A Lie algebra associated with adjoint multiple zeta values

Jarossay (arXiv math.NT1412.5099) introduced adjoint multiple zeta values and, by using Racinet's dual formulation of the generating series of multiple zeta values, found $\mathbb{Q}$-algebraic relations among them, referred to as the \textit{adjoint double shuffle relations}. Additionally, Jarossay defined the affine scheme $\mathrm{AdDMR}_0$ determined by the adjoint double shuffle relations and posed a question whether $\mathrm{AdDMR}_0$ is isomorphic to Racinet's double shuffle group $\mathrm{DMR}_0$ (Publ. Math. Inst. Hautes \'{E}tudes Sci. (2002), no. 95). In this paper, we refine Jarossay's question by introducing the condition referred to as the adjoint conditions, and, based on this refinement, we study the corresponding Lie algebraic aspect. Within this framework, we construct the Lie algebra associated with the adjoint double shuffle relations by imposing Hirose's parity results.