Algebra endomorphisms and derivations of some localized down-up algebras

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.