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Cameron Kemp

Publications and source records attributed to Cameron Kemp.

4 recordsLinked to original sources

arXiv2026

Cartier integration of infinitesimal 2-braidings via 2-holonomy of the CMKZ 2-connection, II: The pentagonator

This is a continuation of the previous paper (arXiv:2508.01944) in this series. We recontextualise Cirio and Martins' work to motivate our fundamental conjecture that the Drinfeld-Kohno (Lie) 2-algebra has trivial cohomology. It is then shown that this conjecture implies the following: given a coherent totally symmetric infinitesimal 2-braiding $t$, every modification endomorphic on the zero transformation vanishes if it is made up of the four-term relationators and whiskerings by $t$. The power of such an implication is that, in our context, one need only construct the data of a braided monoidal 2-category and it will automatically satisfy the axioms. We thus conclude by constructing the pentagonator via Cirio and Martins' Knizhnik-Zamolodchikov 2-connection over the configuration space of 4 distinguishable particles on the complex line, $Y_4$. In particular, we make use of Bordemann, Rivezzi and Weigel's pentagon in $Y_4$.

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arXiv2026

Homotopy Lie algebras and coherent infinitesimal 2-braidings

Given a homotopy Lie algebra (i.e. an $L_\infty$-algebra) $\mathfrak{g}$, we show concretely how the Lada-Markl $\mathfrak{g}$-modules (i.e. representations) assemble into a symmetric monoidal dg-category. Considering the homotopy 2-category of that dg-category, we construct infinitesimal 2-braidings from 2-shifted Poisson structures then show that such infinitesimal 2-braidings are coherent in Cirio and Faria Martins' sense. We then explicitly determine the differential of the Chevalley-Eilenberg algebra associated with a finite-dimensional homotopy Lie algebra and construct the symmetric monoidal dg-equivalence between the category of representations and the category of semi-free dg-modules over the Chevalley-Eilenberg algebra.

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arXiv2024

Shifted Poisson structures on higher Chevalley-Eilenberg algebras

This paper develops a graphical calculus to determine the $n$-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the $(n=1)$- and $(n=2)$-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie $2$-algebra and obtain $n\in\{1,2,3,4\}$ shifted Poisson structures in this case, which we interpret as semi-classical data of `higher quantum groups'.

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arXiv2024

Infinitesimal 2-braidings from 2-shifted Poisson structures

It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free $A$-dg-modules. This provides a concrete realization, to first order in the deformation parameter $\hbar$, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, To\"en, Vaqui\'e and Vezzosi. Of particular interest is the case when $A$ is the Chevalley-Eilenberg algebra of a Lie $N$-algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.

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