Differential Algebraic Closure Framework for Algebraic Representation Theory: Explicit Solutions and Combinatorial Structures
shifa liu · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper extends the differential algebraic closure framework from group representation theory to the broader context of algebraic representation theory. We construct a rigorously defined algebraic representation closure KA for finite-dimensional associative algebras and Lie algebras, demonstrating that character decompositions, Ext groups, and representation-theoretic invariants can be explicitly expressed within this closure.We provide a detailed constructive framework with complete proofs,derive corrected combinatorial expressions with rigorous connections to symmetric group representations and quiver representations, present a complete algorithmic description with complexity analysis, and situate our results within classical algebraic representation theory and differential Galois theory. Numerical experiments across diverse algebra families (path algebras, hereditary algebras, Lie algebras) confirm spectral convergence and demonstrate the necessity of combinatorial corrections for higher-dimensional modules and derived equivalences.This work establishes that explicit analytic expressions exist in the appropriately constructed algebraic representation closure KA for a significant class of algebraic representation problems, providing a new algebraic perspective on representation-theoretic solvability while maintaining consistency with classical complexity results.