Published May 17, 2026 | Version version 1
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Operational Mathematics on Hopf Algebras: Extending the Parameter Domain of Operation Counts to Quantum Algebraic Structures

Authors/Creators

  • 1. ROR icon Peking University

Description

This monograph presents a complete, self-contained extension of Operational Mathematics– the theory of extending the repetition count of basic mathematical operations (addition,multiplication, exponentiation, tetration, and higher hyperoperations) from natural numbers to integers, rational numbers, real numbers, and complex numbers– to the setting of Hopf algebras. Hopf algebras unify algebra and coalgebra, encoding both multiplication (as in groups) and comultiplication (as in tensor products of representations). They form the algebraic backbone of quantum groups, noncommutative geometry, and renormalization in quantum field theory. We establish a fully rigorous axiomatic system consisting of six axioms (H1–H6) that capture the compatibility of the iteration semigroup with the Hopf structure: product, coproduct, antipode, counit, and the convolution algebra of endomorphisms. Unlike the scalar and matrix cases, the Hopf setting forces us to distinguish between composition of endomorphisms (for the iteration parameter) and convolution (for constructing inverses and derivatives). We resolve this by showing that for the relevant classes of Hopf algebras (e.g.,completions of universal enveloping algebras, Hopf algebras of formal power series, and the Connes–Kreimer Hopf algebra of rooted trees), the exponential map from the primitive Lie algebra provides a natural isomorphism between the additive group of parameters and the group of one-parameter automorphisms.The principal results include: 1. Integer-order operations form a pro-p group (in characteristic p) or a one-parameter group over C. 2. Fractional iteration is constructed using Hopf-algebraic Schröder and Abel functions with complete convergence proofs. 3. Real-order tetration is realized by the Hopf-algebraic Kneser construction, proving existence and uniqueness under a convexity condition. 4. Complex-order continuation reveals logarithmic branch points at negative integers and a natural boundary along (−∞,−1]; the Riemann surface is an infinite-sheeted covering branched at the negative integers. 5. Unification with Hopf-algebraic calculus: fractional derivatives, integrals, differences and summations are special cases of a single analytic semigroup. 6. Continuous hyperoperation spectrumisconstructed, and a No-GoTheoremshows the necessity of a piecewise construction. 7. Infinite iteration (t → ∞) converges to a universal attractor L independent of the hyperoperation level. 8. Duality between numbers and operations is formalized as a categorical equivalence, and the Schröder function is classified as a rank-1 étale (ϕ,Γ)-module. 9. Continuous Hasse–Weil zeta function is defined, and the Riemann hypothesis is reformulated in terms of its zeros. 10. Hopf-algebraic fractional calculus of variations yields Euler–Lagrange equations and a Noether theorem. 11. Numerical algorithms with exponential convergence are developed, with rigorous error analysis and verification on the Connes–Kreimer Hopf algebra. 12. Applications include quantum renormalization group, quantum anomalous diffusion,and post-quantum cryptography.

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Alternative title (English)
Operational Mathematics on Hopf Algebras

Dates

Submitted
2025-12-31

References

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