On the socle of a class of Steinberg algebras

TL;DR

The paper develops a socle theory for Steinberg algebras $A_K(\mathcal G)$ of Hausdorff ample groupoids by linking minimal left ideals to open singletons in the unit space with finite isotropy, thereby unifying and extending socle results from Leavitt path algebras and Kumjian-Pask algebras to higher-rank graph settings. It establishes that minimal left ideals are generated by elements supported on compact open corners $\{x\}$ with $x\mathcal G x$ finite; the generator is either a division idempotent when char$(K) \nmid |x\mathcal G x|$ or a nilpotent absolute zero divisor when char$(K) \mid |x\mathcal G x|$. Building on these, the paper characterizes the socle of $A_K(\mathcal G)$ under a potency condition, showing it is the direct sum of matrix algebras over $K$ indexed by orbit classes of open singletons with trivial isotropy, and providing a structural decomposition into homogeneous components $\mathcal M_{|[x]|}(K)$. These results generalize known socle descriptions for Kumjian-Pask and Leavitt path algebras and yield a framework for understanding socles in broad Steinberg algebra settings with applications to finitely aligned $k$-graphs.

Abstract

We study minimal left ideals in Steinberg algebras of Hausdorff groupoids. We establish a relationship between minimal left ideals in the algebra and open singletons in the unit space of the groupoid. We apply this to obtain results about the socle of Steinberg algebras under certain hypotheses. This encompasses known results about Leavitt path algebras and improves on Kumjian-Pask algebra results to include higher-rank graphs that are not row-finite.

Theorems & Definitions (39)

• Lemma 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Remark 3.4
• Corollary 3.5
• Lemma 3.6