Skew PBW Extensions: Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications (Algebra and Applications, 28)

581 Pages·2020·7.351 MB·English

by  William Fajardo, Claudia Gallego, Oswaldo Lezama, Armando Reyes, Héctor Suárez, Helbert Venegas

Algebra and Applications
William Fajardo · Claudia Gallego 
Oswaldo Lezama · Armando Reyes 
Héctor Suárez · Helbert Venegas
Skew PBW
Extensions
Ring and Module-theoretic Properties,
Matrix and Gröbner Methods, and
Applications
Algebra and Applications

Volume 28
Series Editors

Michel Broué, Université Paris Diderot, Paris, France
Alice Fialowski, Eötvös Loránd University, Budapest, Hungary
Eric Friedlander, University of Southern California, Los Angeles, CA, USA
Iain Gordon, University of Edinburgh, Edinburgh, UK
John Greenlees, Sheffield University, Sheffield, UK
Gerhard Hiß, Aachen University, Aachen, Germany
Ieke Moerdijk, Utrecht University, Nijmegen, Utrecht, The Netherlands
Christoph Schwe igert, Hamburg University, Hamburg, Germany
Mina Teicher, Bar-Ilan University, Ramat-Gan, Israel
Alain Verschoren, University of Antwerp, Antwerp, Belgium
Algebra and Applications aims to publish well-written and carefully refereed mono-
graphs with up-to-date expositions of research in all fields of algebra, including its
classical impact on commutative and noncommutative algebraic and differential
geometry, K-theory and algebraic topology, and further applications in related
domains, such as number theory, homotopy and (co)homology theory through to
discrete mathematics and mathematical physics.
Particular emphasis will be put on state-of-the-art topics such as rings of differential
operators, Lie algebras and super-algebras, group rings and algebras, Kac-Moody
theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well
as their applications within mathematics and beyond. Books dedicated to compu-
tational aspects of these topics will also be welcome.

More information about this series at http://www.springer.com/series/6253
William Fajardo • Claudia Gallego
Oswaldo Lezama • Armando Reyes
Héctor Suárez • Helbert Venegas

Skew PBW Extensions

Ring and Module-theoretic Properties,

Matrix and Gröbner Methods,
and Applications
William Fajardo Claudia Gallego
Instituto Politécnico Grancolombiano Universidad Sergio Arboleda
Bogotá, Colombia Bogotá, Colombia

Oswaldo Lezama Armando Reyes
National University of Colombia National University of Colombia
Bogota, Colombia Bogota, Colombia

Héctor Suárez Helbert Venegas
Universidad Pedagógica y Universidad de la Sabana
Tecnológica de Colombia Chía, Colombia
Tunja, Colombia

ISSN 1572-5553 ISSN 2192-2950 (electronic )
Algebra and Applications
ISBN 978-3-030-53377-9 ISBN 978-3-030-53378-6 (eBook)
https://doi.org/10.1007/978-3-030-53378-6

Mathematics Subject Classification (2020): 16S36, 16U20, 16D40, 16E05, 16E65, 16S38, 16S80, 16W70,
16Z05

