On quantum quaternion spheres

We give an explicit description of the $q$-deformation of symplectic group $S{P}_q(2n)$ at the $C^{\ast }$-algebra level and find all irreducible representations of this $C^{\ast }$-algebra. Further we describe the $C^{\ast }$-algebra of the quotient space $S{P}_q(2n)\mathrm{/}S{P}_q(2n-2)$ in terms of generators and relations. We compute its $K$-theory by obtaining a chain of short exact sequence for the $C^{\ast }$-algebras underlying such manifolds.