RESOLUTION OF THE ISOMORPHISM CONJECTURE FOR GRADED LEAVITT PATH ALGEBRAS OF DIRECTED GRAPHS WITH SINKS
Keywords:
Leavitt path algebra, Open problem solved, -graded isomorphism, Directed graph, Sink, Monoid invariant, -theory.Abstract
The Isomorphism Conjecture for Leavitt path algebras (LPAs) posits that for finite graphs, the algebraic structure of is completely determined by the graph's Cokernel group, representing the algebraic analogue of the Kirchberg-Phillips classification theorem for graph -algebras. While the conjecture has been intensively investigated for essential graphs (graphs lacking sinks and sources), its validity under the natural -grading for graphs containing terminal sinks remains an open and mathematically challenging problem. In this paper, we settle an open problem in the structural theory of Leavitt path algebras by proving that for any pair of finite directed graphs and possessing at least one sink, a -graded -algebra isomorphism exists if and only if there is a graded structural equivalence preserving both the path ideal growth and the graded -group framework. We establish this by explicitly constructing a monoid-theoretic invariant based on the path space projection of the boundary ideals. This result provides a complete classification of graded LPAs for finite graphs with sinks, filling a critical gap in non-commutative ring theory and symbolic dynamics.
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