Primitive Ideals Research Articles

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Square-integrable representations and the coadjoint action of solvable Lie groups

Abstract We characterize the square-integrable representations of (connected, simply connected) solvable Lie groups in terms of the generalized orbits of the coadjoint action. We prove that the normal representations corresponding, via the Pukánszky correspondence, to open coadjoint orbits are type I, not necessarily square-integrable representations. We show that the quasi-equivalence classes of type I square-integrable representations are in bijection with the simply connected open coadjoint orbits, and the existence of an open coadjoint orbit guarantees the existence of a compact open subset of the space of primitive ideals of the group. When the nilradical has codimension 1, we prove that the isolated points of the primitive ideal space are always of type I. This is not always true for codimension greater than 2, as shown by specific examples of solvable Lie groups that have dense, but not locally closed, coadjoint orbits.

On simple modules and automorphisms of the quantum Galilei group

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.

The spectrum of the $C^*$-algebra of singular integral operators with semi-almost periodic coefficients

A $C^*$-algebra generated by one-dimensional singular integral operators with semi-almost periodic coefficients is studied. The primitive spectrum of this algebra is described, that is, all of its primitive ideals are listed and the Jacobson topology is described. Bibliography: 27 titles.

The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits

The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits

Open orbits and primitive zero ideals for solvable Lie algebras

Abstract 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.

Classifications of Prime Ideals and Simple Modules of the Weyl Algebra $A_1$ in Prime Characteristic

Let $K$ be an arbitrary field of characteristic $p>0$. Classifications of prime ideals and simple modules are obtained for the Weyl algebra $A_1=K\langle x,\partial \, : \, \partial x-x\partial =1\rangle$, the skew polynomial algebra $\mathbb{A} = K[h][x;\sigma ]$ and the skew Laurent polynomial algebra $\cal{A} := K[h][x^{\pm 1};\sigma ]$ where $\sigma (h) = h-1$. In particular, classifications of prime, completely prime, maximal and primitive ideals are obtained for the above algebras. The quotient ring (of fractions) of each prime factor algebra of $A_1$, $\mathbb{A}$ and $\cal{A}$ is described. It is either a matrix algebra over a field or else a cyclic algebra. These descriptions are a key fact in the classification of completely prime ideals and simple modules for the algebras above.

On the Notion of Metaplectic Barbasch–Vogan Duality

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Let $W = \mathbb {C}[t,t^{-1}]\partial _t$ be the Witt algebra of algebraic vector fields on $\mathbb {C}^\times$ and let $V\!ir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $V\!ir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $\operatorname {S}(V\!ir)$ and $\operatorname {S}(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $\operatorname {S}(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $\chi \in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $\chi$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $V\!ir$ of finite codimension and show, in particular, that any such subalgebra of $V\!ir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension one. As a consequence, we deduce that $\operatorname {S}(V\!ir)/(z-\zeta )$ is Poisson simple if and only if $\zeta \neq ~0$.

Graded irreducible representations of Leavitt path algebras: A new type and complete classification

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 [K. M. Rangaswamy, Theory of prime ideals of Leavitt path algebras over arbitrary graphs, J. Algebra 375 (2013), 73 – 90].

On the nilpotent orbits arising from admissible affine vertex algebras

We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.

Primitive ideals in affinoid enveloping algebras of semisimple Lie algebras

For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions K, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, widehat{U({mathfrak {g}})_{K}}, is the annihilator of an affinoid highest weight module. In the case n>0, we characterise all the primitive ideals in the affinoid algebra widehat{U(mathfrak {{g}})_{n,K}}.

