The Attenuated Space Poset ${\mathcal{A}}_q(N,M)$

In this paper, we study the incidence algebra $T$ of the attenuated space poset ${\mathcal{A}}_q(N,M)$. We consider the following topics. We consider some generators of $T$: the raising matrix $R$, the lowering matrix $L$, and a certain diagonal matrix $K$. We describe some relations among $R,L,K$. We put these relations in an attractive form using a certain matrix $S$ in $T$. We characterize the center $Z(T)$. Using $Z(T)$, we relate $T$ to the quantum group $U_{\tau }({\mathfrak{s}\mathfrak{l}}_2)$ with ${\tau }^2=q$. We consider two elements $A, A^$in$T$ofacertainform\mathrm{.}Wefindnecessaryandsufficientconditionsfor$A, A^$tosatisfythetridiagonalrelations\mathrm{.}Let$W$denoteanirreducible$T$-module\mathrm{.}Wefindnecessaryandsufficientconditionsfortheabove$A, A^*$toacton$W$ as a Leonard pair.