_q(P_J)$ of coinvariants. It was shown that the smash product algebra ${\mathcal O}_q(L_J)\# {\mathcal O}_q(P_J)^{\operatorname{co} θ}$ is isomorphic to ${\mathcal O}_q(P_J)$. In view of this, ${\mathcal O}_q(P_J)^{\operatorname{co} θ}$ -- while it is not a Hopf algebra -- can be viewed as a quantum analogue of the coordinate ring ${\mathcal O}(U_J)$. In this paper we prove that when $q\in \mathbb{K}$ is nonzero and not a root of unity, ${\mathcal O}_q(P_J)^{\operatorname{co} θ}$ is isomorphic to a quantum Schubert cell algebra ${\mathcal U}_q^+[w]$ associated to a parabolic element $w$ in the Weyl group of $\mathfrak{sl}(n)$. An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.
