Ideals of the Enveloping AlgebraU(sl2)

Catoiu, Stefan

doi:10.1006/jabr.1997.7284

Cited by 14 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Analogously as in (10), for 0 β < α we have ω α y β = π α,β ω α−β for some elements π α,β ∈ U ; cf. (11). Now proceed as in (E1) to show that ω is weakly normal.…”

Section: Proposition 51mentioning

confidence: 94%

See 1 more Smart Citation

On noncommutative Noetherian schemes

Širola

2004

Journal of Algebra

View full text Add to dashboard Cite

The main aim of this paper is to better understand the localization technique for certain Noetherian rings like enveloping algebras of nilpotent Lie algebras. For such rings R we also give a conjectural definition of certain sheaves which should be "affine" objects naturally generalizing the classically defined structure sheaves in commutative theory. The corresponding sheaves associated to some R-modules might carry particularly interesting information; e.g., for representation theory of semisimple Lie groups. Next, we generalize one important theorem of P.F. Smith on localization in Noetherian Artin-Rees rings. As an interesting corollary we obtain that every prime ideal of height 1 in the enveloping algebra of the Lie algebra sl(2) over a field of characteristic zero is localizable. Finally, we provide a number of concrete useful calculations for our main example, the enveloping algebra of the three-dimensional Heisenberg Lie algebra; and thus test both the proposed ideas and methods. In particular, we introduce the notion of a weakly normal element, generalizing the notion of a normal element.

show abstract

“…Analogously as in (10), for 0 β < α we have ω α y β = π α,β ω α−β for some elements π α,β ∈ U ; cf. (11). Now proceed as in (E1) to show that ω is weakly normal.…”

Section: Proposition 51mentioning

confidence: 94%

“…Hence, by Corollary 3.5, every I ∈ Id 1 U has the AR property. Next we need a remarkable "arithmetic" property of U , first observed by Bavula ([7]; see also the very nice Catoiu's paper [11]): every two ideals of U commute. This ensures the condition (S U ;1 ).…”

Section: Artin-rees Propertymentioning

confidence: 98%

On noncommutative Noetherian schemes

Širola

2004

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Indeed, since U(sl 2 (C)) is a domain, 0 is a (completely) prime ideal of U(sl 2 (C)). By [5,Remark 4.6], all nonzero prime ideals of U(sl 2 (C)) are primitive, so that prim. deg 0 = 1.…”

Section: Introductionmentioning

confidence: 99%

“…deg C Z(Frac U(sl 2 (C))) = 1. By [5,Theorem 4.5 and Proposition 5.13], there are infinitely many height two prime ideals in U(sl 2 (C)) and their intersection is zero, so that loc. deg 0 > 1.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A strong Dixmier–Moeglin equivalence for quantum Schubert cells

2017

View full text Add to dashboard Cite

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 sl 2 (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 U q [w] of the quantised enveloping K-algebra U q (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.• A prime ideal P of a noetherian K-algebra R is said to be rational if the field extension Z(Frac R/P ) of K is algebraic.Dixmier and Moeglin proved that for a prime ideal of the universal enveloping algebra of a finite-dimensional complex Lie algebra, the properties of being primitive, locally closed, and rational are equivalent. In modern terminology, they proved that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier-Moeglin equivalence.Since the work of Dixmier and Moeglin on universal enveloping algebras of finite-dimensional complex Lie algebras, many more algebras have been shown to satisfy the Dixmier-Moeglin equivalence: [4, Corollary II.8.5] lists several quantised coordinate rings which satisfy the Dixmier-Moeglin equivalence; the first named author, Rogalski, and Sierra [1] have shown that twisted homogeneous coordinate rings of projective surfaces satisfy the Dixmier-Moeglin equivalence. However, Irving [14] and Lorenz [17] have shown that there exist noetherian algebras for which the Dixmier-Moeglin equivalence fails.Our goal is to extend the notion of the Dixmier-Moeglin equivalence to all prime ideals, in a way which captures how "close" they are to being primitive. Of course, not all non-primitive prime ideals are created equal. For example, in the polynomial ring C[x, y], the primitive ideals are the maximal ideals x − α, y − β . For this reason, we think of the prime ideal x as being "closer" to being primitive than the prime ideal 0 , in the same sense that it...

show abstract

Adjoint action for quantum algebra U r,t

Hong

2014

Indian J Pure Appl Math

View full text Add to dashboard Cite

Ideals of the Enveloping AlgebraU(sl2)

Cited by 14 publications

References 8 publications

On noncommutative Noetherian schemes

On noncommutative Noetherian schemes

A strong Dixmier–Moeglin equivalence for quantum Schubert cells

Adjoint action for quantum algebra U r,t

Contact Info

Product

Resources

About