On super Plücker embedding and cluster algebras
On super Plücker embedding and cluster algebras
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关于超级 Plücker 嵌入和簇代数
DOI：
10.1007/s00029-021-00756-w
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
T. Voronov
中科院分区：
文献类型：
--
作者：
E. Shemyakova;T. Voronov
We define a super analog of the classical Plücker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers Λr|s(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }^{r|s}(V)$$\end{document} are not a simple generalization from the completely even case (this works only for r|0 when it is possible to use Λr(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }^r(V)$$\end{document}). To construct the embedding we need to non-trivially combine a super vector space V and its parity-reversion ΠV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi V$$\end{document}. Our “super Plücker map” takes the Grassmann supermanifold Gr|s(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{r|s}(V)$$\end{document} to a “weighted projective space” P1,-1Λr|s(V)⊕Λs|r(ΠV)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1, -1}\left( {\Lambda }^{r|s}(V)\oplus {\Lambda }^{s|r}(\Pi V)\right) $$\end{document} with weights +1,-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1,-1$$\end{document}. A simpler map Gr|0(V)→P(Λr(V))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{r|0}(V)\rightarrow P({\Lambda }^r(V))$$\end{document} works for the case s=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=0$$\end{document}. We construct a super analog of Plücker coordinates, prove that our map is an embedding, and obtain “super Plücker relations”. We analyze another type of relations (due to Khudaverdian) and show their equivalence with the super Plücker relations for r|s=2|0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r|s=2|0$$\end{document}. We discuss application to much sought-after super cluster algebras and construct a super cluster structure for G2(R4|1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2({{\mathbb {R}}}^{4|1})$$\end{document} and G2(R5|1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2({{\mathbb {R}}}^{5|1})$$\end{document}.
We define a super analog of the classical Plücker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers Λr|s(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }^{r|s}(V)$$\end{document} are not a simple generalization from the completely even case (this works only for r|0 when it is possible to use Λr(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }^r(V)$$\end{document}). To construct the embedding we need to non-trivially combine a super vector space V and its parity-reversion ΠV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi V$$\end{document}. Our “super Plücker map” takes the Grassmann supermanifold Gr|s(V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{r|s}(V)$$\end{document} to a “weighted projective space” P1,-1Λr|s(V)⊕Λs|r(ΠV)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1, -1}\left( {\Lambda }^{r|s}(V)\oplus {\Lambda }^{s|r}(\Pi V)\right) $$\end{document} with weights +1,-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1,-1$$\end{document}. A simpler map Gr|0(V)→P(Λr(V))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{r|0}(V)\rightarrow P({\Lambda }^r(V))$$\end{document} works for the case s=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=0$$\end{document}. We construct a super analog of Plücker coordinates, prove that our map is an embedding, and obtain “super Plücker relations”. We analyze another type of relations (due to Khudaverdian) and show their equivalence with the super Plücker relations for r|s=2|0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r|s=2|0$$\end{document}. We discuss application to much sought-after super cluster algebras and construct a super cluster structure for G2(R4|1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2({{\mathbb {R}}}^{4|1})$$\end{document} and G2(R5|1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2({{\mathbb {R}}}^{5|1})$$\end{document}.
具有 Grassmann 变量的聚类代数
DOI：
10.3934/era.2019.26.001
发表时间：
2019
期刊：
Electronic Research Announcements
影响因子：
--
作者：
Ovsienko, Valentin;Shapiro, MichaeL
通讯作者：
Shapiro, MichaeL
超直线、Berezinian 和 Darboux 变换上的微分算子
影响因子：
1.2
作者：
Li, Simon;Shemyakova, Ekaterina;Voronov, Theodore
通讯作者：
Voronov, Theodore