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This paper is devoted to ring theoretic properties and representations of the quantum Galilei group F q (G) and its Hopf dual U q (g) . We classify the primitive ideals and simple modules of F q (G) and U q (g) over an algebraically closed field of arbitrary characteristic. We determine their groups of automorphisms over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

The 3-cyclic quantum Weyl algebra A=A(α,β,γ;q²), where α,β,γ,q²∈K (a ground field), is a quadratic Noetherian domain of Gelfand–Kirillov dimension 3 that is generated by three subalgebras each of them is either a quantum plane or the quantum Weyl algebra. For the algebras A, their prime, completely prime, primitive and maximal spectra are described together with containments of prime ideals (the Zariski–Jacobson topology on the spectrum) and simple A-modules are classified when q2 is not a root of unity. For each prime ideal, an explicit set of ideal generators is given. The centre Z(A) of A is K[Ω] where Ω is a cubic element. A semisimplicity criterion for the category of finite dimensional A-modules is given. Criteria are presented for all ideals of the algebra A to commute and for each ideal of A to be a unique product of primes (up to order). [ABSTRACT FROM AUTHOR]

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial. [ABSTRACT FROM AUTHOR]

For the centrally extended Heisenberg double of SL2, a classification of its prime and primitive ideals is obtained. For each primitive ideal, an explicit set of generators is given. [ABSTRACT FROM AUTHOR]

For the centrally extended Heisenberg double of SL2, a classification of its prime and primitive ideals is obtained. For each primitive ideal, an explicit set of generators is given. [ABSTRACT FROM AUTHOR]

Suppose that (G , G +) is a lattice-ordered abelian group with positive cone G + . Adji and Raeburn have shown a result about the structure of the primitive ideal space of the C ∗ -algebra B G + × α G + for a totally ordered abelian group (Adji and Raeburn in Integral Equ Oper Theory 48:281–293, 2004, Theorem 3.1). In this paper, we show if ∑ (G) is the chain of subgroups H of G, where H : = H + - H + and H + is any hereditary subsemigroup of G + . Then, there exists a well-defined map F from the disjoint union ⨆ { H ^ : H ∈ ∑ G } to the primitive ideals of the C ∗ -algebra B G + × α G + . Our result is interesting and has more challenges, since we are working under the assumption that the group G is lattice-ordered and extends their result to more general cases. [ABSTRACT FROM AUTHOR]

Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$ -graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$. We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$. The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. We prove that there exists a one-to-one correspondence between the set of all $G$ -hereditary and $G$ -saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$ -graph C*-algebras in depth. [ABSTRACT FROM AUTHOR]

In this paper, we describe the primitive ideal space of the $C^*$-algebra $C^*(mathcal G)$ associated to the ultragraph $mathcal{G}$. We investigate the structure of the closed ideals of the quotient ultragraph $ C^* $-algebra $C^*left(mathcal G/(H,S)right)$ which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the Hong and Szyma$ acute{ mathrm { n } } $ski's description of the primitive ideal space of a graph $ C ^ * $-algebra by a simpler method.

Using C*-algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we describe primitive ideal space of the C ⁎ -algebra C ⁎ (Λ) associated to any locally convex row-finite k -graph Λ. To do this, we will apply the Farthing's desourcifying method on a recent result of Carlsen, Kang, Shotwell, and Sims. We also characterize certain maximal ideals of C ⁎ (Λ). Furthermore, we study the decomposability of C ⁎ (Λ). We apply the description of primitive ideals to show that if I is a direct summand of C ⁎ (Λ) , then it is gauge-invariant and isomorphic to a certain k -graph C ⁎ -algebra. So, we may characterize decomposable higher-rank C ⁎ -algebras by giving necessary and sufficient conditions for the underlying k -graphs. Moreover, we determine all such C ⁎ -algebras which can be decomposed into a direct sum of finitely many indecomposable C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

Ideals that share properties with the Frattini ideal of a Leibniz algebra are studied. Similar investigations have been considered in group the- ory. The results will hold for Lie algebras as well. Many of the results involve nilpotency of these algebras. [ABSTRACT FROM AUTHOR]

In this paper, prime as well as primitive Kumjian-Pask algebras of a row-finite -graph over a unital commutative ring are completely characterized in graph-theoretic and algebraic terms. By applying quotient -graphs, these results describe prime and primitive graded basic ideals of Kumjian-Pask algebras. In particular, when is strongly aperiodic and is a field, all prime and primitive ideals of a Kumjian-Pask algebra are determined. [ABSTRACT FROM AUTHOR]

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero. [ABSTRACT FROM AUTHOR]

We present a new class of graded irreducible representations of a Leavitt path algebra. This class is new in the sense that its representation space is not isomorphic to any of the existing simple Chen modules. The corresponding graded simple modules complete the list of Chen modules which are graded, creating an exhaustive class: the annihilator of any graded simple module is equal to the annihilator of either a graded Chen module or a module of this new type. Our characterization of graded primitive ideals of a Leavitt path algebra in terms of the properties of the underlying graph is the main tool for proving the completeness of such classification. We also point out a problem with the characterization of primitive ideals of a Leavitt path algebra in Rangaswamy (2013) [15]. [ABSTRACT FROM AUTHOR]

If X is a compact Hausdorff space and σ is a homeomorphism of X , then a Banach algebra ℓ 1 ( Σ ) of crossed product type is naturally associated with this topological dynamical system Σ = ( X , σ ) . If X consists of one point, then ℓ 1 ( Σ ) is the group algebra of the integers. We study the algebraically irreducible representations of ℓ 1 ( Σ ) on complex vector spaces, its primitive ideals, and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebraically irreducible representations is constructed to be able to conclude that ℓ 1 ( Σ ) is semisimple. All primitive ideals of ℓ 1 ( Σ ) are selfadjoint, and ℓ 1 ( Σ ) is Hermitian if there are only periodic points in X . If X is metrizable or all points are periodic, then all primitive ideals arise as in our construction. A part of the structure space of ℓ 1 ( Σ ) is conditionally shown to be homeomorphic to the product of a space of finite orbits and T . If X is a finite set, then the structure space is the topological disjoint union of a number of tori, one for each orbit in X . If all points of X have the same finite period, then it is the product of the orbit space X / Z and T . For rational rotations of T , this implies that the structure space is homeomorphic to T 2 . [ABSTRACT FROM AUTHOR]

For the algebra $\mathcal {A}$ in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of $\mathcal {A}$ is determined. The simple unfaithful $\mathcal {A}$ -modules and the simple weight $\mathcal {A}$ -modules are classified. [ABSTRACT FROM AUTHOR]

We consider the Deaconu-Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C -algebra. For a special class of actions we describe the Jacobson topology. [ABSTRACT FROM AUTHOR]

For a (countable) graph E und a unitaI commutative rinn analyze the ideal structure of the Leavitt path algebra LR(E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness und the primitivity of LR(E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of LR(E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of LR(E) coincide. [ABSTRACT FROM AUTHOR]