A Lie algebra associated with adjoint multiple zeta values

Jarossay introduced adjoint multiple zeta values, and he found Q-algebraic relations among adjoint multiple zeta values, referred to as the adjoint double shuffle relations, by using Racinet's dual formulation of the generating series of multiple zeta values. Jarossay defined the affine scheme AdDMR₀ determined by the adjoint double shuffle relations and posed a question whether AdDMR₀ is isomorphic to Racinet's double shuffle group DMR₀. In this paper, we refine Jarossay's question by formulating what we call the adjoint conditions and by addressing its Lie algebraic side. Within this framework, we construct the Lie algebra associated with the adjoint double shuffle relations by imposing Hirose's parity results.