On generating series of finitely presented operads

Anton Khoroshkin; Dmitri Piontkovski

doi:10.1016/j.jalgebra.2014.12.012

On generating series of finitely presented operads

Research output: Contribution to journal › Article › peer-review

Abstract

Given an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P(. n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim. . P(. n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(. n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim. . P(. n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

Original language	English
Pages (from-to)	377-429
Number of pages	53
Journal	Journal of Algebra
Volume	426
DOIs	https://doi.org/10.1016/j.jalgebra.2014.12.012
State	Published - 5 Mar 2015
Externally published	Yes

Bibliographical note

Funding Information:
The first author's research is partially supported by NSh-1500.2014.2, by RFBR grants 13-02-00478 , 13-01-12401 , 15-01-09242 , by “The National Research University–Higher School of Economics” Academic Fund Program in 2013–2014 , research grant 14-01-0124 , by Dynasty Foundation and Simons-IUM fellowship . The second author's research was supported by “The National Research University Higher School of Economics Academic Fund Program” in 2013–2014 , research grant 12-01-0134 , and the RFBR project 14-01-00416 .

Publisher Copyright:
© 2014 Elsevier Inc..

Keywords

Algebraic differential equation
Generating series
Gröbner basis for operads
Operad
Variety of algebras

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2014.12.012

Cite this

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title = "On generating series of finitely presented operads",

abstract = "Given an operad P with a finite Gr{\"o}bner basis of relations, we study the generating functions for the dimensions of its graded components P(. n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim. . P(. n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(. n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim. . P(. n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.",

keywords = "Algebraic differential equation, Generating series, Gr{\"o}bner basis for operads, Operad, Variety of algebras",

author = "Anton Khoroshkin and Dmitri Piontkovski",

note = "Funding Information: The first author's research is partially supported by NSh-1500.2014.2, by RFBR grants 13-02-00478 , 13-01-12401 , 15-01-09242 , by “The National Research University–Higher School of Economics” Academic Fund Program in 2013–2014 , research grant 14-01-0124 , by Dynasty Foundation and Simons-IUM fellowship . The second author's research was supported by “The National Research University Higher School of Economics Academic Fund Program” in 2013–2014 , research grant 12-01-0134 , and the RFBR project 14-01-00416 . Publisher Copyright: {\textcopyright} 2014 Elsevier Inc..",

year = "2015",

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doi = "10.1016/j.jalgebra.2014.12.012",

language = "English",

volume = "426",

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journal = "Journal of Algebra",

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AU - Khoroshkin, Anton

AU - Piontkovski, Dmitri

N1 - Funding Information: The first author's research is partially supported by NSh-1500.2014.2, by RFBR grants 13-02-00478 , 13-01-12401 , 15-01-09242 , by “The National Research University–Higher School of Economics” Academic Fund Program in 2013–2014 , research grant 14-01-0124 , by Dynasty Foundation and Simons-IUM fellowship . The second author's research was supported by “The National Research University Higher School of Economics Academic Fund Program” in 2013–2014 , research grant 12-01-0134 , and the RFBR project 14-01-00416 . Publisher Copyright: © 2014 Elsevier Inc..

PY - 2015/3/5

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N2 - Given an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P(. n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim. . P(. n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(. n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim. . P(. n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

AB - Given an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P(. n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim. . P(. n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(. n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim. . P(. n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

KW - Algebraic differential equation

KW - Generating series

KW - Gröbner basis for operads

KW - Operad

KW - Variety of algebras

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JO - Journal of Algebra

JF - Journal of Algebra

ER -

On generating series of finitely presented operads

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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