Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1605.01391

Higher enveloping algebras

Ben Knudsen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincaré-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ben Knudsen. 2018-04-24. Higher enveloping algebras. https://doi.org/10.2140/gt.2018.22.4013

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The fifth algebraic transfer in generic degrees and validation of a localized Kameko's conjecture

This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod--2 Steenrod algebra $\mathscr A$ in a generic family of degrees, together with applications to the fifth Singer algebraic transfer and a localized variation of Kameko's conjecture. As a topological illustration of the usefulness of the Steenrod algebra, we prove that $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ are not homotopy equivalent by showing that their mod--2 cohomologies are not isomorphic as $\mathscr A$-modules, and we further determine the homotopy type of the quotient $\mathbb{C}P^n/\mathbb{C}P^{\,n-2}$ for all $n\ge 3$. For the generic degrees under consideration, we determine the relevant cohit spaces and describe the associated $GL(5,\mathbb F_2)$-module structure. As a consequence, the fifth algebraic transfer is an isomorphism in an explicit infinite family of internal degrees. These results were independently verified by implementations in \texttt{SageMath} and \texttt{OSCAR}. We also study a localized form of Kameko's conjecture concerning the dimensions of the indecomposables $\mathbb F_2\otimes_{\mathscr A}\mathbb F_2[x_1,\ldots,x_m]$ relative to parameter vectors, and prove that this conjecture holds for all $m\ge 1$ in certain degrees.

math.AT↗
arXiv2026

A rigidity theorem for $π_*$-étale $\mathbb E_k$-algebras

We prove a rigidity theorem for $π_*$-étale $\mathbb E_k$-algebras over an $\mathbb E_{k+1}$-ring spectrum: the category of $π_*$-étale extensions of an $\mathbb E_k$-algebra is identified with the ordinary category of étale Dirac algebras over its graded homotopy Dirac ring. The proof develops a relative Goerss-Hopkins type obstruction theory in synthetic spectra, including an $I$-complete version. As an application, the completed obstruction theory constructs the $I_n$-complete $\mathbb E_3$-$MU_{(p)}$-algebra realization of the Lubin-Tate theory, hence an $\mathbb E_4$-orientation $MU_{(p)}\to E_n$.

math.AT↗
arXiv2026

Automated proofs of unstable Adams differentials

We present a computer-based approach to computing differentials in the unstable Adams spectral sequence by systematically applying the unstable Leibniz rule and naturality with respect to maps in the EHP sequence. We record our results in tables of upper and lower bounds on the orders of 2-primary unstable homotopy groups of spheres through the unstable 50-stem. We provide examples of proofs for several differentials and give a guide to interpreting the associated unstable Adams charts and flow chart diagrams for differential proofs.

math.AT↗