Blog

On Weak Lie 3-Algebras

Strongly homotopy Lie algebras or $\mathrm{L}_\infty$-algebras are a generalization of (dg) Lie algebras with better homotopy properties. In particular, a version of the homotopy transfer theorem holds. This is, however, only true when working over a field of characteristic $0.$ When working over an arbitrary unital commutative ring, one must generalize the definition even further, relaxing the skew-symmetry of the brackets up to homotopy in addition to the Jacobi identity.

I just finished my article “On weak Lie $3$-algebras,” in which I consider precisely this problem. I describe a general step by step approach and give explicit definitions for the case of underlying $3$-term complexes. In this post, I’ll try to give a rough idea of the approach used and a quick overview of the main results.

My first article (joint work with Bruno Vallette) has just appeared on arXiv! It deals with various constructions related to the homotopy theory of dg operads over arbitrary unital commutative rings: in particular, we define a higher cobar-bar adjunction and show that it provides us with a new cofibrant replacement functor, and introduce a notion of higher homotopy operads. Here, I would like to take the time to explain the motivation behind this work, sketch the approach we used, and talk about the meaning of the main results.

Algebraic Structures up to Homotopy

Roughly, by an algebraic structure I mean a space equipped with a bunch of structure maps which are subject to certain relations. In general, algebraic structures are rigid and do not behave well with respect to homotopy operations on their underlying space. However, some algebraic structures are sufficiently flexible and much better behaved in this regard. We call these algebraic structure up to homotopy. Sounds vague and confusing? In this post, we’ll consider a concrete and easy example: associative algebras.

Publications

On Weak Lie 3-Algebras

In this article, we introduce a category of weak Lie 3-algebras with suitable weak morphisms. The definition is based on the construction of a partial resolution over $\mathbb{Z}$ of the Koszul dual cooperad $\text{Lie}^\text{¡}$ of the $\text{Lie}$ operad, with free symmetric group action. Weak Lie 3-algebras and their morphisms are then defined via the usual operadic approach—as solutions to Maurer–Cartan equations. As 2-term truncations we recover Roytenberg’s category of weak Lie 2-algebras. We prove a version of the homotopy transfer theorem for weak Lie 3-algebras. A right homotopy inverse to the resolution is constructed and leads to a skew-symmetrization construction from weak Lie 3-algebras to 3-term $\textrm{L}_\infty$-algebras. Finally, we give two applications: the first is an extension of a result of Rogers comparing algebraic structures related to $n$-plectic manifolds; the second is the construction of a weak Lie 3-algebra associated to an LWX 2-algebroid leading to a new proof of a result of Liu–Sheng.

Preprint

The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy operads, for which we establish the relevant homotopy properties. For instance, the higher bar-cobar construction provides us with a cofibrant replacement functor for operads over any ring. All these constructions are produced conceptually by applying the curved Koszul duality for colored operads. This paper is a first step toward a new Koszul duality theory for operads, where the action of the symmetric groups is properly taken into account.

Preprint

Shifted $\text{L}_\infty$-Bialgebras

A Lie bialgebra is a pair consisting of a Lie algebra and a Lie coalgebra structure on a space $\mathfrak{g}$, such that the cobracket is a 1-cocycle for the adjoint representation of $\mathfrak{g}$ on $\bigwedge\mathfrak{g}$. When $\mathfrak{g}$ is finite-dimensional, such Lie bialgebras are in one-to-one correspondence with Manin triples $\mathfrak{d} = \mathfrak{g} \oplus \mathfrak{g}^*$. A third equivalent definition is as Maurer–Cartan elements of $\mathcal{S}(\mathfrak{g}^*)\otimes\mathcal{S}(\mathfrak{g})$ w.r.t. Kosmann-Schwarzbach’s big bracket construction.

We introduce a generalization of Kosmann-Schwarzbach’s construction to chain complexes with a certain shift in degree, i.e. we extend the big bracket to $\mathcal{B}_k(L) = \mathcal{S}(L^*)\otimes\mathcal{S}(L[k])$. We use this construction to give definitions of Lie $k$-bialgebras and Manin $k$-triples on underlying $k$-term complexes $L$ in such a way, that $L$ becomes a Lie $k$-algebra and $L[1-k]$ becomes a Lie $k$-coalgebra resp. $L’=L[1-k]^*$ a Lie $k$-algebra. It is shown that Lie $k$-bialgebras and Manin $k$-triples correspond to Maurer-Cartan elements in certain sub-DGLAs of $\mathcal{B}_k$.

Unpublished

Teaching

I currently have no teaching obligation. In previous semesters I have held teaching positions for the following courses.

Semester Course
2016 Symplectic Geometry
2015–16 Algebra
2014 Functional Analysis
2013–14 Analysis I
2013 Linear Algebra II
2012 Complex Analysis I
2011–12 Differential Geometry I
2011 Numbers and Number Theory
2010–11 Analysis III
2010 Analysis II
2009–10 Analysis I
2009 Precourse in Mathematics