Ukrainian mathematical congress - 2009

Vladimir Dotsenko (Trinity College Dublin & Dublin Institute for Advanced Studies, Ireland)

Gröbner bases for operads

We define a new monoidal structure on (nonsymmetric) collections - shuffle composition of collections. Monoids in the category of collections with this structure (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. These results can be applied to symmetric operads; in particular, we obtain an effective algorithmic criterion of Koszulness for (symmetric) quadratic operads.

The talk is based on a joint work with A. Khoroshkin.