# Algebra, Topology and the Grothendieck-Teichmüller group

August 28, 2022 to September 2, 2022
Europe/Zurich timezone

## The stabilizer Lie algebra of the harmonic coproduct

Aug 30, 2022, 6:45 PM
15m

### Speaker

For a finite abelian group $G$, Racinet constructed a Lie algebra $\mathfrak{dmr}_0^G$, which for $G=\mu_N$ describes double shuffle and regularisation relations between multiple polylogarithm values specialized to $N^{th}$ roots of unity. Enriquez and Furusho then identified this Lie algebra with the stabilizer Lie algebra $\mathfrak{stab}(Delta^M)$ of a coalgebra $(M, Delta^M)$ appearing in Racinet's formalism. On the other hand, Racinet's formalism provides a Hopf algebra $(\mathbb{Q≪Y_G≫, Delta_*^{alg, G})$. When $G=1$, this Hopf algebra is equipped with a Lie algebra action, which gives rise to a stabilizer Lie algebra $\mathfrak{stab}(Delta_*^{alg,1})$, which can be shown to be equal to $\mathfrak{stab}(Delta^M)$. However, when $G neq 1$, no such action exists on $(\mathbb{Q≪Y_G≫, Delta_*^{alg, G})$, making a direct analogue of this construction impossible. We show how Racinet's theory fits in a crossed product formalism, which allows for the construction of an alternative generalisation $(W,Delta^W)$ for general $G$ of the Hopf algebra $(\mathbb{Q≪Y_G≫, Delta_*^{alg, G})$, which is moreover equipped with a Lie algebra action and, therefore, allows for the construction of a stabilizer Lie algebra $\mathfrak{stab}(Delta^W)$, which can be shown to contain $\mathfrak{dmr}_0^G$.