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arXiv · 1810.04042

Gr\"obner bases and dimension formulas for ternary partially associative operads

Fatemeh Bagherzadeh·Murray Bremner
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Abstract

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger's algorithm for Gr\"obner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $({\ast}{\ast}{\ast})$, we compute a Gr\"obner basis for the ideal generated by partial associativity $((abc)de) + (a(bcd)e) + (ab(cde)$. In the category of $\mathbb{Z}$-graded vector spaces with Koszul signs, the (homological) degree of $({\ast}{\ast}{\ast})$ may be even or odd. We use the Gr\"obner bases to calculate the dimension formulas for these operads.

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Fatemeh Bagherzadeh, Murray Bremner. 2018-10-08. Gr\"obner bases and dimension formulas for ternary partially associative operads. https://doi.org/10.1007/978-981-15-1611-5_6

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