scholarly journals GENERALISED QUANTUM DETERMINANTAL RINGS ARE MAXIMAL ORDERS

2020 ◽  
pp. 1-11
Author(s):  
T. H. LENAGAN ◽  
L. RIGAL
Keyword(s):  
Schubert Varieties ◽  
Quantum Matrices ◽  
Maximal Orders ◽  
Noncommutative Version ◽  
Integrally Closed ◽  
Skew Polynomial ◽  
Polynomial Extensions

Abstract Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

scholarly journals Skew Polynomial Extensions over Zip Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-9 ◽  
Author(s):  
Wagner Cortes
Keyword(s):  
Polynomial Extension ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
The Relationship ◽  
Polynomial Extensions

In this article, we study the relationship between left (right) zip property of and skew polynomial extension over , using the skew versions of Armendariz rings.

scholarly journals ORE AND GOLDIE THEOREMS FOR SKEW PBW EXTENSIONS

2013 ◽  
Vol 06 (04) ◽  
pp. 1350061 ◽  
Author(s):  
Oswaldo Lezama ◽  
Juan Pablo Acosta ◽  
Cristian Chaparro ◽  
Ingrid Ojeda ◽  
César Venegas
Keyword(s):  
Commutative Rings ◽  
Polynomial Rings ◽  
Quantum Matrices ◽  
Weyl Algebras ◽  
Skew Polynomial Rings ◽  
Finite Dimensional ◽  
Quantum Version ◽  
Skew Pbw Extensions ◽  
Skew Polynomial ◽  
Finite Dimensional Lie Algebras

Many rings and algebras arising in quantum mechanics can be interpreted as skew Poincaré–Birkhoff–Witt (PBW) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper, we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand–Kirillov conjecture for the skew quantum polynomials.

scholarly journals THE MAXIMAL ORDER PROPERTY FOR QUANTUM DETERMINANTAL RINGS

2003 ◽  
Vol 46 (3) ◽  
pp. 513-529 ◽  
Author(s):  
T. H. Lenagan ◽  
L. Rigal
Keyword(s):  
Maximal Order ◽  
Subject Classification ◽  
Order Property ◽  
Quantum Matrices ◽  
Maximal Orders

AbstractWe develop a method of reducing the size of quantum minors in the algebra of quantum matrices $\mathcal{O}_q(M_n)$. We use the method to show that the quantum determinantal factor rings of $\mathcal{O}_q(M_n)c$ are maximal orders, for $q$ an element of $\mathbb{C}$ transcendental over $\mathbb{Q}$.AMS 2000 Mathematics subject classification: Primary 16P40; 16W35; 20G42

scholarly journals PBW deformations of Artin–Schelter regular algebras

2016 ◽  
Vol 15 (04) ◽  
pp. 1650064 ◽  
Author(s):  
Jason Gaddis
Keyword(s):  
Tensor Products ◽  
Global Dimension ◽  
One Dimensional ◽  
Simple Modules ◽  
Regular Algebras ◽  
Skew Polynomial ◽  
Polynomial Extensions

We consider properties and extensions of PBW deformations of Artin–Schelter regular algebras. PBW deformations in global dimension two are classified and the geometry associated to the homogenizations of these algebras is exploited to prove that all simple modules are one-dimensional in the non-PI case. It is shown that this property is maintained under tensor products and certain skew polynomial extensions.

scholarly journals A noncommutative geometric LR rule

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Edward Richmond ◽  
Vasu Tewari ◽  
Stephanie Van Willigenburg
Keyword(s):  
Schubert Varieties ◽  
Combinatorial Algorithm ◽  
Noncommutative Version ◽  
Geometric Explanation ◽  
International Audience ◽  
Richardson Variety

International audience The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given by Vakil and later generalized by Coskun. In this paper we give a noncommutative version of the geometric LR rule. As a consequence, we establish a geometric explanation for the positivity of noncommutative LR coefficients in certain cases.

scholarly journals QUANTUM ANALOGUES OF SCHUBERT VARIETIES IN THE GRASSMANNIAN

2008 ◽  
Vol 50 (1) ◽  
pp. 55-70 ◽  
Author(s):  
T.H. LENAGAN ◽  
L. RIGAL
Keyword(s):  
General Result ◽  
Minimal Element ◽  
Point Of View ◽  
Schubert Varieties ◽  
Regularity Conditions ◽  
Graded Algebras ◽  
Maximal Orders ◽  
Straightening Law

AbstractWe study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

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