Bigraded Lie algebras related to MZVs

Abstract: We prove that Goncharov's dihedral Lie coalgebra

of the trivial group (

of (arxiv:math/0009121) for

G = e

) is the bigraded dual of Brown's linearized double shuffle Lie algebra

m a t h f r a k l s : = {o p l u s}_{k g e q m g e q 1} m a t h f r a k {l s}_{m}^{k} s u b s e t m a t h b b Q l a n g l e x, z a n g l e

whose Lie bracket is the Ihara bracket initially defined over

m a t h b b Q l a n g l e x, z a n g l e

. This by constructing an explicit isomorphism of bigraded Lie coalgebras

, where

m a t h f r a k {l s}^{v} e e

is the Lie coalgebra dual in the bigraded sense to

m a t h f r a k l s

. The work leads to the equivalence between the two statements: "

is a Lie coalgebra with respect to Goncharov's cobracket formula" and "

m a t h f r a k l s

is preserved by the Ihara bracket". We also prove folklore results (that apparently have no written proofs in the literature) stating that for

m g e q 2

:

is graded isomorphic (dual) to Ihara-Kaneko-Zagier's double shuffle space

m a t h r m {D s h}_{m} : = o p l u s_{k g e q m} m a t h r m {D s h}_{m} (k - m) s u b s e t m a t h b b Q [x_{1}, d o t s, x_{m}]

, and that a given linear map

f_{m} : m a t h b b Q l a n g l e x, z a n g l e_{m} o m a t h b b Q [x_{1}, d o t s, x_{m}]

, where

m a t h b b Q l a n g l e x, z a n g l e_{m}

is the space linearly generated by monomials of

m a t h b b Q l a n g l e x, z a n g l e

of degree

m

with respect to

z

, restricts to a graded isomorphism

. Here, we establish three explicit compatible isomorphisms

and

, where

m a t h r m {D s h}_{m}^{v} e e

is the graded dual of

m a t h r m {D s h}_{m}

.

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