Pedro Tamaroff
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Pedro Tamaroff has a total of 4 co-authors. Here are the top co-authors by number of shared publications:
- Vladimir Dotsenko
Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France
5 shared publications - Sergey Shadrin
Korteweg-de Vriesinstituut voor Wiskunde, Universiteit van Amsterdam, Amsterdam, The Netherlands
2 shared publications - Anton Khoroshkin
Stony Brook University
1 shared publication - Vincent Gélinas
Montreal, Canada
1 shared publication
Latest Publications
- Open Access
- 31 October 2023
- Published by World Scientific Pub Co Pte Ltd in Bulletin of Mathematical Sciences
- Vol. 14 (03)
Tangent complexes and the Diamond Lemma
The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We revisit that result in the context of deformation theory and homotopical algebra; this leads to a new proof using multiplicative free resolutions. Specifically, our main result states that every such resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, where Bergman’s condition of “resolvable ambiguities” is precisely the first nontrivial component of the Maurer–Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing resolutions of algebras with monomial relations.
- 31 October 2023
- Published by World Scientific Pub Co Pte Ltd in Bulletin of Mathematical Sciences
- Vol. 14 (03)
Open AccessTangent complexes and the Diamond Lemma
The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We revisit that result in the context of deformation theory and homotopical algebra; this leads to a new proof using multiplicative free resolutions. Specifically, our main result states that every such resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, where Bergman’s condition of “resolvable ambiguities” is precisely the first nontrivial component of the Maurer–Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing resolutions of algebras with monomial relations.
- 25 January 2023
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 113 (1) , 1-20
Derived Poincaré–Birkhoff–Witt theorems
- 25 January 2023
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 113 (1) , 1-20
Derived Poincaré–Birkhoff–Witt theorems
- 19 December 2022
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 399 (3) , 1439-1500
Generalized Cohomological Field Theories in the Higher Order Formalism
- 19 December 2022
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 399 (3) , 1439-1500
Generalized Cohomological Field Theories in the Higher Order Formalism
- 16 December 2022
- Published by Springer Nature in Selecta Mathematica
- Vol. 29 (1) , 1-45
Finite generation for Hochschild cohomology of Gorenstein monomial algebras
- 16 December 2022
- Published by Springer Nature in Selecta Mathematica
- Vol. 29 (1) , 1-45
Finite generation for Hochschild cohomology of Gorenstein monomial algebras
- Open Access
- 3 March 2022
- Published by Springer Nature in Journal of Homotopy and Related Structures
- Vol. 17 (2) , 175-200
Resolutions of operads via Koszul (bi)algebras
We introduce a construction that produces from each bialgebra H an operad $$\mathsf {Ass}_H$$ controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra $${\mathscr {A}}$$ , then this notion of an associative H-algebra coincides with the usual notion of an $$\mathscr {A}$$ -algebra considered by homotopy theorists. This makes available to us an operad $$\mathsf {Ass}_{{\mathscr {A}}}$$ along with its minimal model that controls the category of associative $${\mathscr {A}}$$ -algebras, and the notion of strong homotopy associative $${\mathscr {A}}$$ -algebras.- 3 March 2022
- Published by Springer Nature in Journal of Homotopy and Related Structures
- Vol. 17 (2) , 175-200
Open AccessResolutions of operads via Koszul (bi)algebras
We introduce a construction that produces from each bialgebra H an operad $$\mathsf {Ass}_H$$ controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra $${\mathscr {A}}$$ , then this notion of an associative H-algebra coincides with the usual notion of an $$\mathscr {A}$$ -algebra considered by homotopy theorists. This makes available to us an operad $$\mathsf {Ass}_{{\mathscr {A}}}$$ along with its minimal model that controls the category of associative $${\mathscr {A}}$$ -algebras, and the notion of strong homotopy associative $${\mathscr {A}}$$ -algebras. - Open Access
Generalized cohomological field theories in the higher order formalism
In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by $\Delta$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $\Delta$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $\Delta$ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s.Open AccessGeneralized cohomological field theories in the higher order formalism
In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by $\Delta$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $\Delta$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $\Delta$ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s. - Open Access
- 6 January 2022
- Published by Taylor & Francis in Communications in Algebra
- Vol. 50 (7) , 2811-2830
The cohomology of coalgebras in species
Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory between twisted algebras and coalgebras on the twisted coalgebra that gives rise to their cohomology theory to give a new alternative description of it which, in particular, allows for its effective computation. We compute it completely in various examples, including those proposed by Aguiar and Mahajan, and obtain structural results: in particular, we study its multiplicative structure and provide a Künneth formula.
- 6 January 2022
- Published by Taylor & Francis in Communications in Algebra
- Vol. 50 (7) , 2811-2830
Open AccessThe cohomology of coalgebras in species
Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory between twisted algebras and coalgebras on the twisted coalgebra that gives rise to their cohomology theory to give a new alternative description of it which, in particular, allows for its effective computation. We compute it completely in various examples, including those proposed by Aguiar and Mahajan, and obtain structural results: in particular, we study its multiplicative structure and provide a Künneth formula.
- Open Access
- 6 February 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2021 (16) , 12670-12690
Endofunctors and Poincaré–Birkhoff–Witt Theorems
We determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a PBW property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.
- 6 February 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2021 (16) , 12670-12690
Open AccessEndofunctors and Poincaré–Birkhoff–Witt Theorems
We determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a PBW property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.