Lie algebras for multiple Eisenstein series and multiple q-zeta values

Abstract

Racinet's double shuffle Lie algebra ${\mathfrak{dm}}_0$ encodes the double shuffle relations of multiple zeta values. We study two analogues of ${\mathfrak{dm}}_0$ for multiple Eisenstein series and $q$-analogues of multiple zeta values, namely the space ${\mathrm{BARI}}_{\mathrm{swap},\mathrm{il}}$ of swap-invariant alternil bimoulds with the uri bracket of Kühn and Schneps, and Burmester's balanced double shuffle space ${\mathfrak{bm}}_0$. Both were conjectured to be Lie algebras. We prove both conjectures and show that ${\mathfrak{bm}}_0$ is isomorphic to the Lie algebra of finite-depth polynomial elements of ${\mathrm{BARI}}_{\mathrm{swap},\mathrm{il}}$.

arXiv comment22 pages. Comments are welcome!