The attenuated space poset Aq(N,M)Aq(N,M)

In this paper, we study the incidence algebra T   of the attenuated space poset Aq(N,M)Aq(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,KR,L,K. We put these relations in an attractive form using a certain matrix S in T  . We characterize the center Z(T)Z(T). Using Z(T)Z(T), we relate T   to the quantum group Uτ(sl2)Uτ(sl2) with τ2=qτ2=q. We consider two elements A,A⁎A,A⁎ in T   of a certain form. We find necessary and sufficient conditions for A,A⁎A,A⁎ to satisfy the tridiagonal relations. Let W denote an irreducible T  -module. We find necessary and sufficient conditions for the above A,A⁎A,A⁎ to act on W as a Leonard pair.