Properads and homological differential operators related to surfaces

Lada Peksová

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 5, page 299-312
  • ISSN: 0044-8753

Abstract

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We give a biased definition of a properad and an explicit example of a closed Frobenius properad. We recall the construction of the cobar complex and algebra over it. We give an equivalent description of the algebra in terms of Barannikov’s theory which is parallel to Barannikov’s theory of modular operads. We show that the algebra structure can be encoded as homological differential operator. Example of open Frobenius properad is mentioned along its specific properties.

How to cite

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Peksová, Lada. "Properads and homological differential operators related to surfaces." Archivum Mathematicum 054.5 (2018): 299-312. <http://eudml.org/doc/294540>.

@article{Peksová2018,
abstract = {We give a biased definition of a properad and an explicit example of a closed Frobenius properad. We recall the construction of the cobar complex and algebra over it. We give an equivalent description of the algebra in terms of Barannikov’s theory which is parallel to Barannikov’s theory of modular operads. We show that the algebra structure can be encoded as homological differential operator. Example of open Frobenius properad is mentioned along its specific properties.},
author = {Peksová, Lada},
journal = {Archivum Mathematicum},
keywords = {properads; Frobenius properad; cobar complex; Barannikov’s type theory; homological differential operators},
language = {eng},
number = {5},
pages = {299-312},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Properads and homological differential operators related to surfaces},
url = {http://eudml.org/doc/294540},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Peksová, Lada
TI - Properads and homological differential operators related to surfaces
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 5
SP - 299
EP - 312
AB - We give a biased definition of a properad and an explicit example of a closed Frobenius properad. We recall the construction of the cobar complex and algebra over it. We give an equivalent description of the algebra in terms of Barannikov’s theory which is parallel to Barannikov’s theory of modular operads. We show that the algebra structure can be encoded as homological differential operator. Example of open Frobenius properad is mentioned along its specific properties.
LA - eng
KW - properads; Frobenius properad; cobar complex; Barannikov’s type theory; homological differential operators
UR - http://eudml.org/doc/294540
ER -

References

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  1. Barannikov, S., Modular operads and Batalin-Vilkovisky geometry, Internat. Math. Res. Notices (2007), Article ID rnm075, 31 pages. (2007) MR2359547
  2. Doubek, M., Jurčo, B., Münster, K., Modular operads and the quantum open-closed homotopy algebra, J. High Energy Phys. (2015), Article ID 158 (2015), arXiv:1308.3223 [math.AT]. (2015) MR3464644
  3. Drummond-Cole, G.C., Terilla, J., Tradler, T., Algebras over Cobar(coFrob), J. Homotopy Relat. Struct. 5 (1) (2010), 15–36, arXiv:0807.1241 [math.QA]. (2010) MR2591885
  4. Getzler, E., Kapranov, M.M., 10.1023/A:1000245600345, Compositio Math. 110 (1) (1998), 65–126, arXiv:dg-ga/9408003. (1998) MR1601666DOI10.1023/A:1000245600345
  5. Hackney, P., Robertson, M., Yau, D., Infinity Properads and Infinity Wheeled Properads, Lecture Notes in Math., Springer International Publishing, 2015. (2015) MR3408444
  6. Markl, M., Shnider, S., Stasheff, J., Operads in algebra, topology and physics, Math. Surveys Monogr., vol. 96, Amer. Math. Soc., Providence, RI, 2002. (2002) Zbl1017.18001MR1898414
  7. Merkulov, S., Vallette, B., Deformation theory of representations of prop(erad)s I, J. Reine Angew. Math. 634 (2009), 51–106. (2009) MR2560406
  8. Münster, K., Sachs, I., 10.1007/s00220-012-1654-1, Comm. Math. Phys. 321 (3) (2013), 769–801, arXiv:1109.4101 [hep-th]. (2013) MR3070036DOI10.1007/s00220-012-1654-1
  9. Peksová, L., Algebras over operads and properads, Master's thesis, Charles Univ. Prague, 2016, https://is.cuni.cz/studium/dipl_uc/index.php?id=40d829716c2891d12550d202e189ef4e&tid=1&do=xdownload&fid=120229648&did=148219&vdetailu=1. (2016) 
  10. Vallette, B., 10.1090/S0002-9947-07-04182-7, Trans. Amer. Math. Soc. 359 (10) (2007), 4865–4943, arXiv:math/0411542. (2007) MR2320654DOI10.1090/S0002-9947-07-04182-7

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