Notes on Cohomologies of Ternary Algebras of Associative Type

(Non-)Koszulness of operads for $$n$$ n -ary algebras, galgalim and other curiosities

“…Operads of this type are called quadratic, or binary quadratic if n = 2. 1 Let E ∨ = {E ∨ (a)} a≥2 be a Σ-module with E ∨ (a) := sgn a ⊗ ↑ a−2 E(a) # , if a = n and 0, otherwise where ↑ a−2 denotes the suspension iterated a−2 times, sgn a the signum representation of the symmetric group Σ a , and # the linear dual of a graded vector space with the induced representation. Recall that V # := Hom(V, k), so (V # ) d = (V −d ) # .…”

“…where R ⊥ ⊂ Γ(E ∨ )(2n − 1) is the annihilator of R ⊂ Γ(E)(2n − 1) in the pairing (1), and (R ⊥ ) the operadic ideal generated by R ⊥ .…”

“…The generators µ 1 5 , µ 2 5 , µ 3 5 , µ 4 5 correspond to these paths. Also for a ≥ 6 the 1-dimensional ∂-cycles in Γ(µ 2 , µ 3 , µ 1 5 , µ 2 5 , µ 3 5 , µ 4 5 )(a) 1 are given by closed edge paths of even length in the associahedron K a but one can show that they are all generated by the squares and the images of the paths as in Figure 5 under the face inclusions K 5 ֒→ K a . Therefore (Γ(µ 2 , µ 3 , µ 1 5 , µ 2 5 , µ 3 5 , µ 4 5 ), ∂) is acyclic in degree 1, so µ 1 5 , µ 2 5 , µ 3 5 , µ 4 5 are the only degree two generators of the minimal model of Ass.…”

“…Let us close this introduction by mentioning a couple of references bearing some relation to the present article, namely the work of H. Ataguema and A. Makhlouf [1], V. Dotsenko and A. Khoroshkin [2], A.V. Gnedbaye [8], E. Hoffbeck [11] and the talk given by J.-L. Loday at the Winter School in Srní, in January 2008.…”

Algebras with ternary law of composition and their realization by cubic matrices

“…The relations for different kinds of partial and total associativity of a ternary algebra and induced by it binary algebras are found in the terms of the structure constants of a ternary algebra. It should be pointed out that the cohomologies of a ternary algebra of associative type are studied in [5].…”