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Medientyp:
E-Artikel
Titel:
LIE ALGEBROIDS AS SPACES
Beteiligte:
Grady, Ryan;
Gwilliam, Owen
Erschienen:
Cambridge University Press (CUP), 2020
Erschienen in:Journal of the Institute of Mathematics of Jussieu
Sprache:
Englisch
DOI:
10.1017/s1474748018000075
ISSN:
1474-7480;
1475-3030
Entstehung:
Anmerkungen:
Beschreibung:
<jats:p>In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique)<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline2.png" /><jats:tex-math>$L_{\infty }$</jats:tex-math></jats:alternatives></jats:inline-formula>space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline3.png" /><jats:tex-math>$L_{\infty }$</jats:tex-math></jats:alternatives></jats:inline-formula>spaces. Then we show that for each Lie algebroid<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline4.png" /><jats:tex-math>$L$</jats:tex-math></jats:alternatives></jats:inline-formula>, there is a fully faithful functor from the category of representations up to homotopy of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline5.png" /><jats:tex-math>$L$</jats:tex-math></jats:alternatives></jats:inline-formula>to the category of vector bundles over the associated<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline6.png" /><jats:tex-math>$L_{\infty }$</jats:tex-math></jats:alternatives></jats:inline-formula>space. Indeed, this functor sends the adjoint complex of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline7.png" /><jats:tex-math>$L$</jats:tex-math></jats:alternatives></jats:inline-formula>to the tangent bundle of the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline8.png" /><jats:tex-math>$L_{\infty }$</jats:tex-math></jats:alternatives></jats:inline-formula>space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748018000075_inline9.png" /><jats:tex-math>$L_{\infty }$</jats:tex-math></jats:alternatives></jats:inline-formula>space.</jats:p>