Quadratic algebras arising from Hopf operads generated by a single element

Anton Khoroshkin

doi:10.1007/s11005-020-01283-z

Quadratic algebras arising from Hopf operads generated by a single element

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Abstract

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in detail Hopf operads generated by a single skew-symmetric element of arbitrary arity. We explain why the dual space to the space of n-ary operations in these operads are quadratic and Koszul algebras. We give a detailed description of generators, relations and monomial bases in these algebras.

Original language	English
Pages (from-to)	2053-2082
Number of pages	30
Journal	Letters in Mathematical Physics
Volume	110
Issue number	8
DOIs	https://doi.org/10.1007/s11005-020-01283-z
State	Published - 1 Aug 2020
Externally published	Yes

Bibliographical note

Funding Information:
I would like to thank Vladimir Dotsenko, Nikita Markarian, Sergei Merkulov, Dmitri Piontkovski and Thomas Willwacher for stimulating discussions. I would like to acknowledge Yurii Ivanovich Manin and Bruno Vallette for the correspondence concerning [] which forces me to finish this paper. My research was carried out within the HSE University Basic Research Program and funded (jointly) by the Russian Academic Excellence Project ’5-100’, Results of Sect. , (in particular, Theorem ) have been obtained under support of the RSF grant No.19-11-00275. I am a Young Russian Mathematics award winner and would like to thank its sponsors and jury.

Funding Information:
I would like to thank Vladimir Dotsenko, Nikita Markarian, Sergei Merkulov, Dmitri Piontkovski and Thomas Willwacher for stimulating discussions. I would like to acknowledge Yurii Ivanovich Manin and Bruno Vallette for the correspondence concerning?[12] which forces me to finish this paper. My research was carried out within the HSE University Basic Research Program and funded (jointly) by the Russian Academic Excellence Project ?5-100?, Results of Sect.?3.4 , (in particular, Theorem?3.22) have been obtained under support of the RSF grant No.19-11-00275. I am a Young Russian Mathematics award winner and would like to thank its sponsors and jury.

Publisher Copyright:
© 2020, Springer Nature B.V.

Keywords

Hopf operads
Orlik–Solomon algebra
Quadratic algebras

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s11005-020-01283-z

Cite this

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abstract = "The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in detail Hopf operads generated by a single skew-symmetric element of arbitrary arity. We explain why the dual space to the space of n-ary operations in these operads are quadratic and Koszul algebras. We give a detailed description of generators, relations and monomial bases in these algebras.",

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Quadratic algebras arising from Hopf operads generated by a single element

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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