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A stabilizer interpretation of the (extended) linearized double shuffle Lie algebra

Annika Burmester, Khalef Yaddaden

Abstract

The linearized double shuffle Lie algebra introduced by Brown reflects the depth-graded structure of multiple zeta values. In a previous paper, the first author introduced an extension of this Lie algebra that accommodates multiple q-zeta values and multiple Eisenstein series. Inspired by the stabilizer interpretation of the double shuffle Lie algebra given by Enriquez and Furusho, we provide in this paper a stabilizer interpretation of both Lie algebras and show that the stabilizers preserve the extension from the first linearized Lie algebra to the second one.

A stabilizer interpretation of the (extended) linearized double shuffle Lie algebra

Abstract

The linearized double shuffle Lie algebra introduced by Brown reflects the depth-graded structure of multiple zeta values. In a previous paper, the first author introduced an extension of this Lie algebra that accommodates multiple q-zeta values and multiple Eisenstein series. Inspired by the stabilizer interpretation of the double shuffle Lie algebra given by Enriquez and Furusho, we provide in this paper a stabilizer interpretation of both Lie algebras and show that the stabilizers preserve the extension from the first linearized Lie algebra to the second one.
Paper Structure (8 sections, 39 theorems, 142 equations)

This paper contains 8 sections, 39 theorems, 142 equations.

Table of Contents

  1. The linearized double shuffle Lie algebra
  2. Algebraic setup
  3. Stabilizer interpretation of the linearized double shuffle Lie algebra
  4. The linearized balanced Lie algebra
  5. Algebraic setup
  6. Stabilizer interpretation of the linearized balanced Lie algebra
  7. Lie algebra injection between the stabilizers
  8. Review of Lie algebra actions

Key Result

Proposition 1

There is a surjective algebra morphism which has a unique extension to $\mathbb{Q}\langle X\rangle$ such that $x_0\mapsto 0$ and $x_1\mapsto 0$.

Theorems & Definitions (77)

  • Proposition 1
  • Proposition 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5
  • Theorem 6: Theorem \ref{['thm:stab_equals_ls']}
  • Proposition 7
  • Conjecture 8
  • Theorem 9: Theorem \ref{['thm:stab_equal_lq']}
  • Theorem 10: Theorem \ref{['thm:theta:stabDeltaY_to_stabtau']}
  • ...and 67 more