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

Markl, Martin; Remm, Elisabeth

doi:10.1007/s40062-014-0090-7

Cited by 21 publications

(40 citation statements)

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“…For the same technical reasons as in [35] we introduce auxiliary operads by means of operadic suspension: tCom n d := S ⊗ tCom n d+n−1 and Lie n d := S ⊗ Lie n d+n−1 . By a direct calculation in the endomorphism operad, we obtain the following result.…”

Section: Definition 35mentioning

confidence: 99%

Veronese powers of operads and pure homotopy algebras

Dotsenko

Markl

Remm

2019

European Journal of Mathematics

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We define the mth Veronese power of a weight graded operad P to be its suboperad P [m] generated by operations of weight m. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads P [m] are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie k-algebras and Lie triple systems. In the case of Lie k-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg-Kapranov power series test is inconclusive.Organisation of the paper. The paper is organised as follows. In Section 1, we recall the key relevant definitions of the theory of operads. In Section 2, we define Veronese powers of weight graded operads, prove some results about them, and provide counterexamples to some celebrated properties of Veronese powers of associative algebras. In Section 3, we relate our work to research of polynomial 2010 Mathematics Subject Classification. 18D50 (Primary), 18G55, 33F10, 55P48 (Secondary).Key words and phrases. Operad, Veronese power, homological purity, Koszul duality, Koszulness, Zeilberger's algorithm. The second author was supported by the EduardČech Institute P201/12/G028 and RVO: 67985840. 1 identities, both classical and recent. In Section 4, we define operads for pure homotopy P-algebras, and explain how they are related to Veronese powers by Koszul duality. Finally, in Section 5, we demonstrate that the ungraded versions of strong homotopy Lie algebras sometimes used in the literature possess worse homotopical properties than the strong homotopy Lie algebras obtained from the minimal model of the Lie operad, and show that those algebras provide an economic example where the positivity criterion for Poincaré series does not settle the matter of non-Koszulness.Acknowledgements. Some of the key results of this paper were obtained in CINVESTAV (Mexico City); the first author and the second author wish to thank that institution for the excellent working conditions. The authors are also grateful to Murray Bremner for his interest in our work (he communicated to us [5] that arrived at the same definition of Veronese powers independently, when trying to define what an n-ary algebra over an operad P is), and to Eric Hoffbeck on comments on the first draft of the paper.

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Section: Definition 35mentioning

confidence: 99%

Veronese powers of operads and pure homotopy algebras

Dotsenko

Markl

Remm

2019

European Journal of Mathematics

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“…Finally, it is known [17] that for a minimal model (T (E ), d) of any operad P, the series t − f E (t ) is the compositional inverse of the series f P (t ), so t exp(−t ) is the compositional inverse of f Com ▽ 0 Lie (t ), and…”

Section: Almost Composite Products and Almost Distributive Lawsmentioning

confidence: 99%

Algebraic structures of F-manifolds via pre-Lie algebras

Dotsenko

2018

Annali di Matematica

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“…Such a (V, m) is sometimes called a dg anti-associative degree +1 algebra [15]. is a derivation of V of degree k + 1.…”

Section: Strong Homotopy Inner Derivationmentioning

confidence: 99%

“…Now we consider (D k P) ¡## , which is isomorphic to (D k P) ¡ . As in (15), its elements arep ∈ P ¡ andφp ∈ ↑ k−1 P ¡ . The partial cocomposition ∆ (1) is dual to the partial composition γ (1) :…”

Section: Ifmentioning

confidence: 99%

Homotopy derivations

Doubek

Lada

2015

J. Homotopy Relat. Struct.

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We define a strong homotopy derivation of (cohomological) degree k of a strong homotopy algebra over an operad P. This involves resolving the operad obtained from P by adding a generator with "derivation relations". For a wide class of Koszul operads P, in particular Ass and Lie, we describe the strong homotopy derivations by coderivations and show that they are closed under the Lie bracket. We show that symmetrization of a strong homotopy derivation of an A ∞ algebra yields a strong homotopy derivation of the symmetrized L ∞ algebra. We give examples of strong homotopy derivations generalizing inner derivations.

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

Cited by 21 publications

References 25 publications

Veronese powers of operads and pure homotopy algebras

Veronese powers of operads and pure homotopy algebras

Algebraic structures of F-manifolds via pre-Lie algebras

Homotopy derivations

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