
Case Study of Nonsymmetric Binary Cubic Operads 291
• the relations
(((∗∗)∗)∗) −φ((∗∗)(∗∗)) −(∗((∗∗)∗)) + φ(∗(∗(∗∗))) = 0,
((∗(∗∗))∗) −((∗∗)(∗∗)) −φ(∗((∗∗)∗)) + φ(∗(∗(∗∗))) = 0
where φ is a root of the polynomial t
2
− t −1.
Proof. Exercise 9.7.
Using operadic Gröbner bases, one can prove the following result on di-
mensions of components of the operads from the previous theorem.
Theorem 9.3.3.2. For the relations
(((∗∗)∗)∗) = 0,
((∗(∗∗))∗) + X((∗∗)(∗∗)) = 0
and
((∗∗)(∗∗)) + X(∗((∗∗)∗)) = 0,
(∗(∗(∗∗))) = 0
with X ∈ F
×
, the dimension of the n-th component of the corresponding operad
is equal to
1, n = 2,
2, n = 3,
3, n = 4,
5, n = 5,
6, n ≥ 6,
and