Gröbner Bases and Dimension Formulas for Ternary Partially Associative Operads

Bagherzadeh, Fatemeh; Bremner, Murray R.

doi:10.1007/978-981-15-1611-5_6

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“…In the odd case, as we have already seen for n = 3, the Gröbner basis is more complicated [2]; in fact, the corresponding operad is not Koszul [7]. LEMMA 4.1.…”

Section: Quaternary Casementioning

confidence: 86%

OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS

Bagherzadeh¹,

Bremner²

2019

Glasgow Math. J.

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The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

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“…In the odd case, as we have already seen for n = 3, the Gröbner basis is more complicated [2]; in fact, the corresponding operad is not Koszul [7]. LEMMA 4.1.…”

Section: Quaternary Casementioning

confidence: 86%