Perron fields and core-preserving homomorphisms of Leavitt path algebras, with an application to the Cuntz splice
Year: 2026
Type: preprint
Abstract: Let E and F be finite graphs, and grade the Leavitt path algebras L_K(E) and L_K(F) by positive integer weights on the edges. Assume that the graphs obtained by subdividing each edge according to its weight are primitive. We prove that if some nonzero ring homomorphism carries the degree-zero subalgebra (the core) of L_K(E) into the core of L_K(F), then two conditions hold. First, the Perron root λ_E of the subdivided graph of E lies in the number field ℚ(λ_F). Second, every prime ideal of the ring of integers of ℚ(λ_F) that divides λ_E also divides λ_F. Neither unitality nor linearity over K is assumed. The proof passes through the K₀-group of the core, the uniqueness of its state, and the group of values of that state, which is a module over ℤ[λ, λ⁻¹] inside ℚ(λ). Consequences: For every n ≥ 2, no nonzero ring homomorphism between the Leavitt algebra L_n and the Leavitt path algebra L_{n⁻} of its Cuntz splice carries the core into the core, in either direction. This extends the case m = ∞ of the theorem of Arnone and Cortiñas to non-unital maps and to maps graded only on the core. A core-preserving homomorphism L_k → L_m exists only if every prime factor of k divides m. An isomorphism that maps the core onto the core forces equal Perron fields and equal prime-ideal divisors. For the Cuntz splice of the rose with two petals, with every edge weight in {1, 2, 3}, the criteria exclude both directions against the standard grading of L₂ for 4,232 of the 4,340 aperiodic weightings. The remaining 108 cases are classified by type. The question whether L₂ and L_{2⁻} are isomorphic, the central open case of the algebraic Kirchberg–Phillips question, remains open. Every deterministic step is verified in Lean 4 with Mathlib or cited to a standard source.
Author: Chuang, William Huanshan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225463158