Bigraded Lie Algebras Related to Multiple Zeta Values

  • Mohamad Maassarani

    Université de Strasbourg, France
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We prove that the dihedral Lie coalgebra corresponding to of Goncharov (Duke Math. J. 110 (2001), 397–487) for is the bigraded dual of the linearized double shuffle Lie algebra of Brown (Compos. Math. 157 (2021), 529–572) whose Lie bracket is the Ihara bracket initially defined over , by constructing an explicit isomorphism of bigraded Lie coalgebras , where is the Lie coalgebra dual (in the bigraded sense) to . The work leads to the equivalence between the two statements “ is a Lie coalgebra with respect to Goncharov’s cobracket formula in Goncharov (Duke Math. J. 110 (2001), 397–487)” and “ 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 , is graded isomorphic (dual) to the double shuffle space (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307–338), and that the linear map , where is the space linearly generated by monomials of of degree with respect to , given by , with the Kronecker delta, restricts to a graded isomorphism (stated in Brown (Compos. Math. 157 (2021), 529–572)). Here, we establish three explicit compatible isomorphisms and , where is the graded dual of .

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