The Alternative Operad Is Not Koszul

Abstract

In the online compendium [Loday 07], it is asked whether the alternative operad is Koszul. The purpose of this note is to demonstrate that the answer to this question is negative. In doing so, we have been aided by the programs Albert and PARI/GP.

2000 AMS Subject Classification:

• Primary 17D05
• Secondary 05A05
• 16N40
• 16S37
• 17-04
• 17DI5
• 18D50
• 65F99

Keywords:

• alternative operad
• right-alternative operad
• Koszulity
• Poincaré series

View correction statement:

Corrigendum

ACKNOWLEDGMENTS

Thanks are due to Vladimir Dotsenko and Leonid Positselski for explanations concerning Koszulity and the Ginzburg–Kapranov criterion, to Alexander Feldman for advice concerning computing issues and comments on the preliminary version of the manuscript, and to the anonymous referee for a very careful reading of the manuscript and useful comments as well.

Notes

1The software Albert, version 4.0, is available online (http://www.cs.clemson.edu/~dpj/albertstuff/albert.html).

²Albert version 4.0M (modified); available online (http://justpasha.org/math/albert/).

³PARI/GP is available at http://pari.math.u-bordeaux.fr/.

⁴The corresponding sequence was submitted to the On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/) as A161391.

⁵The corresponding sequence was submitted to the On-Line Encyclopedia of Integer Sequences as A161392.

⁶The corresponding sequence was submitted to the On-Line Encyclopedia of Integer Sequences as A161393.