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Preface
The usual commutative polynomial ring in several variables over a commu-
tativeringR,R[x ,...,x ],canbegeneralizedtoanoncommutativecontext
1 n
bychangingRtoanynoncommutativering,butpreservingthebasicrulesof
multiplication,i.e.,x x =x x andrx =x rfor1≤i,j ≤nandanyr ∈R.
i j j i i i
However,insomeareasofmathematicsanditsapplications,suchasalgebras
of di(cid:11)erential operators in di(cid:11)erential equations and linear multidimensional
control systems in algebraic analysis (see [91], [92], [93], [95], [316], [318],
[319], [320], [317] and [323]), it is necessary to consider wider classes of rings
of polynomial type in which the variables do not commute or the coe(cid:14)cients
do not commute with the variables. For example, take a homogeneous linear
ordinary di(cid:11)erential equation with coe(cid:14)cients in Q[t],
p (t)y(n)+···+p (t)y0+p (t)y =0, p (t)∈Q[t], 0≤i≤n;
n 1 0 i
considering the di(cid:11)erential operator ∂ := d this equation can be interpreted
dt
as
[p (t)∂n+···+p (t)∂+p (t)](y)=0,
n 1 0
whereyisinthesetS ofsolutionsofthisequation,andinturn,S iscontained
in a D-module F, where D is a Q-algebra generated by two variables t,∂.
Observe that for p∈Q[t] and y ∈F, we must have
(∂p)(y)=∂(py)=p∂(y)+∂(p)y =py0+p0y =(p∂+p0)(y), i.e., ∂p=p∂+p0,
in particular, if p = t, then in D we have the rule of multiplication ∂t =
t∂ +1. Hence, in the new \polynomial ring" D = Q[t,∂], the variables do
not commute. These important types of rings are particular cases of the
skew polynomial rings of injective type, and these last ones are a subclass of
the skew PBW extensions that we will introduce and study in the present
monograph.
Skew PBW extensions represent a generalization of PBW (Poincar(cid:19)e{
Birkho(cid:11){Witt) extensions de(cid:12)ned by Bell and Goodearl ([48]), and include
the usual polynomials rings and a lot of other important classes of rings
vii
viii Preface
such as Weyl algebras, enveloping algebras of Lie algebras, and many exam-
ples of quantum algebras such as the Manin algebra of quantum matrices,
q-Heisenberg algebra, Hayashi algebra, Witten’s deformation of U(sl(2,K),
etc.Mostoftheexamplesofalgebrasandringsdescribedinthepresentwork
as skew PBW extensions have been investigated by many other authors,
but they interpreted them in a di(cid:11)erent way, for example, Levandovsky in
[231]de(cid:12)nedtheG-algebras,whichalsoincludeasimportantparticularcases
the quantum algebras mentioned before. In a similar way, Bueso, G(cid:19)omez-
Torrecillas and Verschoren in [74] introduced the PBW rings, which give
an alternative way of interpreting most of the quantum algebras and rings
arising in mathematical physics. It is also important to note that the Hopf
algebrasareanotherveryusefulwayofinterpretingallofthesenoncommuta-
tive algebras (see [201]). Thus, the skew PBW extensions can be considered
asanalternativetechniqueforstudyingaverywideclassofmodernnoncom-
mutative algebras of polynomial type which have recently arisen in quantum
mechanics and mathematical physics. Our point of view is very general since
for all of these rings of polynomial type the ring of coe(cid:14)cients is not neces-
sarily a (cid:12)eld, but an arbitrary ring.
Thismonographisdividedintofourparts;the(cid:12)rstpartisconcernedwith
the ring-module-theoretic and homological properties of skew PBW exten-
sions.Westartwiththede(cid:12)nitionandtheuniversalpropertythatcharacter-
izes this new class of noncommutative rings. In the second chapter we have
included many important examples of rings and algebras that can be inter-
preted as skew PBW extensions. This chapter contains a new short proof of
thePoincar(cid:19)e{Birkho(cid:11){Witttheoremonthebasesoftheuniversalenveloping
algebraofa(cid:12)nite-dimensionalLiealgebra.Theproofissupportedbytheuni-
versalcharacterizationandthetheoremofexistenceofskewPBW extensions
studied in the (cid:12)rst chapter. Some of the most classical topics of ring-module
theorywillbeconsideredfortheskew PBW extensionsinthis(cid:12)rstpart.We
willstudytheHilbertbasistheorem,ringsoffractionsandtheOreandGoldie
theorems.Aparticularcollectionofprimeidealswillbecomputed,aswellas
the Jacobson and the prime radicals in some special cases. Some dimensions
will be estimated, in particular, the global dimension, the Krull dimension,
the Goldie dimension and the Gelfand{Kirillov dimension. An introduction
to the Gelfand{Kirillov conjecture will be presented, and for this, we will
compute the center of some skew PBW extensions. The theory of regularity
will be considered, in particular, Serre’s theorem, Auslander regularity and
the Cohen{Macaulay condition. We will compute the Quillen K-groups for
bijective skew PBW extensions, in particular, the Grothendieck, Bass and
Milnor groups. Most of the computations and results of the (cid:12)rst part are
supported by the following basic facts: we will prove that skew PBW ex-
tensions are positively (cid:12)ltered rings and the corresponding graded rings are
iterated skew polynomial rings, hence, the well-known results in the litera-
ture (see [159] and [278]) about ring-theoretic and homological properties of
