Twisted unipotent groups
| Autor: | Brown, Ken A., Gelaki, Shlomo |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $S \#_{\sigma}U(\mathfrak{t})$, 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 $\Gamma$, which we prove to be an explicitly determined closed subgroup of $G$. A selection of examples illustrate our results. Comment: Preliminary version, comments welcome. Some overlap with arXiv2009.07760 by the second author, parts of which are extended and corrected here |
| Databáze: | arXiv |
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