Quantum Unique Factorisation Domains

Launois, Stéphane; Lenagan, T. H.; Rigal, Laurent

doi:10.1112/s0024610706022927

Cited by 55 publications

(108 citation statements)

References 18 publications

(38 reference statements)

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“…In this section, we recall the notion of CGL extension that was introduced in [17]. Examples include various quantum algebras in the generic case such as quantum affine spaces, quantum matrices, positive part of quantised enveloping algebras of semisimple complex Lie algebras, etc.…”

Section: Primitive Ideals Of Cgl Extensionsmentioning

confidence: 99%

“…Moreover, as q is not a root of unity, R endowed with this action of H is a CGL extension (see for instance [17]). This implies in particular that H-Spec(R) is finite and that every H-prime is completely prime.…”

Section: Quantum Matrices As a Cgl Extensionmentioning

confidence: 99%

“…Thus R is a noetherian domain. Henceforth, we assume that, in the terminology of [17], R is a CGL extension. …”

Section: H -Stratification Theory Of Goodearl and Letzter And Cgl Exmentioning

confidence: 99%

“…In the first section, we recall the notion of CGL extension that was introduced in [17]. The advantage of these algebras is that one can use both the H -stratification theory of Goodearl and Letzter, and the deletingderivations theory of Cauchon to study their prime and primitive spectra.…”

Section: Introductionmentioning

confidence: 99%

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Dimension and enumeration of primitive ideals in quantum algebras

2008

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In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2 × n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the "variety of 2 × n quantum matrices".

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Section: Primitive Ideals Of Cgl Extensionsmentioning

confidence: 99%

Section: Quantum Matrices As a Cgl Extensionmentioning

confidence: 99%

“…Thus R is a noetherian domain. Henceforth, we assume that, in the terminology of [17], R is a CGL extension. …”

Section: H -Stratification Theory Of Goodearl and Letzter And Cgl Exmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dimension and enumeration of primitive ideals in quantum algebras

2008

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“…More recently, such phenomena have been studied; for semifirs and in particular 2-firs by P. M. Cohn [Coh85,Coh06], for the ring of Hurwitz and Lipschitz quaternions by Conway and Smith [CS03] and by H. Cohn and Kumar [CK15], for quaternion orders by Estes and Nipp [EN89,Est91], and in a more general setting by Brungs [Bru69]. Somewhat different notions of unique factorization domains and unique factorization rings were introduced by Chatters and Jordan [Cha84,CJ86,Jor89], and have found applications in [JW01,LLR06,GY12].…”

Section: Introductionmentioning

confidence: 99%

Factorization theory: From commutative to noncommutative settings

2015

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Abstract. We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n × n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.

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Factorizations of Elements in Noncommutative Rings: A Survey

Smertnig

2016

Springer Proceedings in Mathematics &Amp; Statistics

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We survey results on factorizations of non-zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.

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Quantum Unique Factorisation Domains

Cited by 55 publications

References 18 publications

Dimension and enumeration of primitive ideals in quantum algebras

Dimension and enumeration of primitive ideals in quantum algebras

Factorization theory: From commutative to noncommutative settings

Factorizations of Elements in Noncommutative Rings: A Survey

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