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On quantizable odd Lie bialgebras,
Letters in Mathematical Physics. 2016. Vol. 106. P. 1199–1215.
Motivated by the universal obstruction to the deformation quantization of Poisson structures in infinite dimensions we introduce the notion of quantizable odd Lie bialgebra. The main result of the paper is a construction of a highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so called quantizable Poisson structures.
Merkulov S., Communications in Mathematical Physics 1999 Vol. 205 P. 369–375
Added: October 1, 2026
Merkulov S., Compositio Mathematica 2005 Vol. 141 No. 5 P. 1238–1254
Added: October 1, 2026
Merkulov S., Communications in Mathematical Physics 2006 Vol. 262 P. 117–135
Added: October 1, 2026
Merkulov S., Vallette B., Journal fuer die reine und angewandte Mathematik 2009 Vol. 634 P. 51–106
Added: October 1, 2026
Merkulov S., Vallette B., Journal fuer die reine und angewandte Mathematik 2009 Vol. 636 P. 123–174
Added: October 1, 2026
Merkulov S., Shadrin S., Markl M., Journal of Pure and Applied Algebra 2009 Vol. 213 P. 496–535
Added: October 1, 2026
Merkulov S., Alm J., Journal of Noncommutative Geometry 2015 Vol. 9 No. 1 P. 185–215
Added: October 1, 2026
Merkulov S., Letters in Mathematical Physics 2016 Vol. 106 No. 2 P. 169–195
Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras. ...
Added: September 30, 2026
Merkulov S., Willwacher T., Campos R., Duke Mathematical Journal 2016 Vol. 165 No. 1 P. 2921–2989
We show the Koszulness of the properad governing involutive Lie bialgebras and also of the properads governing nonunital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, ...
Added: September 30, 2026
Merkulov S., Willwacher T., Communications in Mathematical Physics 2018 Vol. 364 P. 505–578
We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation ...
Added: September 30, 2026
Merkulov S., Willwacher T., Compositio Mathematica 2020 Vol. 156 P. 2111–2148
We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a ...
Added: September 30, 2026
Merkulov S., Journal of Pure and Applied Algebra 2026 Vol. 230 P. 1–19
We study the dual cyclic Hochschild complex $Cyc(A,\K)$ of a
(possibly, infinite-dimensional) $A_\infty$-algebra $(A,\mu)$ and prove
that any pre-Calabi-Yau extension $\pi$ of the given $A_\infty$ structure $\mu$ in $A$
induces on the cyclic cohomology of $(A,\mu)$ a representation of a new dg properad of {\em oriented}\, ribbon graphs. We compute the cohomology of that properad in terms of ...
Added: September 29, 2026
Khoroshkin A., Merkulov S., Communications in Mathematical Physics 2023 Vol. 404 No. 2 P. 597–628
We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck–Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application ...
Added: December 19, 2025
Merkulov S., Letters in Mathematical Physics 2023 Vol. 113 No. 3 Article 62
Added: December 19, 2025
Merkulov S., International Mathematics Research Notices 2023 No. 7 P. 5685–5727
Added: December 19, 2025
Merkulov S., Živković M., Letters in Mathematical Physics 2022 Vol. 112 No. 13
We prove that the action of the Grothendieck–Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. ...
Added: December 19, 2025
Talalaev D., Шарыгин Г. И., Journal of Noncommutative Geometry 2017 Vol. 11 No. 2 P. 741–756
In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra A so that certain Poisson-commutative subalgebra Cin it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of C with coefficients in A. In some ...
Added: October 28, 2020
Springer Nature Switzerland AG, 2018.
Our book is a selection of works presented at the Conference of Mathematical Physics “Kezenoi-Am 2016”. The Organising and Programme Committee of the conference tried to create a programme which embraces the variety of research directions inspired by the modern developments in Mathematical Physics and the theory of Integrable Systems. The authors of the included papers are well known mathematicians ...
Added: January 29, 2019
Khoroshkin A., Mekulov S., Willwacher T., Letters in Mathematical Physics 2016 Vol. 106 No. 9 P. 1199–1215
Motivated by the obstruction to the deformation quantization of Poisson structures in infinitedimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures. ...
Added: September 11, 2016