Related papers: Double shuffle Lie algebra and special derivations
According to Racinet's work, the scheme of double shuffle and regularization relations between cyclotomic analogues of multiple zeta values has the structure of a torsor over a pro-unipotent $\mathbb Q$-algebraic group $\sf{DMR}_0$, which…
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…
To describe the double shuffle relations between multiple polylogarithm values at $N$th roots of unity, Racinet attached to each finite cyclic group $G$ of order $N$ and each group embedding $\iota : G \to \mathbb{C}^{\times}$, a…
It is proved by L.~Schneps that the double shuffle Lie algebra $\mathfrak{dmr}_0$ injects to the Kashiwara-Vergne Lie algebra $\mathfrak{krv}_2$ in \cite{Schneps2012,Schneps2025}. We show that $\mathfrak{dmr}_0$ with the infinitesimal…
The real multiple zeta values $\zeta(k_1,\ldots,k_r)$ are known to form a ${\bf Q}$-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. In order to study the algebraic properties…
We explain the current situation of the relationship between the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ and the double shuffle Lie algebra $\mathfrak{dmr}$. We also show the validity of Ecalle's senary relation for small depths.
We prove that the double shuffle Lie algebra ds, dual to the space of new formal multiple zeta values, injects into the Kashiwara-Vergne Lie algebra krv defined and studied by Alekseev-Torossian. The proof is based on a reformulation of the…
In this article we prove that there exists an injective Lie morphism from the double shuffle Lie algebra ${\frak{ds}}$ into the Kashiwara-Vergne Lie algebra ${\frak{krv}}$, forming a commutative triangle with the known Lie injections of the…
It is proved that Drinfel'd's pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the Grothendieck-Teichm\"uller group $GRT_1$ into Racinet's double shuffle group $DMR_0$ is obtained, which…
This paper is the first in a series which aims at: (a) giving a proof that the associator relations between multizeta values imply the double shuffle and regularization (DSR) ones, alternative to that of the second-named author's 2010…
We study the reduced coaction Lie algebra $\mathfrak{rc}_0$, which is defined by an algebraic equation satisfied by the reduced coaction (an upgraded version of the necklace cobracket) and the skew-symmetric condition. We prove that the…
This paper draws connections between the double shuffle equations and structure of associators; universal mixed elliptic motives as defined by Hain and Matsumoto; and the Rankin-Selberg method for modular forms for $SL_2(\mathbb{Z})$. We…
The linearized double shuffle Lie algebra $\mathfrak{ls}$ is a well-studied Lie algebra, which reflects the depth-graded structure of multiple zeta values. We introduce a generalization $\mathfrak{lq}$, which is motivated from the…
A double extension ($\mathscr{D}$ extension) of a Lie (super)algebra $\mathfrak a$ with a non-degenerate invariant symmetric bilinear form $\mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $\mathfrak a$ by means of a…
In this article we define an elliptic double shuffle Lie algebra $ds_{ell}$ that generalizes the well-known double shuffle Lie algebra $ds$ to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the…
We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal…
The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The…
The Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ encodes symmetries of the Kashiwara-Vergne problem on the properties of the Campbell-Hausdorff series. It is conjectures that $\mathfrak{krv} \cong \mathbb{K}t \oplus \mathfrak{grt}_1$,…
We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $C_n$ and $D_n$, as well as their Lusztig and RTT integral forms, in the new Drinfeld realization. We also establish a shuffle algebra…
These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a…