What does this research mean for the field?

The iterations of the adjoint representation and its inverse on Lie algebras can be rigorously extended to fractional, real, and complex orders, with their existence and uniqueness guaranteed under specific spectral conditions. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

This work aims to extend the theory of operational mathematics on Lie algebras by examining fractional iterations of the adjoint representation.

May 20, 2026Open Access

Operational Mathematics on Lie Algebras: Fractional Iterations of the Adjoint Representation and Its Inverse

Key Points

This work aims to extend the theory of operational mathematics on Lie algebras by examining fractional iterations of the adjoint representation.
Developed a theory encompassing integers, rationals, and complex numbers for the adjoint representation.
Utilized frameworks such as scalar and matrix representations along with p-adic and Hopf-algebraic methods.
Employed spectral decomposition and various equations to rigorously define powers associated with the adjoint operator.
Established existence and uniqueness of iterations across different orders (integer, fractional, real, complex).
Characterized the collapse phenomenon into subalgebras using spectral radius and weighted parameters.
Developed numerical algorithms with complexity analysis demonstrating exponential convergence.

Abstract

This monograph systematically establishes a theory of Operational Mathematics on Lie algebras, extending the number of repetitions of the Lie bracket (the adjoint representation) and its inverse (the anti-Lie bracket) from natural numbers to integers, rational numbers, real numbers, complex numbers, and even infinity.Using the scalar, matrix, p-adic and Hopf-algebraic frameworks as methodological blueprints, we exploit the exponential–logarithm correspondence between Lie algebras and Lie groups to translate iterations of the Lie bracket into iterations of group multiplication. Spectral decomposition, Schröder’s equation, Abel’s equation and Kneser’s construction are employed to rigorously define fractional-, real- and complex-order powers of ada. A weighted parametrisation is introduced to control the convergence/divergence threshold, and the resulting “collapse phenomenon”where the iteration sequence degenerates into a proper subalgebra (center, nilpotent ideal, or abelian subalgebra)– is characterized in terms of the spectral radius and the weight. We prove existence and uniqueness of integer-, fractional-, real and complex-order iterations under the essential assumption that ada is invertible or that the zero eigenvalue is semi-simple. The singularity structure (poles at negative integers) is described, and exponentially convergent numerical algorithms with detailed complexity analysis are developed. Deep connections with representation theory (fractional Casimir operators, fractional Weyl character formula), quantum groups (q-adjoint operators), integrable systems (fractional Lax pairs), gauge theory (fractional curvature), and the renormalisation group (infrared attractors) are explored. All open problems are treated rigorously: some are unconditionally proved, others are proved under standard conjectures (e.g., Schanuel’s conjecture), and the rest are formulated as precise conjectures with supporting evidence.

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