Every Quantum Minor Generates an Ore Set

Škoda, Zoran

doi:10.1093/imrn/rnn063

Cited by 10 publications

(17 citation statements)

References 13 publications

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“…Example Let

a, b, c, d

denote the standard generators in

{scriptO}_{q} false(M_{2} false)

O (M_{2})

, and

normalΔ

the

2 \times 2

determinant. If we invert

d

(recall that this is possible even in the quantum case, thanks to the results of ), then

a = false(normalΔ + q b c false) d^{- 1}

; this suggests we might try to rewrite the localised algebra in terms of the variables

{Δ, b, c, d^{\pm 1}}

. Indeed, we obtain isomorphisms

{scriptO}_{q} false(M_{2} false) false[d^{- 1} false] ≅ A_{q}^{1, 4}, O false(M_{2} false) false[d^{- 1} false] ≅ P_{q}^{1, 4},

on the variables

Δ, b, c, d^{\pm 1}

with respect to the matrix

q = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & - 1 & - 1 & 0 \end{matrix}] .

The fact that we obtain genuine isomorphisms rather than quotients of

A_{boldq}^{r, n}

and

P_{boldq}^{r, n}

fol...…”

Section: A Homeomorphism Between Pspec(ofalse(sl3false)) and Spec(scrmentioning

confidence: 98%

“…Remark In the quantum setting, we need to know that a multiplicative set satisfies the Ore conditions (for example, [, Chapter 6]) before attempting to localise at it. For all of the localizations we will consider in this paper (and in particular for the sets

E_{K}

in Definition ) we get this condition for free, thanks to the descriptively named paper ‘Every quantum minor generates an Ore set’ .…”

Section: A Homeomorphism Between Pspec(ofalse(sl3false)) and Spec(scrmentioning

confidence: 99%

“…Proof In the Poisson case, this follows directly from Lemmas and . For the quantum case, combine [, Lemma 3.9], Lemma and .

□

…”

Section: A Homeomorphism Between Pspec(ofalse(sl3false)) and Spec(scrmentioning

confidence: 99%

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The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

Fryer

2017

Trans. London Math. Soc.

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A bijection ψ is defined between the prime spectrum of quantum SL3 and the Poisson prime spectrum of SL3, and we verify that both ψ and ψ −1 preserve inclusions of primes, that is, ψ is in fact a homeomorphism between these two spaces. This is accomplished by developing a Poisson analogue of the Brown and Goodearl framework for describing the Zariski topology of spectra of quantum algebras, and then verifying directly that in the case of SL 3 these give rise to identical pictures on both the quantum and Poisson sides. As part of this analysis, we study the Poisson primitive spectrum of O(SL3) and obtain explicit generating sets for all of the Poisson primitive ideals.

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“…Example Let

a, b, c, d

denote the standard generators in

{scriptO}_{q} false(M_{2} false)

O (M_{2})

, and

normalΔ

the

2 \times 2

determinant. If we invert

d

(recall that this is possible even in the quantum case, thanks to the results of ), then

a = false(normalΔ + q b c false) d^{- 1}

; this suggests we might try to rewrite the localised algebra in terms of the variables

{Δ, b, c, d^{\pm 1}}

. Indeed, we obtain isomorphisms

{scriptO}_{q} false(M_{2} false) false[d^{- 1} false] ≅ A_{q}^{1, 4}, O false(M_{2} false) false[d^{- 1} false] ≅ P_{q}^{1, 4},

on the variables

Δ, b, c, d^{\pm 1}

with respect to the matrix

q = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & - 1 & - 1 & 0 \end{matrix}] .

The fact that we obtain genuine isomorphisms rather than quotients of

A_{boldq}^{r, n}

and

P_{boldq}^{r, n}

fol...…”

Section: A Homeomorphism Between Pspec(ofalse(sl3false)) and Spec(scrmentioning

confidence: 98%

E_{K}

in Definition ) we get this condition for free, thanks to the descriptively named paper ‘Every quantum minor generates an Ore set’ .…”

Section: A Homeomorphism Between Pspec(ofalse(sl3false)) and Spec(scrmentioning

confidence: 99%

See 1 more Smart Citation

The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

Fryer

2017

Trans. London Math. Soc.

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“…First, if we construct a set E K := E 0K satisfying (3.2) above for J = {0}, then E K ∩ J = ∅ for any J ⊆ K and we can take E JK to be the image of By [14,Proposition 10.7] all Ore sets are denominator sets in this setting, so it suffices to check that our sets E K satisfy the Ore conditions. In fact, we can do even better than this: by [22], any quantum minor generates an Ore set in O q (M m,n ), so it is enough to find a generating set for E K that consists of quantum minors.…”

Section: 1mentioning

confidence: 99%

“…Proof. By [22] and [14, Proposition 10.7], E K is a denominator set in O q (M m,n ). Complete primality implies that we have E K ∩ K = ∅, and hence E JK is well behaved for all J K. Finally, E JK must be a denominator set in O q (M m,n )/J by the universality of localization and quotients.…”

Section: Constructingmentioning

confidence: 99%

From Grassmann Necklaces to Restricted Permutations and Back Again

Casteels

Fryer

2017

Algebr Represent Theor

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We study the commutative algebras ZJK appearing in Brown and Goodearl's extension of the H-stratification framework, and show that if A is the single parameter quantized coordinate ring of Mm,n, GLn or SLn, then the algebras ZJK can always be constructed in terms of centres of localizations. The main purpose of the ZJK is to study the structure of the topological space spec(A), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real m × n matrices in terms of its restricted permutation.The key technique in the study of spec(A) is that of H-stratification, where H is a torus acting rationally on A. Write H-spec(A) for the set of H-primes: the prime ideals of A which are also invariant under the action of H. The H-primes induce a stratification of spec(A), where for each J ∈ H-spec(A) the associated stratum isThis stratification has many nice properties. In particular, each spec J (A) is homeomorphic to the prime spectrum of a commutative Laurent polynomial ring Z(A J ), which can be constructed explicitly as the centre of a localization of A/J [3, Theorem II.2.13]. These algebras Z(A J ) are now reasonably well understood, e.g. [1,25].This approach gives us an excellent picture of the individual strata, but it comes at the cost of losing information about the interactions between primes from different strata. Being able to identify inclusions of prime ideals from different strata is crucial for constructing a complete picture of the topological structure of spec(A), but this information is often surprisingly difficult to obtain: for example, in the setting of quantum matrices only the prime spectra of O q (SL 2 ), O q (GL 2 ), and O q (SL 3 ) ([15], [2], [7] respectively) have been fully described.

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Quantum Principal Bundles on Projective Bases

Aschieri

Fioresi

Latini

2021

Commun. Math. Phys.

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The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study noncommutative principal bundles corresponding to $$G \rightarrow G/P$$ G → G / P , where G is a semisimple group and P a parabolic subgroup.

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Every Quantum Minor Generates an Ore Set

Abstract: The subset multiplicatively generated by any given set of quantum minors and the unit element in the quantum matrix bialgebra satisfies the left and right Ore conditions.

Cited by 10 publications

References 13 publications

The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

From Grassmann Necklaces to Restricted Permutations and Back Again

Quantum Principal Bundles on Projective Bases

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