# Bigraded Lie Algebras Related to Multiple Zeta Values

### Mohamad Maassarani

Université de Strasbourg, France

## Abstract

We prove that the dihedral Lie coalgebra $D_{∙∙}:=⨁_{k≥m≥1}D_{m,k}$ corresponding to $D_{∙∙}(G)$ of Goncharov (Duke Math. J. **110** (2001), 397–487) for $G={e}$ is the bigraded dual of the linearized double shuffle Lie algebra $ls:=⨁_{k≥m≥1}ls_{m}⊂Q⟨x,z⟩$ of Brown (Compos. Math. **157** (2021), 529–572) whose Lie bracket is the Ihara bracket initially defined over $Q⟨x,z⟩$, by constructing an explicit isomorphism of bigraded Lie coalgebras $D_{∙∙}→ls_{∨}$, where $ls_{∨}$ is the Lie coalgebra dual (in the bigraded sense) to $ls$. The work leads to the equivalence between the two statements “$D_{∙∙}$ is a Lie coalgebra with respect to Goncharov’s cobracket formula in Goncharov (Duke Math. J. **110** (2001), 397–487)” and “$ls$ is preserved by the Ihara bracket''. We also prove folklore results from Brown (Compos. Math. **157** (2021), 529–572) and Ihara et al. (Compos. Math. **142** (2006), 307–338) (which apparently have no written proofs in the literature) stating that for $m≥2$, $D_{m,∙}:=⨁_{k≥m}D_{m,k}$ is graded isomorphic (dual) to the double shuffle space $Dsh_{m}:=⨁_{k≥m}Dsh_{m}(k−m)⊂Q[x_{1},…,x_{m}]$ (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307–338), and that the linear map $f_{m}:Q⟨x,z⟩_{m}→Q[x_{1},…,x_{m}]$, where $Q⟨x,z⟩_{m}$ is the space linearly generated by monomials of $Q⟨x,z⟩$ of degree $m$ with respect to $z$, given by $x_{n_{1}}z⋯x_{n_{m}}zx_{n_{m+1}}↦δ_{0,n_{m+1}}x_{1}⋯x_{n_{m}}$, with $δ_{a,b}$ the Kronecker delta, restricts to a graded isomorphism $fˉ _{m}:ls_{m}:=⨁_{k≥m}ls_{m}→Dsh_{m}$ (stated in Brown (Compos. Math. **157** (2021), 529–572)). Here, we establish three explicit compatible isomorphisms $D_{∙∙}→ls_{∨},D_{m∙}→Dsh_{m}$ and $fˉ _{m}:ls_{m}→Dsh_{m}$, where $Dsh_{m}$ is the graded dual of $Dsh_{m}$.

## Cite this article

Mohamad Maassarani, Bigraded Lie Algebras Related to Multiple Zeta Values. Publ. Res. Inst. Math. Sci. 58 (2022), no. 4, pp. 757–791

DOI 10.4171/PRIMS/58-4-4