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

Fryer, Siân

doi:10.1112/tlm3.12004

Cited by 6 publications

(11 citation statements)

References 28 publications

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“…It is worth mentioning that our first main theorem is very close in spirit to a conjecture of Hodges and Levasseur [21, § 2.8, Conjecture 1] which seeks to relate the primitive spectrum of the quantised coordinate ring of a complex simple algebra group O q (G) in the case where q is a generic parameter, to the Poisson spectrum of the classical limit O(G); see [16, § 4.4] for a survey of results. Although the spectra are always known to lie in natural bijection, this bijection is only known to be a homeomorphism in case G = SL 2 (C) and SL 3 (C) [12]. By contrast, our bijection is always a homeomorphism, however our results only apply to these families of algebras when the parameter is a root of unity.…”

Section: A Discussion Of Related Results and New Directionsmentioning

confidence: 76%

The orbit method for Poisson orders

Launois

Topley

2019

Proc. London Math. Soc.

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A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation A of a Poisson algebra Z should correspond bijectively to the symplectic leaves of Spec(Z). In this article, we consider a Poisson order A over a complex affine Poisson algebra Z. We stratify the primitive spectrum Prim(A) into symplectic cores, which should be thought of as families of non‐commutative symplectic leaves. We then introduce a category A-P-Mod of A‐modules adapted to the Poisson structure on Z, and we show that when Spec(Z) is smooth with locally closed symplectic leaves, there is a natural homeomorphism from the spectrum of annihilators of simple objects in A-P-Mod to the set of symplectic cores in Prim(A) with its quotient topology. Several applications are given to Poisson representation theory. Our main tool is the Poisson enveloping algebra Ae of a Poisson order A, which captures the Poisson representation theory of A. For Z regular and affine, we prove a Poincarée–Birkhoff–Witt (PBW) theorem for Ae and use this to characterise the annihilators of simple Poisson modules: they coincide with the Poisson weakly locally closed, the Poisson primitive and the Poisson rational ideals. We view this as a generalised weak Poisson Dixmier–Mœglin equivalence.

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Section: A Discussion Of Related Results and New Directionsmentioning

confidence: 76%

The orbit method for Poisson orders

Launois

Topley

2019

Proc. London Math. Soc.

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“…In [7], the second author constructed denominator sets E JK for all pairs of H -primes J K in O q (SL 3 ) in order to describe the topological structure of spec(O q (SL 3 )). These were constructed using lacunary sequences from white squares rather than chains rooted at squares along the southeast border of the Cauchon diagram, so the sets in [7, Table 3] are different from those constructed in Corollary 5.4.…”

Section: 2mentioning

confidence: 99%

“…These were constructed using lacunary sequences from white squares rather than chains rooted at squares along the southeast border of the Cauchon diagram, so the sets in [7, Table 3] are different from those constructed in Corollary 5.4. As Example 3.15 indicates, we expect that Corollary 5.4 will be the better method for generalising the results of [7] to larger algebras.…”

Section: 2mentioning

confidence: 99%

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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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“…For instance it is conjectured in [10, Section 9.1] that there should be a homeomorphism between the spectrum of the generic quantised coordinate ring of an affine algebraic variety V and the Poisson spectrum of its semiclassical limit O(V ) when K is algebraically closed of characteristic zero. This conjecture has been investigated for several algebras, for instance we refer to the recent works [8] and [24]. In particular, building on previous work of Hodges-Levasseur and Joseph, progress have been made by Yakimov [24] towards obtaining a homeomorphism between the symplectic leaves of a connected, simply connected complex algebraic group G and the primitive spectrum of the quantized coordinate ring R q [G].…”

Section: Introductionmentioning

confidence: 99%

Poisson deleting derivations algorithm and Poisson spectrum

Launois

Lecoutre

2016

Communications in Algebra

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In [5] Cauchon 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 nonnegative cells in totally nonnegative matrix varieties [12]. This led to recent progress in the study of totally nonnegative matrices such as new recognition tests, see for instance [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.

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

Cited by 6 publications

References 28 publications

The orbit method for Poisson orders

The orbit method for Poisson orders

From Grassmann Necklaces to Restricted Permutations and Back Again

Poisson deleting derivations algorithm and Poisson spectrum

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