A Hopf Algebraic Theory of the Quantum Magnusian

Source abstract

We develop a Hopf-algebraic theory of the graph coefficients arising in the quantum Magnus expansion. At the classical level, the graph expansion is governed by directed trees, whereas its quantum counterpart involves loop graphs, multiple edges, and different types of edge data. We introduce a contraction Hopf algebra of mixed quivers containing these physical diagrams and closed under contraction. On the Hopf subalgebra spanned by quivers, we define a character $e$ from normalized linear-extension data and a tadpole prescription, and let $ω$ be its convolution inverse. Our main result is a universal closed formula for the connected function $ω_c$ on every finite quiver. The formula is a finite sum over ordered set partitions of the vertex set and is valid without acyclicity or simplicity assumptions. We also give a local characterization: a two-vertex contraction identity, together with tadpole factorization, vanishing on disconnected graphs, and boundary data, determines $ω_c$ uniquely. For acyclic quivers, the general formula reduces to a permutation formula whose coefficients depend only on descent numbers. We derive the same formula independently from the quantum Magnus expansion via operator products and Wick contractions, showing that its physical graph coefficients are governed by the convolution structure of the contraction Hopf algebra. As further consequences, we obtain a refined quantum Murua formula and orientation-sum identities related to Tutte and chromatic polynomials.