On the homogeneous zero components of Leavitt algebras

We prove that the zero component $L(m,n)0$ of a Leavitt algebra $L(m,n)$ with respect to the canonical grading is a direct limit $\varinjlim{z}L(m,n){0,z}$, where each algebra $L(m,n){0,z}$ is a free product of two Bergman algebras. For the special case $m=1,n>1$, one recovers the known result that the zero component $L(1,n)_0$ is a direct limit of matrix algebras. Moreover, we show that $L(m,n)_0$ has the IBN property.

💡 Research Summary

The paper investigates the zero‑degree component of Leavitt algebras (L(m,n)) under the canonical (\mathbb Z)-grading, where each generator (x_{ij}) is assigned degree +1 and each (y_{ji}) degree –1. This grading is coarser than the standard (\mathbb Z^{m})-grading that arises from viewing (L(m,n)) as a weighted Leavitt path algebra, and it makes (L(m,n)) a strongly graded ring. The authors’ main achievement is a structural description of the zero component (L(m,n)_{0}).

First, they construct a linear basis for (L(m,n){0}) using a reduction system that eliminates the forbidden subwords (x{i n}y_{n i’}) and (y_{j m}x_{m j’}). By applying Bergman’s Diamond Lemma they prove that the set of irreducible words with equal numbers of (x)- and (y)-letters forms a basis. Every such word can be uniquely factored into “prime” subwords, each of which starts with an (x)-letter and ends with a (y)-letter (an “xy‑word”) or starts with a (y)-letter and ends with an (x)-letter (a “yx‑word”). This factorisation yields a decomposition
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