Homology of generalized partition posets

By Bruno Vallette. 2004

237 citations recorded by the source corpus.

Abstract We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen–Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/968d2776aaa9f04b1f87af00be09e34b02bb807a

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Citations in the corpus

Bar constructions for topological operads and the Goodwillie derivatives of the identity (2005, background): “ also the restriction of that on B(H∗P), it follows that H∗B(P) ∼=K(H∗P) is an isomorphism of cooperads. The dual result is proved similarly. Remark 9.49 Proposition 9.48 extends a result of Vallette [18] for discrete operads. Recall from Remark 8.7 that he constructs the ‘order complex’ for an oper”
Splitting of operations, Manin products and Rota-Baxter operators (2011, background): “A PostLie algebra[41] is a vector spaceA with a product◦ and a skew-symmetric operation [, ] satisfying the relations:”
On the Lie Enveloping Algebra of a Post-Lie Algebra (2014, unknown): “Post-Lie algebras appear in both geometry and algebra [22, 31].”
On the algebra of quasi-shuffles (2005, background): “iversalproperty: it is the free “CommutativeTriDendriform algebra”. This notion of CTD-algebra is a commutative version of the notion of dendriform algebras, which plays a role in several themes (cf. [2, 14, 21]). A CTD-algebra is determined by two operations, denoted ≺ (left) and · (dot), related b”
Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras (2017, methodology): “Post-Lie algebra Post-Lie algebras have been introduced by Valette in connection with the homology of partition posets and the study of Koszul operads [31].”
Generalized bialgebras and triples of operads (2006, methodology): “If the operad P is Koszul, then Vallette proved in [80]) the formula: f ! (−g(−t)) = t.”
Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, methodology): “algebra [43, 56] generated by one generator { }.”
Embedding of dendriform algebras into Rota-Baxter algebras (2011, background): “s e(n) i , i = 1,...,n. Every e(n) i can be identiﬁed with an associative and commutative poly-linear monomial in x 1,...,xn with one emphasized variable xi. Example 2. Operad CommTrias introduced in [36] is governing the variety of associative and commutative trialgebras, see [39, p. 25]. Namely, C”
Commutative post-Lie algebra structures on Lie algebras (2015, methodology): “Introduction Post-Lie algebras have been introduced by Valette in connection with the homology of partition posets and the study of Koszul operads [18].”
Affine actions on Lie groups and post-Lie algebra structures (2011, background): “Introduction Post-Lie algebras have been introduced recently by Vallette in [20], in connection with homology of partition posets, and the study of Koszul operads.”
Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras (2019, background): “The notion of a PostLie algebra arising from an operad study [42] and finding quite broad applications.”
Rota-Baxter operators on cocommutative Hopf algebras. (2020, unknown): “Definition [16].”
Operads of compatible structures and weighted partitions (2007, methodology): “on associated to bj 1 ...b j |Bj ai 1 1 ...a it mit ! . We call ΠP together with this partial order the operadic partition poset of P. Remark 2.7. We deﬁne the order in the opposite way to the one in [Val07] to make it correspond to the way it is deﬁned in [CV06]. Note that with this in mind our deﬁ”
Rota—Baxter groups, skew left braces, and the Yang—Baxter equation (2021, background): “A space A with two bilinear operations [, ] and · is called a post-Lie algebra [8, 37] if [, ] is a Lie bracket and the next two identities hold”
Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad (2006, methodology): “Definition 7. A finite poset M is said to be Cohen–Macaulay if Hi(M ) = 0 for all positive i less than the length of maximal chains in M . The following result appears in [ 17 ].”
Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras (2009, background): “(i) The notion of PostLie algebra was introduced in [51] (in i ts “right version”), where it is pointed out that PostLie, the operad of PostLie algebras, is the Koszul dual of ComTrias, the operad of commutative trialgebras.”