Duke Mathematical Journal

Gröbner bases for operads

Vladimir Dotsenko and Anton Khoroshkin

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Abstract

We define a new monoidal structure on the category of collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads and present operadic versions of Bergman's diamond lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's Poincaré-Birkhoff-Witt criterion of Koszulness for (symmetric) quadratic operads

Article information

Source
Duke Math. J., Volume 153, Number 2 (2010), 363-396.

Dates
First available in Project Euclid: 26 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1274902083

Digital Object Identifier
doi:10.1215/00127094-2010-026

Mathematical Reviews number (MathSciNet)
MR2667136

Zentralblatt MATH identifier
0895.16020

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 46F10: Operations with distributions
Secondary: 20C99: None of the above, but in this section 20G05: Representation theory 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 14L24: Geometric invariant theory [See also 13A50] 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]

Citation

Dotsenko, Vladimir; Khoroshkin, Anton. Gröbner bases for operads. Duke Math. J. 153 (2010), no. 2, 363--396. doi:10.1215/00127094-2010-026. https://projecteuclid.org/euclid.dmj/1274902083


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