Integral quantum cluster structures

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the forms of nilpotent always possess algebra structures. These are then shown to be isomorphic corresponding upper algebras, again defined Z[q±1/2]. Previously, this was only known acyclic algebras. The is applied that, every symmetrizable Kac–Moody g and Weyl group element w, dual canonical form Aq(n+(w))Z[q±1] unipotent cell has property Aq(n+(w))Z[q±1]⊗Z[q±1]Z[q±1/2] Z[q±1/2]