[Paper Review] Quantum quasi-shuffle algebras II. Explicit formulas, dualization, and representations

Run-Qiang Jian|arXiv (Cornell University)|Feb 24, 2013

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper provides explicit formulas for the quantum quasi-shuffle algebra using mixable shuffles, constructs a dual braided coalgebra structure, and establishes representations on commutative braided Rota-Baxter algebras. A key contribution is the reformulation of multiple $q$-zeta values as evaluations of formal power series derived from these representations, linking quantum quasi-shuffle algebras to $q$-analogues of multiple zeta values.

ABSTRACT

Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product, as well as the subalgebra generated by primitive elements of the quantum quasi-shuffle bialgebra. We construct a braided coalgebra structure which is dual to the quantum quasi-shuffle algebra. We provide representations of quantum quasi-shuffle algebras on commutative braided Rota-Baxter algebras. As an application, we establish a formal power series whose terms come from a special representation of some kind of quasi-shuffle algebra and whose evaluation at 1 is the multiple $q$-zeta values.

Motivation & Objective

• To provide explicit formulas for the quantum quasi-shuffle product using the concept of mixable shuffles.
• To construct a braided coalgebra dual to the quantum quasi-shuffle algebra via the universal property of tensor algebras.
• To develop representations of quantum quasi-shuffle algebras on commutative braided Rota-Baxter algebras.
• To apply these representations to express multiple $q$-zeta values as evaluations of formal power series.

Proposed method

• Use mixable shuffles to explicitly describe the quantum quasi-shuffle product on the tensor algebra $T(V)$, generalizing the classical quasi-shuffle product.
• Define the subalgebra generated by primitive elements using mixable shuffles, showing its structure within the quantum quasi-shuffle bialgebra.
• Construct a braided coalgebra structure on $T(C)$ for a braided coalgebra $C$ using the universal property of tensor algebras.
• Prove that the dual of this braided coalgebra is isomorphic to the quantum quasi-shuffle algebra.
• Define an algebra map $\overline{f}$ from the positive part of the quantum quasi-shuffle algebra to a commutative braided Rota-Baxter algebra, enabling module actions.
• Use the map $\overline{f}$ to define formal power series whose evaluations at 1 yield $q$-analogues of multiple zeta values.

Experimental results

Research questions

• RQ1 How can the quantum quasi-shuffle product be explicitly formulated using mixable shuffles?
• RQ2 What is the structure of the subalgebra generated by primitive elements in the quantum quasi-shuffle bialgebra?
• RQ3 Can a dual braided coalgebra structure be constructed for the quantum quasi-shuffle algebra?
• RQ4 How do quantum quasi-shuffle algebras act on commutative braided Rota-Baxter algebras?
• RQ5 Can multiple $q$-zeta values be expressed as evaluations of formal series derived from these representations?

Key findings

• The quantum quasi-shuffle product is explicitly described using mixable shuffles, providing a constructive formula beyond inductive or universal property definitions.
• The subalgebra generated by primitive elements is characterized via mixable shuffles, offering a concrete realization within the bialgebra.
• A braided coalgebra structure on $T(C)$ is constructed as the dual of the quantum quasi-shuffle algebra, using the universal property of tensor algebras.
• The quantum quasi-shuffle algebra acts on commutative braided Rota-Baxter algebras via an algebra map $\overline{f}$, generalizing known actions on polynomial algebras.
• Formal power series $Z_q(i_1,\ldots,i_k;t)$ are defined such that their evaluation at $t=1$ yields Zudilin’s $q$-analogue of multiple zeta values.
• For admissible indices, the series $\zeta_q(i_1,\ldots,i_k;t)$ is constructed using $\overline{f}$ applied to $((1-q)t^n)^{*i}$, and its evaluation at $t=1$ recovers Bradley’s multiple $q$-zeta values.