A note on the Dixmier-Moeglin equivalence in Leavitt path algebras of arbitrary graphs over a field

The Dixmier-Moeglin Equivalence (for short, the DM-equivalence) is the equivalence of three distinguishing properties of prime ideals in a non-commutative algebra A. These properties are of (i) being primitive, (ii) being rational and (iii) being locally closed in the Zariski topology of Spec(A). The DM-equivalence holds in many interesting algebras over a field. Recently, it was shown that the prime ideals of a Leavitt path algebra of a finite graph satisfy the DM-equivalence. In this note, we investigate the occurrence of the DM-equivalence in a Leavitt path algebra L of an arbitrary directed graph E. Our analysis shows that locally closed prime ideals of L satisfy interesting equivalent properties such as being strongly primitive and completely irreducible. This leads to new characterizations of primitive ideals in a Leavitt path algebra of a finite graph. Examples illustrate the results and the constraints.

Primitive ideals in rational, nilpotent Iwasawa algebras

Given a p-adic field K and a nilpotent uniform pro-p group G, we prove that all primitive ideals in the K-rational Iwasawa algebra KG are maximal, and can be reduced to a particular standard form. Setting L as the associated Zp-Lie algebra of G, our approach is to study the action of KG on a Dixmier module D(λ)ˆ over the affinoid envelope U(L)ˆK, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.

On prime and primitive ideals of the centrally extended Heisenberg double of SL2

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.

Deformations of symplectic singularities and orbit method for semisimple Lie algebras

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.

Classification of Irreducible (𝔤, 𝔨)-Modules Associated to the Ideals of Minimal Nilpotent Orbits for Simple Lie Groups of TypeA

AbstractWe classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent coadjoint orbit for $\mathfrak {g}=\mathfrak {s}\mathfrak {l}(n,\mathbb {C})$ and classify irreducible $(\mathfrak {g},\mathfrak {k})$-modules, which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak {k}$-modules.

Principal Primitive Ideals in Quadratic Orders and Pell’s Equations

In this paper, we introduce a method of determining whether the primitive ideal is principal in a real quadratic order, depending on the solvability of Pell’s equation.

Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence

The affinoid enveloping algebra widehat {U({mathscr{L}})}_{K} of a free, finitely generated mathbb {Z}_{p}-Lie algebra {mathscr{L}} has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over widehat {U({mathscr{L}})}_{K}, a generalisation of the Verma module, and we prove that when {mathscr{L}} is nilpotent, all primitive ideals of widehat {U({mathscr{L}})}_{K} can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.

N-Homogeneous C*-algebras

The classical results by J. Fell, J. Tomiyama, M. Takesaki describe n-homogeneous С*-algebras as algebras of all continuous sections for an appropriate algebraic bundle. By using this realization, several authors described the set of n-homogeneous С*-algebras with different spaces of primitive ideals. In 1974 F. Krauss and T. Lawson described the set of all n-homogeneous С*-algebras whose space Prim of primitive ideals is homeomorphic to the sphere S2. Suppose the space PrimA of primitive ideals is homeomorphic to the sphere S3 for some n-homogeneous С*-algebra A. In this case, these authors showed that the algebra A is isomorphic to the algebra C(S3,Cn×n). If n ≥ 2 then there are countably many pairwise non-isomorphic n-homogeneous С*-algebras A such that PrimA ≅ S 4. Further, let n ≥ 3. There is only one n-homogeneous С*-algebra A such that PrimA ≅ S 5. There are two non-isomorphic 2-homogeneous С*-algebras A and B with space PrimA ≅ S 5. On the other hand, algebraic bundles over the torus T 2 are described by a residue class [p] in Z/nZ = π1(PUn). Two such bundles with classes [pi] produce isomorphic С*-algebras if and only if [p1] = ±[p2]. An algebraic bundle over the torus T 3 is determined by three residue classes in Z/nZ. Anatolii Antonevich and Nahum Krupnik introduced some structures on the set of algebraic bundles over the sphere S2. Algebraic bundles over the compact connected two-dimensional oriented manifolds were considered by the author. In this case, the set of non-equivalent algebraic bundles over such space is like the set of algebraic bundles over the torus T2. Further advances could be in describing the set of algebraic bundles over the 3-dimensional manifolds.