Algebra endomorphisms and derivations of some localized down-up algebras

作者：Tang Xin^*

来源：Journal of Algebra and Its Applications, 2015, 14(3): 1550034.

DOI：10.1142/S0219498815500346

• Down-up algebras
• algebra endomorphisms
• algebra automorphisms
• derivations
• Hochschild cohomology

We study algebra endomorphisms and derivations of some localized down-up algebras AS(r + s, -rs). First, we determine all the algebra endomorphisms of A(S)(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A(S)(r + s, -rs) is an algebra automorphism if r(m)s(n) = 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 A(S)(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A(S)(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 A(S)(r + s, (-)rs) and calculate its first degree Hochschild cohomology group.

• 出版日期2015-4