Derived Poincaré–Birkhoff–Witt theorems

Khoroshkin, Anton; Tamaroff, Pedro

doi:10.1007/s11005-022-01617-z

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…This was resolved by Shoikhet in [Sho01] by showing that wheels can be set to zero and thus we get a map of the desired form. This result fully agrees with the PBW theorem obtained in [KT23] using the bar-cobar duality in the homotopy theory.…”

Section: It Has Been Shownsupporting

confidence: 90%

“…Moreover, any other attempt to construct such a generalization satisfying some natural conditions must be gauge equivalent to these ones as the cohomology of the complex Def (Ass ∞ → O(Lie ∞ )) is one-dimensional (see Corollary 7.8). This cohomology result is in a full agreement with the derived Poincaré-Birkhoff-Witt theorem established by A. Khoroshkin and P. Tamaroff in [KT23].…”

Section: Introductionsupporting

confidence: 89%

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From the Lie operad to the Grothendieck-Teichmüller group

Wolff¹

2023

Preprint

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We study the deformation complex of the standard morphism from the degree d shifted Lie operad to its polydifferential version, and prove that it is quasi-isomorphic to the Kontsevich graph complex GC d . In particular, we show that in the case d = 2 the Grothendieck-Teichmüller group GRT 1 is a symmetry group (up to homotopy) of the aforementioned morphism. We also prove that in the case d = 1 corresponding to the usual Lie algebras the standard morphism admits a unique homotopy non-trivial deformation which is described explicitly with the help of the universal enveloping construction. Finally we prove the rigidity of the strongly homotopy version of the universal enveloping functor in the Lie theory.

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Section: It Has Been Shownsupporting

confidence: 90%

Section: Introductionsupporting

confidence: 89%

From the Lie operad to the Grothendieck-Teichmüller group

Wolff¹

2023

Preprint

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“…This result ultimately leads to a question of a definition of a homotopy invariant notion of the universal enveloping pre-Lie algebra (of a Lie algebra up to homotopy). A general framework for addressing questions like that is discussed in a paper of Khoroshkin and Tamaroff [ 22 ]; it would be interesting to apply their methods in this particular case.…”

Section: The Universal Enveloping Pre-lie Algebra Of a Lie Algebramentioning

confidence: 99%

Three Schur functors related to pre-Lie algebras

DOTSENKO,

FLYNN-CONNOLLY

2023

Math. Proc. Camb. Phil. Soc.

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We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the Poincaré-Birkhoff-Witt (PBW) theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.

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From the Lie Operad to the Grothendieck–Teichmüller Group

Wolff

2023

International Mathematics Research Notices

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We study the deformation complex of the natural morphism from the degree $d$ shifted Lie operad to its polydifferential version, and prove that it is quasi-isomorphic to the Kontsevich graph complex $\textbf {GC}_{d}$. In particular, we show that in the case $d=2$ the Grothendieck–Teichmüller group $\textbf {GRT}_{1}$ is a symmetry group (up to homotopy) of the aforementioned morphism. We also prove that in the case $d=1$, corresponding to the usual Lie algebras, the natural morphism admits a unique homotopy non-trivial deformation, which is described explicitly with the help of the universal enveloping construction. Finally, we prove the rigidity of the strongly homotopy version of the universal enveloping functor in the Lie theory.

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Derived Poincaré–Birkhoff–Witt theorems

Cited by 3 publications

References 18 publications

From the Lie operad to the Grothendieck-Teichmüller group

From the Lie operad to the Grothendieck-Teichmüller group

Three Schur functors related to pre-Lie algebras

From the Lie Operad to the Grothendieck–Teichmüller Group

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