An extension of the linearized double shuffle Lie algebra

TL;DR

This paper introduces a new Lie algebra $raklq$ extending the linearized double shuffle Lie algebra $rakls$, inspired by the algebraic structures of multiple q-zeta values and Eisenstein series, and relates it to Ecalle's ari bracket.

Contribution

It defines a generalized Lie algebra $raklq$ motivated by multiple q-zeta values, proves it is a Lie algebra, and embeds the classical $rakls$ into it.

Findings

$raklq$ is a Lie algebra related to multiple q-zeta values.

An embedding of $rakls$ into $raklq$ is established.

The Lie bracket in $raklq$ is connected to Ecalle's ari bracket.

Abstract

The linearized double shuffle Lie algebra $\mathfrak{ls}$ is a well-studied Lie algebra, which reflects the depth-graded structure of multiple zeta values. We introduce a generalization $\mathfrak{lq}$, which is motivated from the $\mathbb{Q}$-algebraic structure of multiple q-zeta values and multiple Eisenstein series. Precisely, we show that $\mathfrak{lq}$ is a Lie algebra, where the Lie bracket is related to Ecalle's ari bracket on bimoulds, and give an embedding of $\mathfrak{ls}$ into $\mathfrak{lq}$.