scholarly journals PRIME SPECTRA OF AMBISKEW POLYNOMIAL RINGS

2018 ◽  
Vol 61 (1) ◽  
pp. 49-68
Author(s):  
CHRISTOPHER D. FISH ◽  
DAVID A. JORDAN
Keyword(s):  
Positive Integer ◽  
Polynomial Algebra ◽  
Central Element ◽  
Universal Enveloping Algebra ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Closed Field ◽  
Prime Spectrum ◽  
Enveloping Algebra ◽  
Root Of Unity

AbstractWe determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.

scholarly journals CONFORMAL $\mathcal{s}$$\mathcal{l}$2 ENVELOPING ALGEBRAS AS AMBISKEW POLYNOMIAL RINGS

2002 ◽  
Vol 45 (1) ◽  
pp. 91-115 ◽  
Author(s):  
Martin O’Neill
Keyword(s):  
Characteristic Zero ◽  
Subject Classification ◽  
Polynomial Rings ◽  
Irreducible Representations ◽  
Prime Ideals ◽  
Closed Field ◽  
Enveloping Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Parameter Deformation

AbstractWe study a three parameter deformation $\mathcal{U}_{abc}$ of $\mathcal{U}(\mathfrak{sl}_2)$ introduced by Le Bruyn in 1995. Working over an arbitrary algebraically closed field of characteristic zero, we determine the centres, the finite-dimensional irreducible representations, and, when the parameter $a$ is not a non-trivial root of unity, the prime ideals of those $\mathcal{U}_{abc}$, with $ac\neq0$, which are conformal as ambiskew polynomial rings.AMS 2000 Mathematics subject classification: Primary 16W35; 17B37. Secondary 16S36; 16S80

scholarly journals The prime spectrum of the universal enveloping algebra of the 1-spatial ageing algebra and of U(gl2)

2021 ◽  
Vol 31 (1) ◽  
pp. 1-16
Author(s):  
Volodymyr Bavula ◽  
◽  
Tao Lu ◽  
Keyword(s):  
Prime Ideal ◽  
Universal Enveloping Algebra ◽  
Explicit Description ◽  
Prime Spectrum ◽  
Enveloping Algebra

For the algebras in the title, their prime, primitive and maximal spectra are explicitly described. For each prime ideal an explicit set of generators is given. An explicit description of all the containments between primes is obtained.

THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(𝔟⋉V2)

2019 ◽  
Vol 62 (S1) ◽  
pp. S77-S98 ◽  
Author(s):  
VOLODYMYR V. BAVULA ◽  
TAO LU
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Direct Product ◽  
Prime Factor ◽  
Universal Enveloping Algebra ◽  
Explicit Description ◽  
Prime Ideals ◽  
Two Dimensional ◽  
Enveloping Algebra ◽  
Torsion Modules

AbstractLet 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.

scholarly journals On representation theory of total (co)integrals

2016 ◽  
Vol 15 (10) ◽  
pp. 1650195
Author(s):  
Mohammad Hassanzadeh
Keyword(s):  
Representation Theory ◽  
Hopf Algebra ◽  
Polynomial Algebra ◽  
Universal Enveloping Algebra ◽  
Drinfeld Modules ◽  
Enveloping Algebra

In this paper, we show that total integrals and cointegrals are new sources of stable anti-Yetter–Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter–Drinfeld and Yetter–Drinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions of the Connes–Moscovici Hopf algebra, and some examples of universal enveloping algebra and polynomial algebra.

scholarly journals Classification theorem on irreducible representations of theq-deformed algebraU′q(son)

2005 ◽  
Vol 2005 (2) ◽  
pp. 225-262 ◽  
Author(s):  
N. Z. Iorgov ◽  
A. U. Klimyk
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
Classical Type ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Complete Reducibility ◽  
Finite Dimensional

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandardq-deformationU′q(son)(which does not coincide with the Drinfel'd-Jimbo quantum algebraUq(son)) of the universal enveloping algebraU(son(ℂ))of the Lie algebrason(ℂ)whenqis not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. The theorem on complete reducibility of finite-dimensional representations ofU′q(son)is proved.

scholarly journals The Universal Enveloping Algebra of the Schrödinger Algebra and its Prime Spectrum

2018 ◽  
Vol 61 (4) ◽  
pp. 688-703 ◽  
Author(s):  
V. V. Bavula ◽  
T. Lu
Keyword(s):  
Universal Enveloping Algebra ◽  
Prime Spectrum ◽  
Enveloping Algebra ◽  
Whittaker Modules ◽  
Schrödinger Algebra

AbstractThe prime, completely prime, maximal, and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. The explicit generators are given for all of these ideals. A counterexample is constructed to the conjecture of Cheng and Zhang about nonexistence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds ‘generically’.

ON THE PRIME SPECTRUM OF THE ENVELOPING ALGEBRA AND CHARACTERISTIC VARIETIES

2007 ◽  
Vol 06 (03) ◽  
pp. 369-383 ◽  
Author(s):  
NIKOLAOS PAPALEXIOU
Keyword(s):  
Topological Space ◽  
Lie Algebra ◽  
Prime Ideals ◽  
Prime Spectrum ◽  
Semisimple Lie Algebra ◽  
Characteristic Variety ◽  
Enveloping Algebra ◽  
Commutative Algebras ◽  
Order Relations ◽  
Characteristic Varieties

Let 𝔤 be a semisimple Lie algebra and U(𝔤), its enveloping algebra. A problem in the theory of non-commutative algebras is the description of the set Spec U(𝔤) of prime ideals in U(𝔤) as topological space. Using the notion of the characteristic variety as introduced by Joseph, we compute some order relations between prime ideals.

scholarly journals DG Poisson algebra and its universal enveloping algebra

2016 ◽  
Vol 59 (5) ◽  
pp. 849-860 ◽  
Author(s):  
JiaFeng Lü ◽  
XingTing Wang ◽  
GuangBin Zhuang
Keyword(s):  
Poisson Algebra ◽  
Universal Enveloping Algebra ◽  
Enveloping Algebra
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