Canonical Gelfand–Zeitlin Modules over Orthogonal Gelfand–Zeitlin Algebras

Early, Nick; Mazorchuk, Volodymyr; Vishnyakova, Elizaveta

doi:10.1093/imrn/rnz002

Cited by 19 publications

(21 citation statements)

References 25 publications

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“…The first main result of this paper provides a sufficient condition for a ring to be a Galois order and simultaneously provides canonical simple Gelfand-Zeitlin modules. This generalizes the construction of modules from [EMV17], and can also be viewed as a partial generalization of the statement about non-empty fibers in [FO14].…”

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confidence: 83%

“…In [EMV17] the authors constructed canonical simple Gelfand-Zeitlin modules was defined in the setting of orthogonal Gelfand-Zeitlin algebras. In this section we generalize this to an arbitrary principal Galois Γ-order U in K .…”

Section: Gelfand-zeitlin Modulesmentioning

confidence: 99%

“…It is conjectured that any simple Gelfand-Zeitlin module over U (gl n ) will be a subquotient of one of those. In the latest development [EMV17], the authors moved beyond U (gl n ) and constructed canonical Gelfand-Zeitlin modules for so called orthogonal Gelfand-Zeitlin (OGZ) algebras [M99], and constructed bases for some of them.…”

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confidence: 99%

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Principal Galois orders and Gelfand-Zeitlin modules

Hartwig

2020

Advances in Mathematics

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We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order.As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type A, and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative.Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if q is not a root of unity, the Gelfand-Zeitlin subalgebra of Uq(gl n ) is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.Notation. The multiplicative identity element in a monoid or ring S is denoted 1 S . All rings are assumed to have an identity. Frac A denotes the field of fractions of an Ore domain A. For a group G acting by automorphisms on a ring A we let A G = {a ∈ A | ∀g ∈ G : g(a) = a} denote the subring of G-invariants.

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confidence: 83%

Section: Gelfand-zeitlin Modulesmentioning

confidence: 99%

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Principal Galois orders and Gelfand-Zeitlin modules

Hartwig

2020

Advances in Mathematics

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“…In the case of gl n further developments led to recent breakthrough results in its representation theory (see [8], [16], [10] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Algebras of invariant differential operators

Futorny

Schwarz

2020

Journal of Algebra

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We prove that an invariant subalgebra A W n of the Weyl algebra An is a Galois order over an adequate commutative subalgebra Γ when W is a two-parameters irreducible unitary reflection group G(m, 1, n), m ≥ 1, n ≥ 1, including the Weyl group of type Bn, or alternating group An, or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group in [15]. In each of the cases above, except for the alternating groups, we show that A W n is free as a right (left) Γ-module. Similar results are established for the algebra of W -invariant differential operators on the n-dimensional torus where W is a symmetric group Sn or orthogonal group of type Bn or Dn. As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for Uq(sl 2 ), the first Witten deformation and the Woronowicz deformation.

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“…Following the techniques in [EMV17] and [Har20], we construct canonical simple Gelfand-Tsetlin modules over A (gl n ). We need the following additional assumptions for these next two results: (A4) Λ is finitely generated over an algebraically closed field k of characteristic 0, (A5) G and M act by k-algebra homomorphisms on Λ.…”

Section: Some General Resultsmentioning

confidence: 99%

A study of Galois and flag orders

Jauch¹

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Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class of associative algebras that contain many important examples, such as the enveloping algebra of gl n (as well as its quantum deformation), generalized Weyl algebras, and shifted Yangians. The main motivation for introducing Galois orders is they provide a setting for studying certain infinite dimensional irreducible representations, called Gelfand-Tsetlin modules. Principal Galois orders, defined by J. Hartwig in 2017, are Galois orders with an extra property, which makes them easier to study. All of the mentioned examples are principal Galois orders. In 2019, B. Webster defined principal flag orders which are Morita equivalent to principal Galois orders and further simplifies their study.The purpose of this dissertation is twofold:(1) To introduce a new example of a Galois order, A (gl n ), which is an extension of the enveloping algebra of gl n such that the "Weyl group" of A (gl n ) is the alternating group;(2) To describe some techniques to study such objects including tensor products and morphisms between standard flag orders with conjectured application to the orthogonal Lie algebra.

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Canonical Gelfand–Zeitlin Modules over Orthogonal Gelfand–Zeitlin Algebras

Cited by 19 publications

References 25 publications

Principal Galois orders and Gelfand-Zeitlin modules

Principal Galois orders and Gelfand-Zeitlin modules

Algebras of invariant differential operators

A study of Galois and flag orders

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