The Prime Spectrum and Simple Modules Over the Quantum Spatial Ageing Algebra

Abstract

For the algebra A in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of A is determined. The simple unfaithful A-modules and the simple weight A-modules are classified.

The prime spectrum of the algebra $\mathbb K_q[X,Y]\rtimes U_q(\mathfrak {sl}_2)$ and a classification of simple weight modules

“…Let A be the subalgebra of A generated by the elements K ±1 , E, X and Y . The properties of this algebra were studied in [8] where the prime, maximal and primitive spectrum of A were found. In particular, the algebra…”

“…All the terms in this equality belong to the algebra A. Recall that X is a normal element in A such that A/AX is a domain (see [8]) and the element K −1 EY 2 does not belong to the ideal AX. Hence we have g 0 ∈ AX, i.e., g 0 = Xh 0 for some h 0 ∈ A.…”

“…The element ϕ ∈ A is a normal element such that the factor algebra A/Aϕ is a domain (see [8]) and the element KXY does not belong to the ideal Aϕ. Therefore, g 0 ∈ Aϕ, i.e., g 0 = ϕh 0 for some element h 0 ∈ A.…”

The Universal Enveloping Algebra of the Schrödinger Algebra and its Prime Spectrum

“…It is straightforward to verify that the commutation relations in the lemma hold. The fact that the elements E ′ , F ′ and H ′ commute with the elements X and Y follows from (5), (6) and (9), respectively. Let U be the universal enveloping algebra of the Lie algebra sl 2 = ⟨F ′ , H ′ , E ′ ⟩.…”

The quantum euclidean algebra and its prime spectrum

“…[7] Let R be a Noetherian ring and s be an element of R such that S s := {si | i ∈ N}is a left denominator set of the ring R and (s i ) = (s) i for all i 1 (e.g., s is a normal element such that ker(•s R ) ⊆ ker(s R •)). Then Spec (R) = Spec(R, s) ⊔ Spec s (R) where Spec(R, s) := {p ∈ Spec (R) | s ∈ p}, Spec s (R) := {q ∈ Spec (R) | s / ∈ q} and (a) the map Spec(R, s) → Spec (R/(s)), p → p/(s), is a bijection with inverse q → π −1 (q) where π : R → R/(s), r → r + (s), (b) the map Spec s…”