Quantizations of multiplicative hypertoric varieties at a root of unity

Ganev, Iordan

doi:10.48550/arxiv.1412.7211

Cited by 3 publications

(3 citation statements)

References 10 publications

(12 reference statements)

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“…. = ǫ N = ǫ for a root of unity ǫ = 1 and χ(e i , e l ) = ǫ m il for some m il ∈ Z was previously obtained by Ganev [17] and Cooney [12]. The special case of Theorem 6.2(a) when k = 1 was obtained in [21].…”

Section: Equivalent Characterizations Of the Equality In Corollary 51(c)mentioning

confidence: 78%

Azumaya loci and discriminant ideals of PI algebras

Brown

Yakimov

2018

Advances in Mathematics

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We prove that, under mild assumptions, for all positive integers ℓ, the zero set of the discriminant ideal D ℓ (R/Z(R), tr) of a prime affine polynomial identity (PI) algebra R coincides with the zero set of the modified discriminant ideal M D ℓ (R/Z(R), tr) of R, and give an explicit description of this set in terms of the dimensions of the irreducible representations of R. Furthermore, we prove that, when ℓ is the square of the PI-degree of R, this zero set is precisely the complement of the Azumaya locus of R. This description is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime affine PI algebra R coincides with the singular locus of the center of R, provided that the discriminant ideal has height at least 2, R has finite global dimension and R is a Cohen-Macaulay module over its center.

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Section: Equivalent Characterizations Of the Equality In Corollary 51(c)mentioning

confidence: 78%

Azumaya loci and discriminant ideals of PI algebras

Brown

Yakimov

2018

Advances in Mathematics

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“…The proofs of our main results are rooted in a collection of beautiful ideas emerging from the literature on quantum groups and geometric representation theory, most notably the seminal paper [BFG06] where the Hamiltonian reduction of Azumaya algebras in characteristic p was first carried out for differential operators with applications to Cherednik algebras, and [VV10], where a q-analog was developed to study q-difference operators and double affine Hecke algebras at roots of unity. Similar techniques were used in the study of hypertoric varieties in positive characteristic [Sta13], and of their q-analogs, quantum multiplicative hypertoric varieties [Gan18].…”

Section: Main Results: Methods and General Resultsmentioning

confidence: 99%

The quantum Frobenius for character varieties and multiplicative quiver varieties

Ganev,

Jordan,

Safronov

2019

Preprint

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We develop a general mechanism for constructing sheaves of Azumaya algebras on moduli spaces obtained by Hamiltonian reduction, via their quantizations at roots of unity. We achieve this by exploiting a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson orders. We use these tools to construct canonical central subalgebras of quantum algebras, and explicitly compute the resulting Azumaya loci we encounter, using a natural nondegeneracy assumption. As our main applications, we prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and finally we compute the Azumaya locus of the Kauffman bracket skein algebras, proving a strong form of the Unicity Conjecture of Bonahon and Wong.

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“…Quantized Weyl algebras. The quantized Weyl algebras and their generalizations have been studied from many different points of view: quantum groups and Hecke type quantizations [20,30], structure of prime spectra and representations [6,21,22,27], automorphism and isomorphism problems [3,16,23,28,33,34], homological and ring theoretic dimensions [15], quantizations of multiplicative hypertoric varieties [12,18] and others. Most of these results concern the generic case when the algebras are not polynomial identity (PI).…”

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confidence: 99%

Quantized Weyl algebras at roots of unity

Levitt

Yakimov

2018

Isr. J. Math.

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We classify the centers of the quantized Weyl algebras that are polynomial identity algebras and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.2000 Mathematics Subject Classification. Primary: 17B37; Secondary: 53D17, 16W20, 15A69. Key words and phrases. Quantized Weyl algebras, discriminants of PI algebras, Poisson prime elements, quantum cluster algebras, automorphism and isomorphism problems.

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Quantizations of multiplicative hypertoric varieties at a root of unity

Cited by 3 publications

References 10 publications

Azumaya loci and discriminant ideals of PI algebras

Azumaya loci and discriminant ideals of PI algebras

The quantum Frobenius for character varieties and multiplicative quiver varieties

Quantized Weyl algebras at roots of unity

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