FROM MATRICES TO LEAVITT PATH ALGEBRAS: BUILDING MAXIMAL COMMUTATIVE SUBALGEBRAS

Speaker(s)

MICHAŁ ZIEMBOWSKI

Affiliation

Politechnika Warszawska, Warszawa, Poland

Language of the talk

English

Date

Oct. 8, 2025, 5:15 p.m.

Information about the event

IMPAN - Room 405

Title in Polish

FROM MATRICES TO LEAVITT PATH ALGEBRAS: BUILDING MAXIMAL COMMUTATIVE SUBALGEBRAS

Seminar

North Atlantic Noncommutative Geometry Seminar

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We move from the classical matrix setting to new constructions in Leavitt path algebras to investigate maximal commutative subalgebras. As a starting point, we recall the Schur-Jacobson viewpoint on maximal commutative subalgebras of matrix algebras, and note how the matrix picture connects to graphs through the standard graph-to-matrix correspondence. Our main focus is on two recent developments for Leavitt path algebras. First, for prime Leavitt path algebras, we construct a broad family of maximal commutative subalgebras determined by a partition of the set of vertices, thus recovering the expected matrix-type extremal behavior via the path-matrix isomorphism. On the way, we clarify how vertex partitions govern maximal commutativity. Next, we extend the aforementioned method to row-finite graphs, again obtaining maximal commutative subalgebras through vertex partitions. To this end, first we make robust ``tree-intersection'' assumptions, but then we find out how ro remove them to obtain the result in full generality. Summarizing, we show that simple vertex partition data provides a unified and effective tool for constructing and understanding maximal commutative subalgebras in Leavitt path algebras.

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