Given a directed graph, one can associate two algebraic entities: the Leavitt path algebra and the talented monoid. The Graded Classification conjecture states that the talented monoid could be a graded invariant for the Leavitt path algebra, i.e., isomorphism in the talented monoids reflects as graded equivalence in the category of graded modules over the Leavitt path algebra of the corresponding directed graphs. In this talk, we shall see confirmations of this invariance in the ideal structure of the talented monoid with the so-called Gelfand-Kirillov Dimension of the Leavitt path algebra. The last part of the talk is an affirmation of the Graded classification conjecture in the finite-dimensional case.
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