Algebra endomorphisms and derivations of some localized down-up algebras

Abstract

We study algebra endomorphisms and derivations of some localized down-up algebras A S (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.

The automorphism groups for a family of generalized Weyl algebras

“…It was also proved that a quantum analogue of the Dixmier conjecture holds for some quantum generalized Weyl algebras in [15,16]. Similar results were established for the simple localizations of some two-parameter down-up algebras in [25] and some simple localizations of the multiparameter quantized Weyl algebras in [26]. Since the generalized Weyl algebra A p (λ, µ, K q [s, t]) has a simple localization, it is natural to ask whether a quantum analogue of the Dixmier conjecture holds for the simple localization.…”

Endomorphisms of Quantum Generalized Weyl Algebras

“…Originally the following result was proved in a more general setting, namely for twisted group algebras. Theorem 2.1 has been useful in computing derivations in a number of settings, see for example [13] or [17]. Likewise, we will use this to explicitly classify the derivations of k[h ±1 ](σ q , a).…”

“…Theorem 2.1. (Passman, Osbourn [14, Corollary 2.3]) Every derivation of T q can be uniquely written as ad t + δ α,β where t ∈ T q and δ α,β has action δ α,β (u) = αu and δ α,β (v) = βv where α, β ∈ k. Theorem 2.1 has been useful in computing derivations in a number of settings, see for example [13] or [17]. Likewise, we will use this to explicitly classify the derivations of k[h ±1 ](σ q , a).…”

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

“…The Dixmier conjecture has been proved to be stably equivalent to the Jacobian conjecture [21,7,34]. There have been several works studying a quantum analogue of the Dixmier conjecture for some quantum algebras [3,30,22,23,33]. In particular, it has recently been proved in [23] that each K−algebra endomorphism of a simple localization of A…”