Jordan Triples and Operads

Gnedbaye, Allahtan Victor; Wambst, Marc

doi:10.1006/jabr.2000.8368

Cited by 9 publications

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“…Gnedbaye [31], A.V. Gnedbaye and M. Wambst [32], P. Hanlon and M. Wachs [35], Y. Manin [56]). The theory of V. Ginzburg and M. Kapranov does not work for these examples, but the generalized construction gives the right results.…”

Section: Introductionmentioning

confidence: 99%

Koszul duality of operads and homology of partition posets

Fresse¹

2004

Homotopy Theory: Relations With Algebraic Geometry, Group Cohomology, and Algebraic 𝐾-Theory

117

114

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Introduction Conventions Chapter 1. Composition products and operad structures 1.1. Introduction: monads and operads 1.2. Composition products and operad structures 1.3. The composition product of symmetric modules Chapter 2. Chain complexes of modules over an operad 2.1. Summary 2.2. Digression: model categories of modules over an operad 2.3. On composite symmetric modules in the differential graded framework 45 2.4. The spectral sequence of a quasi-free module 2.5. Proof of comparison theorems Chapter 3. The reduced bar construction 3.1. Summary: quasi-free operads and reduced bar constructions 57 3.2. Digression: the model category of operads 3.3. The language of trees 3.4. Trees and free operads 3.5. Trees and reduced bar constructions 3.6. Comparison of quasi-free operads: proof of theorem 3.2.1 Chapter 4. Bar constructions with coefficients 4.1. Summary 4.2. Composite symmetric modules and trees with levels 4.3. The simplicial bar construction 4.4. The differential graded bar construction 4.5. The levelization morphism 4.6. Proofs 4.7. Quasi-free resolutions of operads and bar constructions Chapter 5. Koszul duality for operads iii iv CONTENTS 5.1. Weight graded operads 5.2. Koszul operads 5.3. Koszul complexes and characterization of Koszul operads Chapter 6. Epilog: partition posets Bibliography Glossary Notation PROLOGOne can deduce from results of A. Björner that the (reduced) homology of partition posets vanishes in degree * = r − 1 (cf. [14]). Then, in [5], H. Barcelo defines an isomorphism of representations Hr−1 ( K(r), K) ≃ L(r) ∨ ⊗ sgn r , based on the Lyndon basis of the Lie operad (cf. M. Lothaire [50], C. Reutenauer [70]). This result is improved by P. Hanlon and M. Wachs in [35]. Namely, these authors define a natural morphism (which does not involve the choice of a basis of the Lie operad) from the dual of the Lie operad to the chain complex of the partition posetThis morphism fixes a representative of the homology class associated to a given element of the Lie operad. In addition, P. Hanlon and M. Wachs generalize the theorem above and give a relationship between partition posets and structures of Lie algebras with kary brackets.On the other hand, a topological proof of the theorem above, based on calculations of F. Cohen (cf. [20]), is given by G. Arone and M. Kankaanrinta in [2]. In connection with this result, we should mention that an article of G. Arone and M. Mahowald (cf. [3]) sheds light on the importance of partition posets in homotopy theory. Namely, these authors prove that the Goodwillie tower of the identity functor on topological spaces is precisely determined by partition posets. * (S(C)(V )) vanishes for general reasons while the Harrison homology H Harr * (S(C)(V )) does not (cf. M. Barr [6], D. Harrison [36], S. Whitehouse [85]). Consequently, the comparison morphism * ∈N V * equipped with a differential δ : V * → V * −1 which decreases degrees by 1. A dg-module is equivalent to a chain complex

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Section: Introductionmentioning

confidence: 99%

Koszul duality of operads and homology of partition posets

Fresse¹

2004

Homotopy Theory: Relations With Algebraic Geometry, Group Cohomology, and Algebraic 𝐾-Theory

117

114

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“…We denote by J ord 3 the operad of algebras defined by a Jordan triple system. In [5] one proves that this quadratic operad is of Koszul computing its dual. But the definition of the dual is based of the nongraded version of [4].…”

Section: Propositionmentioning

confidence: 99%

“…In [5] one proves that this quadratic operad is of Koszul computing its dual. But the definition of the dual is based of the nongraded version of [4].…”

Section: Propositionmentioning

confidence: 99%

“…Definition 1 Let E be a K[Σ 3 ]-module and F (E) the free operad over (5) and if (R) is the ideal generated by R, then the ternary quadratic operad P(E, R) generated by E and R is the quotient (5) generated by the vectors…”

Section: The Operad 3assmentioning

confidence: 99%

“…When computing the free algebras associated to (2p + 1)-ary partially associative algebras, using different arguments in [11] and [9], we saw that the odd and even cases behave in a completely different way. This approach is a little bit different to [4], [1] and [5] where odd and even cases have been studied in the same way and then contain some misunderstandings in the odd case. In fact, contrary to what we find in the previous papers, the operads associated to (2p + 1)-ary partially associative algebras are non Koszul so there is no operadic cohomology associated to these algebras (we call it operadic because it uses the dual operad to define a cohomology).…”

Section: Introductionmentioning

confidence: 99%

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On the non-Koszulity of ternary partially associative operad

Remm¹

2010

Proc. Estonian Acad. Sci.

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We want to present here the part of the work in common with Martin Markl [11] which concerns quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n = 3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework.We prove that the operad associated to (2p + 1)-ary partially associative algebra is not Koszul. Recall that, in the even case, this operad is Koszul.

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𝒪-operators of ternary Jordan algebras and ternary Jordan Yang–Baxter equations

Chtioui

Hajjaji

Mabrouk

2022

Asian-European J. Math.

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In this paper, we study the representation of ternary Jordan algebras which allows us to introduce the notion of coherent ternary Jordan algebras. Then the [Formula: see text]-operators of ternary Jordan algebras are introduced and the solutions of ternary Jordan Yang–Baxter equation are discussed involving [Formula: see text]-operators. Moreover, ternary pre-Jordan algebras are studied as the algebraic structure behind the [Formula: see text]-operators. Finally, the relations among ternary Jordan algebras and ternary pre-Jordan algebras are established and illustrated by examples.

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Jordan Triples and Operads

Cited by 9 publications

References 10 publications

Koszul duality of operads and homology of partition posets

Koszul duality of operads and homology of partition posets

On the non-Koszulity of ternary partially associative operad

𝒪-operators of ternary Jordan algebras and ternary Jordan Yang–Baxter equations

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