arXiv subjects

Annika Burmester

Publications and source records attributed to Annika Burmester.

7 recordsLinked to original sources

arXiv2026

Combinatorial multiple Eisenstein series

We construct a family of $q$-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given $\mathbb{Q}$-valued solution of the extended double shuffle equations. These $q$-series will be called combinatorial (bi-)multiple Eisenstein series, and in depth one they are given by Eisenstein series. The combinatorial multiple Eisenstein series can be seen as an interpolation between the given $\mathbb{Q}$-valued solution of the extended double shuffle equations (as $q\rightarrow 0$) and multiple zeta values (as $q\rightarrow 1$). In particular, they are $q$-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould.

math.NT↗

arXiv2026

A stabilizer interpretation of the (extended) linearized double shuffle Lie algebra

The linearized double shuffle Lie algebra introduced by Brown reflects the depth-graded structure of multiple zeta values. In a previous paper, the first author introduced an extension of this Lie algebra that accommodates multiple q-zeta values and multiple Eisenstein series. Inspired by the stabilizer interpretation of the double shuffle Lie algebra given by Enriquez and Furusho, we provide in this paper a stabilizer interpretation of both Lie algebras and show that the stabilizers preserve the extension from the first linearized Lie algebra to the second one.

math.NT↗

arXiv2025

Balanced multiple q-zeta values

We introduce the balanced multiple q-zeta values. They give a new model for multiple q-zeta values, whose product formula combines the shuffle and stuffle product for multiple zeta values in a natural way. Moreover, the balanced multiple q-zeta values are invariant under a very explicit involution. Thus, all relations among the balanced multiple q-zeta values are conjecturally of a very simple shape. Examples of the balanced multiple q-zeta values are the classical Eisenstein series, and they also contain the combinatorial multiple Eisenstein series. The construction of the balanced multiple q-zeta values is done on the level of generating series. We introduce a general setup relating Hoffman's quasi-shuffle products to explicit symmetries among generating series of words, which gives a clarifying approach to Ecalle's theory of bimoulds. This allows us to obtain an isomorphism between the underlying Hopf algebras of words related to the combinatorial bi-multiple Eisenstein series and the balanced multiple q-zeta values.

math.NT↗

arXiv2025

A generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values

We present the $τ$-invariant balanced quasi-shuffle algebra $\mathcal{G}^{\operatorname{f}}$, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular, $\mathcal{G}^{\operatorname{f}}$ has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra $\mathcal{Z}^f$ of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra $\mathcal{G}^{\operatorname{f}}$. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra $\mathcal{G}^{\operatorname{f}}$ onto $\mathcal{Z}^f$. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit $q\to1$ of multiple q-zeta values.

math.NT↗

arXiv2025

An extension of the linearized double shuffle Lie algebra

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 $\mathbb{Q}$-algebraic structure of multiple q-zeta values and multiple Eisenstein series. Precisely, we show that $\mathfrak{lq}$ is a Lie algebra, where the Lie bracket is related to Ecalle's ari bracket on bimoulds, and give an embedding of $\mathfrak{ls}$ into $\mathfrak{lq}$.

math.NT↗

arXiv2025

On post-Lie structures for free Lie algebras

We study post-Lie structures on free Lie algebras, the Grossman-Larson product on their enveloping algebras, and provide an abstract formula for its dual coproduct. This might be of interest for the general theory of post-Hopf algebras. Using a magmatic approach, we explore post-Lie algebras connected to multiple zeta values and their $q$-analogues. For multiple zeta values, this framework yields an algebraic interpretation of the Goncharov coproduct. Assuming that the Bernoulli numbers satisfy the so called threshold shuffle identities, we present a post-Lie structure, whose induced Lie bracket we expect to restrict to the dual of indecomposables of multiple $q$-zeta values. Our post-Lie algebras align with Ecalle's theory of bimoulds: we explicitly identify the ari bracket with a post-Lie structure on a free Lie algebra, and conjecture a correspondence for the uri bracket.

math.NT↗

arXiv2024

AGZT-Lectures on formal multiple zeta values

Formal multiple zeta values allow to study multiple zeta values by algebraic methods in a way that the open question about their transcendence is circumvented. In this note we show that Hoffman's basis conjecture for formal multiple zeta values is implied by the free odd generation conjecture for the double shuffle Lie algebra. We use the concept of a post-Lie structure for a convenient approach to the multiplication on the double shuffle group. From this, we get a coaction on the algebra of formal multiple zeta values. This in turn allows us to follow the proof of Brown's celebrated and unconditional theorem for the same result in the context of motivic multiple zeta values. We need the free odd generation conjecture twice: at first it gives a formula for the graded dimensions and secondly it is a key to derive a lift of the Zagier formula to the formal context.

math.NT↗