Cuntz-Krieger algebras and a generalization of Catalan numbers

Details Cuntz-Krieger algebras and a generalization of Catalan numbers Cuntz-Krieger algebras and a generalization of Catalan numbers

2006-07-21

arxiv.org/abs/math/0607517v2

Mathematics

Operator Algebras

24pages, 5figures

Scientific paper

We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$. Its generating functions will be studied.

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