Endofunctors and Poincaré–Birkhoff–Witt Theorems

Dotsenko, Vladimir; Tamaroff, Pedro

doi:10.1093/imrn/rnz369

Cited by 14 publications

(30 citation statements)

References 43 publications

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“…This approach allows vast generalizations, and new universal enveloping algebras can be constructed for other types of algebraic structures in this way. It can also generalize well-known theorems such as the Poincaré-Birkoff-Witt theorem in a functorial way, see [DT20].…”

Section: Introductionmentioning

confidence: 82%

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Curved operadic calculus

Lucio¹

2022

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Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational article is to introduce the notion of a curved operad, and to develop the operadic calculus at this new level. The algebraic side of the curved operadic calculus provides us with universal constructions: using a new notion of curved operadic bimodules, we construct curved universal enveloping algebras. Since there is no notion of quasi-isomorphism in the curved context, we develop the homotopy theory of curved operads using new methods. This approach leads us to introduce the new notion of a curved absolute operad, which is the notion Koszul dual to counital cooperads nonnecessarily conilpotent, and we construct a complete Bar-Cobar adjunction between them. We endow curved absolute operads with a suitable model category structure. We establish a magical square of duality functors which intertwines this complete Bar-Cobar construction with the Bar-Cobar adjunction between unital operads and conilpotent curved cooperads. This allows us to compute minimal cofibrant resolutions for various curved absolute operads. Using the complete Bar construction, we show a general Homotopy Transfer Theorem for curved algebras. Along the way, we construct the cofree cooperad not necessarily conilpotent. CONTENTSIntroduction 1 1. Recollections on different types of (co)operads 6 2. Curved operads and their curved algebras 13 3. Curved bimodules and universal functors 17 4. Curved cooperads and curved partial cooperads 23 5. The groupoid-colored level 25 6. Curved twisting morphisms and Bar-Cobar adjunctions at the operadic level 38 7. Counital partial cooperads up to homotopy and transfer of model structures 48 8. Duality functors and Koszul duality 53 9. Application: Homotopy transfer theorem for curved algebras 59 10. Appendix: What is an absolute partial operad ? 63 References 71

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Section: Introductionmentioning

confidence: 82%

“…Thus the universal enveloping algebra of a curved Lie algebra also satisfies the Poincaré-Birkoff-Witt property. See [DT20].…”

Section: Curved Bimodules and Universal Functorsmentioning

confidence: 99%

Curved operadic calculus

Lucio¹

2022

Preprint

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“…Freeness of the left module structure allows one to prove that free PreLie C -algebras are free as PreLie B -algebras, generalising known results like the second author's theorem [16]. Freeness of the right module can be used in the categorical framework for Poincaré-Birkhoff-Witt theorems developed in the first author's joint work with Tamaroff [14], implying a functorial PBW type theorem for universal enveloping PreLie C -algebras of PreLie B -algebras.…”

Section: Introductionmentioning

confidence: 91%

Enriched pre-Lie operads and freeness theorems

Dotsenko¹,

Foissy²

2020

Preprint

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In this paper, we study the operad of C-enriched pre-Lie algebras, defined for any Hopf cooperad C; it slightly generalises a similar notion defined by Calaque and Willwacher to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding operads of enriched pre-Lie algebras; we prove a criterion for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré-Birkhoff-Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.

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“…The theorem asserts that as a coalgebra, U(g) is isomorphic to S c (g), the cofree cocommutative coassociative conilpotent coalgebra generated by g (note that since we work over Q, the underlying object of S c (X) is the same as that of S(X) for all X in C). To prove the Poincaré-Birkhoff-Witt theorem in C, one may use the methods of [2] or [11] for the first version and the methods of [48,Appendix B] or [40] for the second version. We note that for an abelian group L equipped with a homomorphism p : L → Z 2 (for example, the parity Z → Z 2 ), one normally thinks of the category of L-graded vector spaces with the braiding morphism (the Koszul sign rule)…”

Section: Conventionsmentioning

confidence: 99%

“…It was observed in[34, §2.6] that it is possible to modify multiplication in H Q (non-canonically) to make it super-commutative. For that, one defines the parity map p : L → Z 2 , d → χ(d, d) (mod 2) and chooses a group homomorphism ε : L × L → µ {±1} such that(11) ε(d, e) = (−1) p(d)p(e)+χ(d,e) ε(e, d), d, e ∈ L.…”

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confidence: 99%

DT invariants from vertex algebras

Dotsenko¹,

Mozgovoy²

2021

Preprint

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We obtain a new interpretation of the cohomological Hall algebra H Q of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that H Q is naturally identified with the graded dual vector space of the principal free vertex algebra associated to the Euler form of Q; the product of H Q arises from an isomorphism of the latter with the universal enveloping vertex algebra of a certain vertex Lie algebra. This leads to a new interpretation of Donaldson-Thomas invariants of Q (and, in particular, re-proves their positivity), and to a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

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Endofunctors and Poincaré–Birkhoff–Witt Theorems

Cited by 14 publications

References 43 publications

Curved operadic calculus

Curved operadic calculus

Enriched pre-Lie operads and freeness theorems

DT invariants from vertex algebras

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