Under the hood of a carbon footprint calculator
We explain the mathematical theory of the Input-Output method for carbon footprints computations.
math.OC↗
arXiv subjects
Elise Raphael
Publications and source records attributed to Elise Raphael.
On the elliptic Kashiwara-Vergne Lie algebra
We recall the definitions of two independently defined elliptic versions of the Kashiwara-Vergne Lie algebra $\frak{krv}$, namely the Lie algebra $\frak{krv}^{(1,1)}$ constructed by A.Alekseev, N.Kawazumi, Y.Kuno and F.Naef arising from the study of graded formality isomorphisms associated to topological fundamental groups of surfaces, and the Lie algebra $\frak{krv}_{ell}$ defined using mould theoretic techniques arising from multiple zeta theory by E.Raphael and L.Schneps, and show that they coincide.
math.QA↗
On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra
The goal of this article is to define a linearized or depth-graded version $\mathfrak{lkv}$, and a closely related elliptic version $\mathfrak{krv}_{ell}$, of the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra $\mathfrak{krv}_{ell}$ is related to earlier constructions of elliptic versions $\mathfrak{grt}_{ell}$ and $\mathfrak{ds}_{ell}$ of the Grothendieck-Teichm\"uller Lie algebra $\mathfrak{grt}$ and the double shuffle Lie algebra $\mathfrak{ds}$. In particular we show that there is an injective Lie morphism $\mathfrak{ds}_{ell}\hookrightarrow \mathfrak{krv}_{ell}$, and an injective Lie algebra morphism $\mathfrak{krv}\rightarrow \mathfrak{krv}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow\mathfrak{grt}_{ell}$ (Enriquez section) and $\mathfrak{ds}\rightarrow\mathfrak{ds}_{ell}$ (\'Ecalle map).
math.QA↗
Divergence and q-divergence in depth 2
The Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ encodes symmetries of the Kashiwara-Vergne problem on the properties of the Campbell-Hausdorff series. It is conjectures that $\mathfrak{krv} \cong \mathbb{K}t \oplus \mathfrak{grt}_1$, where $t$ is a generator of degree 1 and $\mathfrak{grt}_1$ is the Grothendieck-Teichmüller Lie algebra. In the paper, we prove this conjecture in depth 2. The main tools in the proof are the divergence cocycle and the representation theory of the dihedral group $D_{12}$. Our calculation is similar to the calculation by Zagier of the graded dimensions of the double shuffle Lie algebra in depth 2. In analogy to the divergence cocycle, we define the super-divergence and $q$-divergence cocycles (here $q^l=1$) on Lie subalgebras of $\mathfrak{grt}_1$ which consist of elements with weight divisible by $l$. We show that in depth $2$ these cocycles have no kernel. This result is in sharp contrast with the fact that the divergence cocycle vanishes on $[\mathfrak{grt}_1, \mathfrak{grt}_1]$.
math.QA↗