scholarly journals Bott–Samelson Varieties and Poisson Ore Extensions

2019 ◽  
Author(s):  
Balázs Elek ◽  
Jiang-Hua Lu
Keyword(s):  
Total Positivity ◽  
Cluster Algebras ◽  
Poisson Brackets ◽  
Ore Extension ◽  
Bruhat Cells ◽  
Toric Degenerations ◽  
Ore Extensions ◽  
Structure Constants ◽  
Poisson Varieties ◽  
Coordinate Rings

Abstract We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

scholarly journals Affine cluster monomials are generalized minors

2019 ◽  
Vol 155 (7) ◽  
pp. 1301-1326
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Lie Groups ◽  
Cluster Algebras ◽  
Quiver Representations ◽  
Group Representations ◽  
Finite Dimensional ◽  
Bruhat Cells ◽  
Level Zero ◽  
Coordinate Rings ◽  
Affine Type

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.

Ore extensions of quasitriangular Hopf group coalgebras

2014 ◽  
Vol 13 (06) ◽  
pp. 1450016 ◽  
Author(s):  
Daowei Lu ◽  
Dingguo Wang
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Baxter Equation ◽  
Ore Extension ◽  
Ore Extensions ◽  
Necessary And Sufficient

In this paper, we mainly consider some special Ore extension of quasitriangular Hopf group coalgebra, and give the necessary and sufficient conditions when the Ore extension of quasitriangular Hopf group coalgebras will preserve the same quasitriangular structure. Furthermore, in the two examples given at the end, we construct new solutions of Yang–Baxter equation of Hopf group coalgebras version.

On Ore extension and skew power series rings with some restrictions on zero-divisors

2016 ◽  
Vol 16 (09) ◽  
pp. 1750164
Author(s):  
E. Hashemi ◽  
A. As. Estaji ◽  
A. Alhevaz
Keyword(s):  
Power Series ◽  
Left Zero ◽  
Ore Extension ◽  
Series Ring ◽  
Zero Divisors ◽  
Open Questions ◽  
Ore Extensions ◽  
Power Series Rings ◽  
Extension Ring ◽  
The Right

The study of rings with right Property ([Formula: see text]), has done an important role in noncommutative ring theory. Following literature, a ring [Formula: see text] has right Property ([Formula: see text]) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ([Formula: see text]) of Ore extensions as well as skew power series rings. We will show that if [Formula: see text] is a right duo ring, then the skew power series ring [Formula: see text] has right Property ([Formula: see text]), when [Formula: see text] is right Noetherian and [Formula: see text]-compatible. Moreover, for a right duo ring [Formula: see text] which is [Formula: see text]-compatible, it is shown that (i) the Ore extension ring [Formula: see text] has right Property ([Formula: see text]) and (ii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip. As a corollary of our results, we provide answers to some open questions related to Property [Formula: see text], raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ([Formula: see text]) and their extensions, J. Algebra 315 (2007) 612–628].

scholarly journals Cluster algebras III: Upper bounds and double Bruhat cells

2005 ◽  
Vol 126 (1) ◽  
pp. 1-52 ◽  
Author(s):  
Arkady Berenstein ◽  
Sergey Fomin ◽  
Andrei Zelevinsky
Keyword(s):  
Upper Bounds ◽  
Cluster Algebras ◽  
Bruhat Cells

scholarly journals PS-Modules over Ore Extensions and Skew Generalized Power Series Rings

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Refaat M. Salem ◽  
Mohamed A. Farahat ◽  
Hanan Abd-Elmalk
Keyword(s):  
Power Series ◽  
Associative Ring ◽  
Ore Extension ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Monoid Homomorphism ◽  
Ore Extensions ◽  
Power Series Rings ◽  
Unitary Right

A rightR-moduleMRis called a PS-module if its socle,SocMR, is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. LetRbe an associative ring with identity,MRa unitary rightR-module,O=Rx;α,δOre extension,MxOa rightO-module,S,≤a strictly ordered additive monoid,ω:S→EndRa monoid homomorphism,A=RS,≤,ωthe skew generalized power series ring, andBA=MS,≤RS,≤, ωthe skew generalized power series module. Then, under some certain conditions, we prove the following: (1) IfMRis a right PS-module, thenMxOis a right PS-module. (2) IfMRis a right PS-module, thenBAis a right PS-module.

scholarly journals ON q-SKEW ITERATED ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

2011 ◽  
Vol 10 (04) ◽  
pp. 771-781
Author(s):  
ANDRÉ LEROY ◽  
JERZY MATCZUK
Keyword(s):  
Polynomial Identity ◽  
Ore Extension ◽  
Ore Extensions

For iterated Ore extensions satisfying a polynomial identity (PI) we present an elementary way of erasing derivations. As a consequence we recover some results obtained by Haynal. [PI degree parity in q-skew polynominal rings, J. Algebra319 (2008) 4199–4221]. We also prove that under mild assumptions on Rn = R[x1; σ1, δ1]⋯ [xn;σn;δn], the Ore extension R[x1;σ1]⋯[xn;σn] exists and is PI if Rn is PI.

scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

scholarly journals Toric degenerations of cluster varieties and cluster duality

2020 ◽  
Vol 156 (10) ◽  
pp. 2149-2206
Author(s):  
Lara Bossinger ◽  
Bosco Frías-Medina ◽  
Timothy Magee ◽  
Alfredo Nájera Chávez
Keyword(s):  
Toric Variety ◽  
Mirror Symmetry ◽  
Duke Math ◽  
Cluster Algebras ◽  
Canonical Bases ◽  
Toric Degeneration ◽  
The Family ◽  
Toric Degenerations ◽  
Cluster Varieties ◽  
Natural Way

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Simplicity of iterated Ore extensions

2021 ◽  
pp. 2250052
Author(s):  
Mamta Balodi ◽  
Sumit Kumar Upadhyay
Keyword(s):  
Polynomial Ring ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Commutative Domain ◽  
Necessary And Sufficient ◽  
Commutative Subring

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

Some types of ring elements in Ore extensions over noncommutative rings

2017 ◽  
Vol 16 (11) ◽  
pp. 1750201 ◽  
Author(s):  
E. Hashemi ◽  
M. Hamidizadeh ◽  
A. Alhevaz
Keyword(s):  
Ore Extension ◽  
Von Neumann ◽  
Noncommutative Rings ◽  
Regular Elements ◽  
Derivation Formula ◽  
Von Neumann Regular ◽  
Ore Extensions ◽  
Base Ring ◽  
Extension Ring ◽  
Ring Elements

Let [Formula: see text] be an associative unital ring with an endomorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text]. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the [Formula: see text]-regular elements and also the von Neumann local elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible. As an application, we completely characterize the clean elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible.

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