arXiv subjects

Silvan Schwarz

Publications and source records attributed to Silvan Schwarz.

2 recordsLinked to original sources

arXiv2022

An Obstruction Theory for the Existence of Maurer-Cartan Elements in curved $L_\infty$-algebras and an Application in Intrinsic Formality of $P_\infty$-Algebras

Let $\mathfrak{g}$ be a curved $L_\infty$-algebra endowed with a complete filtration $\mathfrak{F}\mathfrak{g}$. Suppose there exists an integer $r \in \mathbb{N}_0$ for which the curvature $μ_0$ satisfies $μ_0 \in \mathfrak{F}_{2r+1} \mathfrak{g}$ and the spectral sequence yields $E_{r+1}^{p,q} =0$ for $p,q$ with $p+q=2$. We prove that then a Maurer-Cartan element exists. In addition, we show, as a typical application, that for $P$ a possibly inhomogeneous Koszul operad with generating set in arities 1,2 (e.g. $P$=Com,As,BV,Lie,Ger), a $P_\infty$-algebra $A$ is intrinsically formal if its twisted deformation complex $\mathrm{Def}(H(A)\stackrel{\mathrm{id}}{\to} H(A))$ is acyclic in total degree 1.

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arXiv2022

A Variation of the Goldman-Millson Theorem for Filtered $L_\infty$ Algebras

In this paper, we extend the Goldman-Millson Theorem for $L_\infty$ algebras. We consider two $L_\infty$ algebras $L$ and $\tilde{L}$ endowed with descending, bounded above and complete filtrations compatible with the $L_\infty$ structures and ${U:L \rightarrow \tilde{L}}$ an $\infty$-morphism respecting the filtrations. We prove that in the setting of the linear part of $U$, say $ψ$, being a quasi-isomorphism on the r-1st page of the spectral sequences and ${H^1 ((\mathfrak{F}_{2^q} L)/(\mathfrak{F}_{\mathrm{min}(2^{q+1},r)} L))=0}$ for every $q$ with $2^q < r$ and ${H^i((\mathfrak{F}_1 \tilde{L}) / (\mathfrak{F}_q \tilde{L}))=0}$ for $i=0,1$ and $q$ every power of 2 smaller than $r$ and $q=r$ this induces a weak homotopy equivalence of the simplicial sets $\mathfrak{MC}_\bullet (L)$ and $\mathfrak{MC}_\bullet (\tilde{L})$.

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