# Lecture series: An algebraic approach to multiple q-zeta values

Time: 13th (13:00-16:30) & 14th (9:30 - 16:30) February 2023.

Place: Room 409, Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya, 464-8602 Japan.

## Abstract: The goal of this lecture series is to present the results obtained in the speaker's thesis [B].

Part 1: Elementary theory of multiple q-zeta values (13th February)

We will introduce the algebra of multiple q-zeta values and explain its relations to partition functions and multiple zeta values. At first we present two spanning sets of this algebra, the Schlesinger-Zudilin multiple q-zeta values and the bi-brackets obtained by Bachmann. This will motivate our rough overview on quasi-shuffle algebras and the theory of bimoulds. Then by using the bi-brackets and a rational solution of the double shuffle equations, we construct the combinatorial (bi-)multiple Eisenstein series. Those q-series also span the algebra of multiple q-zeta values and can be seen as a weight-graded version of the bi-brackets. In particular, the combinatorial (bi-)multiple Eisenstein series give a simple description of a conjectural weight-grading on the algebra of multiple q-zeta values. Moreover, we will obtain also a weight-graded version of the Schlesinger-Zudilin multiple q-zeta values, the so-called balanced multiple q-zeta values. Those satisfy very simple relations and their product formula is a combination of the two product formulas of multiple zeta values. We will also give a regularized version of the balanced multiple q-zeta values, which will be needed to define formal multiple q-zeta values.

1.1. The algebra of multiple q-zeta values

1.2. Quasi-shuffle products

1.3. Schlesinger-Zudilin multiple q-zeta values

1.4. Bi-brackets

1.5. Bimoulds and their symmetries

1.6. Combinatorial (bi-)multiple Eisenstein series

1.7. Balanced multiple q-zeta values

1.8. Regularization and formal multiple q-zeta values

Part 2: Multiple q-zeta values in a broader perspective (14th February)

We start with a short introduction into affine group schemes and Hopf algebras, which give the basics for Racinet’s approach to multiple zeta values. More precisely, Racinet showed that the algebra of formal multiple zeta values essentially represents a pro-unipotent affine group scheme DM_0. A key ingredient was to show the corresponding linearized space dm_0 is a Lie subalgebra of the twisted Magnus Lie algebra. A consequence of this approach is that the algebra of formal multiple q-zeta values is a free polynomial algebra.

We will give a similar approach to the algebra of formal multiple q-zeta values, which formalizes the expected algebraic structure of the balanced multiple q-zeta values. In particular, we observe that there is an affine scheme BM_0, which is essentially represented by the algebra of formal multiple q-zeta values. Linearizing the defining equations of the affine scheme BM_0 leads to a space bm_0 of non-commutative polynomials, which we expect to be a Lie algebra. We will present some details how to compute this space, this allows to test some conjectures regarding the dimensions of the homogeneous subspace and particular families of relations. Moreover, we will introduce the q-twisted Magnus Lie algebra, which should be seen as a generalization of the twisted Magnus Lie algebra and which contains the conjectural Lie algebra bm_0. We will end by a comparison with a second approach to this Lie algebra related to multiple q-zeta values, which was proposed by Kühn and uses the theory of bimoulds.

2.1. Affine group schemes and Hopf algebras

2.2. Racinet’s approach to multiple zeta values

2.3. Formal multiple q-zeta values and the corresponding affine scheme BM_0

2.4. A conjectural Lie algebra bm_0 to multiple q-zeta values

2.5. Explicit computations in bm_0

2.6. The q-twisted Magnus Lie algebra

2.7. A conjectural Lie algebra of bimoulds related to multiple q-zeta values

2.8. Comparison of the approaches via non-commutative polynomials and bimoulds

## Schedule: All talks will take place in Room 409 (4th floor) of the Graduate School of Mathematics

Monday 13th February 2023

13:00 - 14:30: Lecture 1 (Elementary theory of multiple q-zeta values I)

15:00 - 16:30: Lecture 2 (Elementary theory of multiple q-zeta values II)

Tuesday 14th February 2023

9:30 - 11:00: Lecture 3 (Multiple q-zeta values in a broader perspective I)

13:00 - 14:30: Lecture 4 (Multiple q-zeta values in a broader perspective II)

15:00 - 16:30: Lecture 5 (Multiple q-zeta values in a broader perspective III)

Notice that from Thursday 16th February, there will be the 16th Young Mathematicians Conference on Zeta Functions held at Nagoya University.

## References:

[B] A. Burmester: An algebraic approach to multiple q-zeta values, PhD Thesis, Universität Hamburg, 2022.

[BB] H. Bachmann, A. Burmester: Combinatorial multiple Eisenstein series, preprint, 2022.

A detailed list of references in given in [B].