arXiv subjects

Pedro Tamaroff

Publications and source records attributed to Pedro Tamaroff.

10 recordsLinked to original sources

arXiv2022

The Tamarkin--Tsygan calculus of an algebra a la Stasheff

We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a `small' $\mathsf{Calc}_\infty$-model of its calculus, which we make somewhat explicit at the level of $\mathsf{Calc}$-algebras. To do this, we prove that the operad $\mathsf{Calc}$ is inhomogeneous Koszul; to our best knowledge, this result is new. We illustrate our technique by carrying out some computations for two monomial associative algebra using the cofibrant replacement obtained by the author in 1804.01435.

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arXiv2022

Generalized cohomological field theories in the higher order formalism

In the classical Batalin--Vilkovisky formalism, the BV operator $Δ$ 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 $Δ$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $Δ$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $Δ$ 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.

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arXiv2021

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.

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arXiv2020

Differential forms on smooth operadic algebras

The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace algebras, respectively. We also prove an HKR theorem for operads obtained from a filtered distributive law, which recovers, in particular, all the aspects of the classical HKR theorem. Finally, we show that this property is Koszul dual to the operadic PBW property defined by V. Dotsenko and the second author (1804.06485).

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arXiv2020

Minimal models for monomial algebras

Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_\infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jöllenbeck, Welker and Sköldberg. We then show how this result can be used to obtain models for algebras with a chosen Gröbner basis, and briefly outline how to compute some classical homological invariants with it.

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arXiv2020

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 present a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of "resolvable ambiguities" becomes the first non-trivial 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 multiplicative free resolutions of algebras with monomial relations.

math.RA↗

arXiv2020

A spectral sequence for tangent cohomology of algebras over algebraic operads

Operadic tangent cohomology generalizes the existing cohomology theories of Chevalley--Eilenberg, Hochschild, and Harrison to address the deformation theory of general types of algebras through gadgets known as deformation complexes. The cohomology of these is in general very non-trivial to compute, and in this paper we complement the existing computational techniques by producing a spectral sequence that converges to the operadic cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. We also consider relative Sullivan--de Rham models of a fibration $p$, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber-homotopy equivalences of $p$.

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arXiv2020

Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)

We define derived Poincaré--Birkhoff--Witt maps of dg operads or derived PBW maps, for short, which extend the definition of PBW maps between operads of V.~Dotsenko and the second author in 1804.06485, with the purpose of studying the universal enveloping algebra of dg Lie algebras as a functor on the homotopy category. Our main result shows that the map from the homotopy Lie operad to the homotopy associative operad is derived PBW, which gives us an amenable description of the homology of the universal envelope of an $L_\infty$-algebra in the sense of Lada--Markl. We deduce from this several known results involving universal envelopes of $L_\infty$-algebras of V. Baranovsky and J. Moreno-Fernández, and extend D. Quillen's classical quasi-isomorphism $\mathcal C \longrightarrow BU$ from dg Lie algebras to $L_\infty$-algebras; this confirms a conjecture of J. Moreno-Fernández.

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arXiv2019

Endofunctors and Poincaré-Birkhoff-Witt theorems

We determine what appears to be the bare-bones categorical framework for Poincaré-Birkhoff-Witt 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 Poincaré-Birkhoff-Witt 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 Poincaré-Birkhoff-Witt type theorems. In particular, we prove a PBW type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.

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arXiv2019

Finite generation for Hochschild cohomology of Gorenstein monomial algebras

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann-Holloway-Snashall-Solberg-Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

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