A review of Lie superalgebra cohomology for pseudoforms

Carlo Alberto Cremonini

Archivum Mathematicum (2022)

  • Volume: 058, Issue: 5, page 269-286
  • ISSN: 0044-8753

Abstract

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This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of 𝔬𝔰𝔭 ( 1 ∣ 4 ) and choose 𝔬𝔰𝔭 ( 1 ∣ 2 ) × 𝔰𝔭 ( 2 ) as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [10].

How to cite

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Cremonini, Carlo Alberto. "A review of Lie superalgebra cohomology for pseudoforms." Archivum Mathematicum 058.5 (2022): 269-286. <http://eudml.org/doc/298938>.

@article{Cremonini2022,
abstract = {This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of $\mathfrak \{osp\}(1\mid 4)$ and choose $\mathfrak \{osp\}(1\mid 2) \times \mathfrak \{sp\} (2)$ as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [10].},
author = {Cremonini, Carlo Alberto},
journal = {Archivum Mathematicum},
keywords = {Lie superalgebras; cohomology; pseudoforms; integral forms; infinite-dimensional representations},
language = {eng},
number = {5},
pages = {269-286},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A review of Lie superalgebra cohomology for pseudoforms},
url = {http://eudml.org/doc/298938},
volume = {058},
year = {2022},
}

TY - JOUR
AU - Cremonini, Carlo Alberto
TI - A review of Lie superalgebra cohomology for pseudoforms
JO - Archivum Mathematicum
PY - 2022
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 058
IS - 5
SP - 269
EP - 286
AB - This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of $\mathfrak {osp}(1\mid 4)$ and choose $\mathfrak {osp}(1\mid 2) \times \mathfrak {sp} (2)$ as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [10].
LA - eng
KW - Lie superalgebras; cohomology; pseudoforms; integral forms; infinite-dimensional representations
UR - http://eudml.org/doc/298938
ER -

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