Iterated Ore Extensions Research Articles

Coideal subalgebras of pointed and connected Hopf algebras

Let H H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B A\supseteq B be left (or right) coideal subalgebras of H H that contain the coradical of H H . We show that A A has a PBW basis over B B , provided that H H satisfies certain mild conditions. In the case that H H is a connected graded Hopf algebra of characteristic zero and A A and B B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A A is a graded iterated Ore extension of B B . These results turn out to be conceptual consequences of a structure theorem for each pair S ⊇ T S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S S over T T . The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don’t priorilly assume R R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.

Iterated Ore polynomial maps

We study a natural notion of polynomial maps attached to elements of an iterated Ore extension [Formula: see text]. We develop some tools to analyze these maps such as good points, multilinear transformations and P-independence. We also present polynomial maps arising in the Multivariate extensions [Formula: see text], introduced by Martínez-Peñas and Kschischang [Evaluation and interpolation over multivariate skew polynomial rings, J. Algebra 525 (2019) 111–139], through the use of Multilinear polynomial maps.

Simplicity of iterated Ore extensions

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.

Ore extensions and infinite triangularization

We give infinite triangularization and strict triangularization results for algebras of operators on infinite-dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore extensions but need not be free as modules over the intermediate subrings. Ore-solvable algebras include many examples as particular cases, such as group algebras of polycyclic groups or finite solvable groups, enveloping algebras of solvable Lie algebras, quantum planes and quantum matrices. We prove both triangularization and strict triangularization results for this class, and show how they generalize and extend classical simultaneous triangularization results such as the Lie and Engel theorems. We show that these results are, in a sense, the best possible, by showing that any finite-dimensional triangularizable algebra must be of this type. We also give connections between strict triangularization and nil and nilpotent algebras, and prove a very general result for algebras defined via a recursive “Ore” procedure starting from building blocks which are either nil, commutative or finite-dimensional algebras.

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

ABSTRACTGiven a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.

On the ACCP in skew Poincaré–Birkhoff–Witt extensions

The objective of this paper is to investigate the ascending chain condition on principal left (resp. right) ideals of noncommutative rings known as skew PBW extensions. We consider these extensions over domains and over \(\varSigma \)-rigid rings. We unify and extend several results in the literature for Ore extensions and another families of rings which can not be expressed as iterated Ore extensions.

Actions of Ore extensions and growth of polynomial H-identities

ABSTRACTWe show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexpH(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

A note on a paper by Cuadra, Etingof and Walton

ABSTRACTWe analyze the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebra H on the nth Weyl algebra A = An(K) over a field K of characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebra H on an iterated Ore extension of derivation type in characteristic zero factors through a group algebra.

Classification of relation types of Ore extensions of dimension 5

ABSTRACTTo study AS-regular algebras of dimension 5, we consider dimension 5 graded iterated Ore extensions generated in degree one. We classify the possible degrees of relations and structure of the free resolution for extensions with 3 and 4 generators. We show that every known type of algebra of dimension 5 can be realized by an Ore extension and we consider which of these types cannot be realized by an enveloping algebra.

Universal enveloping algebras of Poisson Ore extensions

We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra. As a consequence, we observe certain ring-theoretic invariants of the universal enveloping algebras that are preserved under iterated Poisson-Ore extensions. We apply our results to iterated quadratic Poisson algebras arising from semiclassical limits of quantized coordinate rings and a family of graded Poisson algebras of Poisson structures of rank at most two.

Connected Hopf algebras and iterated Ore extensions

We investigate when a skew polynomial extension T=R[x;σ,δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand–Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.

On the indecomposable modules of U q (sl(2)) constructed via ore-extension

Let A be a subalgebra of U q (sl(2)) generated by K,K −1 and F and A δ be a subalgebra of U q (sl(2)) generated by K,K −1 (and also F d if q is a primitive d-th root of unity with d an odd number). Given an A δ -module M, a U q (sl(2))-module \(A \otimes _{A^\delta } M\) is constructed via the iterated Ore extension of U q (sl(2)) in a unified framework for any q. Then all the submodules of \(A \otimes _{A^\delta } M\) are determined for a fixed finite-dimensional indecomposable A δ -module M. It turns out that for some indecomposable A δ -module M, the U q (sl(2))-module \(A \otimes _{A^\delta } M\) is indecomposable, which is not in the BGG-categories O q associated with quantum groups in general.

Free subalgebras of quotient rings of Ore extensions

Let $K$ be a field, let $\sigma$ be an automorphism of $K$, and let $\delta$ be a derivation of $K$. We show that if $D$ is one of $K(x;\sigma)$ or $K(x;\delta)$, then $D$ either contains a free algebra over its center on two generators, or every finitely generated subalgebra of $D$ satisfies a polynomial identity. As a corollary, we are able to show that the quotient division ring of any iterated Ore extension of an affine domain satisfying a polynomial identity either again satisfies a polynomial identity or it contains a free algebra over its center on two variables.

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6] introduced an algebra extension A P [y 1, y 2; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y 1; σ1, δ1][y 2; σ2, δ2]. We also give partial answers to some questions posed in Zhang and Zhang [6].

ON q-SKEW ITERATED ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

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.

Cauchon diagrams for quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra, K a commutative field and q a nonzero element of K which is not a root of unity. To each reduced decomposition of the longest element w 0 of the Weyl group W corresponds a PBW basis of the quantised enveloping algebra U q + ( g ) , and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of w 0 , this theory constructs a bijection between the set of prime ideals in U q + ( g ) that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of w 0 corresponds to a good ordering (in the sense of Lusztig (1990) [Lus90]) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes Lusztig (1990) [Lus90]: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitly describe the set of Cauchon diagrams for explicit reduced decomposition of w 0 in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of W. In Cauchon and Mériaux (2008) [CM08], we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of w 0 . So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of w 0 corresponding to a good ordering.

Generic Lie colour algebras

We describe a type of Lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Gröbner basis methods can be used to study universal enveloping algebras of factors of generic Lie colour algebras.

Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R = k [ X 1 ][ X 2 ; σ 2 , δ 2 ]⋯[ X N ; σ N , δ N ] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F =Fract( R ), starting with the indeterminates ( X 1 ,…, X N ) and terminating with new variables ( T 1 ,…, T N ). These new variables generate some quantum-affine space R such that F= Fract ( R ) . This algorithm induces a natural embedding ϕ : Spec (R)→ Spec ( R ) which satisfies the following property: If P is in Spec (R), then Fract (R/P)≃ Fract ( R /ϕ(P)) . We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: U q w ( g ) , B q w ( g ) ( w ∈ W ), R q [ G ] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra g and Weyl group W ), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.