扭化幺幂群
Twisted unipotent groups
AI总结：
本文研究仿射代数幺幂群上带Hopf 2-上循环形变的余三角Hopf代数的结构与表示，证明其为迭代导子型Hopf Ore扩张，给出交叉积分解、单模分层及有限维单模的群结构。
摘要：
我们研究当$G$是复数域$\u2102$上维数为$n$的仿射代数幺幂群、$J$是$G$的Hopf 2-上循环时，Hopf代数${}_J\mathcal{O}(G)_J$的代数结构与表示理论。余三角Hopf代数${}_J\mathcal{O}(G)_J$与$\mathcal{O}(G)$具有相同的余代数结构，但乘法发生形变。我们证明它们是对合的$n$步迭代导子型Hopf Ore扩张。2-上循环$J$的支撑集是$G$的一个闭子群$T$，且${}_J\mathcal{O}(G)_J$是交叉积$S \\#_σU(\mathfrak{t})$，其中$\mathfrak{t}$是$T$的李代数，$S$是形变余理想子代数。单${}_J\mathcal{O}(G)_J$-模由一族因子代数${}_J\mathcal{O}(Z_g)_J$分层，该族因子代数以$T$在$G$中的双陪集$TgT$为参数。有限维单${}_J\mathcal{O}(G)_J$-模均为1维，因此构成一个群$Γ$，我们证明它是$G$的一个可显式确定的闭子群。若干实例阐释了我们的结果。
英文摘要：
We study the algebraic structure and representation theory of the Hopf algebras ${}_J\mathcal{O}(G)_J$ when $G$ is an affine algebraic unipotent group over $\mathbb{C}$ with $\mathrm{dim}(G) = n$ and $J$ is a Hopf $2$-cocycle for $G$. The cotriangular Hopf algebras ${}_J\mathcal{O}(G)_J$ have the same coalgebra structure as $\mathcal{O}(G)$ but a deformed multiplication. We show that they are involutive $n$-step iterated Hopf Ore extensions of derivation type. The 2-cocycle $J$ has as support a closed subgroup $T$ of $G$, and ${}_J\mathcal{O}(G)_J$ is a crossed product $S \#_σU(\mathfrak{t})$, where $\mathfrak{t}$ is the Lie algebra of $T$ and $S$ is a deformed coideal subalgebra. The simple ${}_J\mathcal{O}(G)_J$-modules are stratified by a family of factor algebras ${}_J\mathcal{O}(Z_g)_J$, parametrised by the double cosets $TgT$ of $T$ in $G$. The finite dimensional simple ${}_J\mathcal{O}(G)_J$-modules are all 1-dimensional, so form a group $Γ$, which we prove to be an explicitly determined closed subgroup of $G$. A selection of examples illustrate our results.
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