Dixmier Moeglin Equivalence Research Articles

Commutative bidifferential algebra

Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a biderivation, namely a binary operation that is a derivation in each argument, is here begun, with an eye toward the geometry of the corresponding B-varieties. Foundational results about extending biderivations to localisations, algebraic extensions and transcendental extensions are established. Resolving a deficiency in Poisson algebraic geometry, a theory of base extension is achieved, and it is shown that dominant B-morphisms admit generic B-fibres. A bidifferential version of the Dixmier-Moeglin equivalence problem is articulated.

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.

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.

Poisson Dixmier-Moeglin equivalence from a topological point of view

A complex affine Poisson algebra A is said to satisfy the Poisson Dixmier-Moeglin equivalence if the Poisson cores of maximal ideals of A are precisely those Poisson prime ideals that are locally closed in the Poisson prime spectrum P.spec A and if, moreover, these Poisson prime ideals are precisely those whose extended Poisson centers are exactly the complex numbers. In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for A in terms of the poset (P.spec A, ⊆) and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O. L. Sanchez and B. Moosa, Poisson algebras via model theory and differential-algebraic geometry, J. Eur. Math. Soc. (JEMS) 19 (2017), 2019–2049] to the general context of a commutative differential algebra.

Hopf Algebra Actions and Rational Ideals

This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H. In particular, we study a notion of H-rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H-prime ideals of A.

The Dixmier-Moeglin equivalence, Morita equivalence, and homeomorphism of spectra

Let k be a field and let R be a left noetherian k-algebra. The algebra R satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the Spec(R) and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if R and S are two left noetherian k-algebras with dimk(R),dimk(S)<|k| then if R and S have homeomorphic spectra then R satisfies the Dixmier-Moeglin equivalence if and only if S does. In particular, the topology of Spec(R) can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if k is uncountable and R is affine noetherian and its prime spectrum is a finite disjoint union of locally closed subspaces that are each homeomorphic to the spectrum of an affine commutative ring then R satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if k is countable and R is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that R and S are left noetherian k-algebras that satisfy the Dixmier-Moeglin equivalence then R⊗kS does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that eRe also satisfies the Dixmier-Moeglin equivalence, where e is a nonzero idempotent of R.

Isolated types of finite rank: an abstract Dixmier–Moeglin equivalence

Suppose T is a totally transcendental first-order theory and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank type p=tp(a/A) is isolated if and only if Open image in new window for every b∈acl(Aa) and q∈S(Ab) nonisolated and minimal. This applies to the theory of differentially closed fields—where it is motivated by the differential Dixmier–Moeglin equivalence problem—and the theory of compact complex manifolds.

On the Dixmier–Moeglin equivalence for Poisson–Hopf algebras

We prove that the Poisson version of the Dixmier–Moeglin equivalence holds for cocommutative affine Poisson–Hopf algebras. This is a first step towards understanding the symplectic foliation and the representation theory of (cocommutative) affine Poisson–Hopf algebras. Our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations. As an application, we show that the symmetric algebra of a finite dimensional Lie algebra, equipped with its natural Poisson structure, satisfies the Poisson Dixmier–Moeglin equivalence.

The prime spectrum and representation theory of the 2 × 2 reflection equation algebra

The theory of generalized Weyl algebras is used to study the 2 × 2 reflection equation algebra in the case that q is not a root of unity, where the R-matrix used to define is the standard one of type A. Simple finite-dimensional -modules are classified, finite-dimensional weight modules are shown to be semisimple, is computed, and the prime spectrum of is computed along with its Zariski topology. Finally, it is shown that satisfies the Dixmier–Moeglin equivalence.

D-groups and the Dixmier–Moeglin equivalence

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if $R$ is a commutative affine Hopf algebra over a field of characteristic zero, and $A$ is an Ore extension to which the Hopf algebra structure extends, then $A$ satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such $A$ are Hopf Ore extensions

A strong Dixmier–Moeglin equivalence for quantum Schubert cells

Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier–Moeglin equivalence.We define quantities which measure how “close” an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier–Moeglin equivalence. Using the example of the universal enveloping algebra of sl2(C), we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence.For a simple complex Lie algebra g, a non-root of unity q≠0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subalgebra Uq[w] of the quantised enveloping K-algebra Uq(g). These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.

Poisson algebras via model theory and differential-algebraic geometry

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.

Poisson Hopf algebra related to a twisted quantum group

ABSTRACTA Poisson algebra ℂ[G] considered as a Poisson version of the twisted quantized coordinate ring ℂq,p[G], constructed by Hodges et al. [11], is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of ℂ[G] are characterized. Further it is shown that ℂ[G] satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of ℂ[G] agrees with the quotient topology induced by the natural surjection from the maximal ideal space of ℂ[G] onto the Poisson primitive ideal space.

Poisson deleting derivations algorithm and Poisson spectrum

ABSTRACTCauchon [5] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.

The Dixmier-Moeglin Equivalence for Cocommutative Hopf Algebras of Finite Gelfand-Kirillov Dimension

Let k be an algebraically closed field of characteristic zero and let H be a noetherian cocommutative Hopf algebra over k. We show that if H has polynomially bounded growth then H satisfies the Dixmier-Moeglin equivalence. That is, for every prime ideal P in Spec(H) we have the equivalences $$P\; \text{primitive}\iff P\; \text{rational}\iff P\; \text{locally closed in}~\text{Spec}(H).$$ We observe that examples due to Lorenz show that this does not hold without the hypothesis that H have polynomially bounded growth. We conjecture, more generally, that the Dixmier-Moeglin equivalence holds for all finitely generated complex noetherian Hopf algebras of polynomially bounded growth.

Poisson spectra in polynomial algebras

A significant class of Poisson brackets on the polynomial algebra C[x1,…,xn] is studied and, for this class of Poisson brackets, the Poisson prime ideals, Poisson primitive ideals and symplectic cores are determined. Moreover it is established that these Poisson algebras satisfy the Poisson Dixmier–Moeglin equivalence.

The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras

We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of n × n n\times n matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras A A over infinite fields k k of arbitrary characteristic. Our main result is the verification, for A A , of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal P P of A A is primitive if and only if the center of the Goldie quotient ring of A / P A/P is algebraic over k k , if and only if P P is a locally closed point – with respect to the Jacobson topology – in the prime spectrum of A A . These equivalences are established with the aid of a suitable group H \mathcal {H} acting as automorphisms of A A . The prime spectrum of A A is then partitioned into finitely many “ H \mathcal {H} -strata” (two prime ideals lie in the same H \mathcal {H} -stratum if the intersections of their H \mathcal {H} -orbits coincide), and we show that a prime ideal P P of A A is primitive exactly when P P is maximal within its H \mathcal {H} -stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which H \mathcal {H} acts transitively on the set of rational ideals within each H \mathcal {H} -stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of A A . For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.