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Bergman realizability of Kumjian–Pask algebras

Determine the class of k-graphs Λ for which the Kumjian–Pask algebra KP_k(Λ) can be realised as a Bergman algebra; equivalently, characterize precisely which Kumjian–Pask algebras arise from Bergman’s universal algebra construction.

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Background

In contrast to the 1-graph (Leavitt path algebra) case, where many algebras can be expressed as Bergman algebras and their K_0 groups described systematically, the situation for higher-rank graphs is unclear. The authors point out that no general description is known of which Kumjian–Pask algebras admit a Bergman realization.

This gap has concrete consequences: without such a realization, there is no general, systematic method to compute the (non-graded) Grothendieck group K_0(KP_k(Λ)) directly from a higher-rank graph Λ, even though the graded version K_0{gr} can be approached via the talented monoid and groupoid techniques.

References

The class of Kumjian–Pask algebras that can be realised as Bergman algebras remains unknown. Therefore, there is not yet a systematic way to describe the Grothendieck group $K_0(KP_k(\Lambda))$ for a $k$-graph $\Lambda$ (see Problem~\ref{prob For which k-graphs KP-algebras are Bergman algebras}).

— Higher-rank graphs and the graded $K$-theory of Kumjian-Pask Algebras (2507.19879 - Hazrat et al., 26 Jul 2025) in Section 2.3 (Graded K-theory of Kumjian–Pask algebra)