Derived coisotropic structures I: affine case
Derived coisotropic structures I: affine case
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导出的各向同性结构 I：仿射情况
DOI：
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发表时间：
2016
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通讯作者：
P. Safronov
中科院分区：
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作者：
Valerio Melani;P. Safronov
We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. Pndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {P}_n$$end{document}-algebras. Roughly speaking, a coisotropic morphism is given by a Pn+1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {P}_{n+1}$$end{document}-algebra acting on a Pndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {P}_n$$end{document}-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer–Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.
We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. Pndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {P}_n$$end{document}-algebras. Roughly speaking, a coisotropic morphism is given by a Pn+1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {P}_{n+1}$$end{document}-algebra acting on a Pndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {P}_n$$end{document}-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer–Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.