skew polynomials, and the transfer theorems from graded to (cid:12)ltered, can be
applied. The (cid:12)rst important theorem that we will prove using this technique
Preface ix
is the Hilbert basis theorem, which states if the ring of coe(cid:14)cients of a bi-
jective skew PBW extensions is left noetherian, then the extension is left
noetherian.
Thesecondpartisdedicatedtotheinvestigationof(cid:12)nitelygeneratedpro-
jective modules over skew PBW extensions from a matrix point of view.
Aspreparatorymaterial,weconsider(cid:12)rststablyfreeness,Sta(cid:11)ord’stheorem
aboutthestablerank,andtheHermiteconditionforarbitrarynoncommuta-
tiverings,andthen,weapplythesepreliminariestostudyextendedmodules
over skew PBW extensions. An elementary matrix-constructive proof of the
Quillen{Suslin theorem for single Ore extensions of bijective type over (cid:12)elds
is included.
In order to make constructive the theory of projective modules studied in
the second part, in the third part we will construct the theory of Gr(cid:127)obner
bases of left (right) ideals and modules for bijective skew PBW extensions.
Wewillextendsomeresultsof[421](comparealsowith[73],[231]and[257]).
Wewillpresentsomeapplicationsinnoncommutativehomologicalalgebra,as
was done in [239] for commutative polynomial rings (see also [236],[237] and
[238]). For example, we will compute syzygies and the Ext and Tor modules
over bijective skew PBW extensions. Matrix computations using Gr(cid:127)obner
bases are included in this part. We will calculate inverses of matrices as well
as algorithms for testing stably-freeness, and we will compute minimal pre-
sentations of stably-free modules over skew PBW extensions. The Gr(cid:127)obner
theory and some of the mentioned computations have been implemented in
Maple in [117] and [118]. This implementation is based on a library special-
izedforworkingwithbijectiveskewPBW extensions.Thelibraryhasutilities
to calculate Gr(cid:127)obner bases, and it includes some functions that compute the
moduleofsyzygies,freeresolutionsandleftinversesofmatrices,amongother
things. In addition, another independent library was created that makes it
possible to execute the Quillen{Suslin theorem for K[x;σ,δ], with K a (cid:12)eld,
σ a K-automorphism and δ a σ-derivation. The reader interested in these
computational aspects of the skew PBW extensions can consult Appendices
C, D and E at the end of the monograph (see also Chapter 4 and Appendix
A in [117]).
The last part of the monograph is dedicated to the application of skew
PBW extensions in the investigation of some key topics of the noncommu-
tative algebraic geometry of quantum algebras. For this purpose, we will
introduce the semi-graded rings, and for them we will study the noncommu-
tativeversionoftheSerre{Artin{Zhang{Verevkintheoremontheequivalence
ofacertainquotientabeliancategoryof(cid:12)nitelygeneratedsemi-gradedmod-
ules with the noncommutative version of the category of coherent sheaves.
The semi-graded Koszul algebras and the semi-graded Artin{Schelter regu-
lar algebras will be de(cid:12)ned and investigated. In order to understand better
the semi-graded rings, we include in Appendix A a quick review of the non-
commutativealgebraicgeometryof(cid:12)nitelygradedalgebrasandinAppendix
B a review of Koszul and Artin{Schelter regular algebras. Another classical
topic arising in commutative algebraic geometry is the Zariski cancellation
x Preface
problem.Wewillinvestigateitfornoncommutativeringsandalgebrasinthe
context of skew PBW extensions. We will identify families of skew PBW
extensions that are cancellative.
Aswasmentionedabove,thepresentmonographisbasedonknownresults
inthetheoryofskewpolynomialrings,andalsoonalgebraicandhomological
propertiesof(cid:12)lteredandgradedrings.Moreover,forthelastpartconcerning
the noncommutative algebraic geometry of skew PBW extensions, we use
basic properties of (cid:12)nitely graded algebras over (cid:12)elds. Thus, in general, the
monograph is not self-contained, and the wide list of references included at
the end contains many theoretical facts that we have used in the proofs of
propertiesandresultsontheskew PBW extensions.However,theusedfacts
have been properly inserted and cited in the text. A very reduced list of the
most used references, and that have mainly in(cid:13)uenced the present work, is
[20], [22], [23], [30], [51], [52], [74], [130], [138], [139], [140], [151], [159], [215],
[216],[231],[257],[264],[278],[281],[291],[292],[300],[311],[322],[323],[344],
[373], [379], [421].
The four parts of the monograph are almost independent, but we recom-
mend the readers to cover the (cid:12)rst three chapters in order to understand
betteranyofthetopicsstudied.Someproblems(probablyopen)whicharose
during the writing of this monograph have been included. The readers are
invited to work on them.
In this monograph we will use the following notation: in general, all rings
are noncommutative, but always with unity; if nothing contrary is said, all
modulesareleftmodules;K isa(cid:12)eld,S isanarbitraryring;S∗representsthe
group of invertible elements of S; M (S) is the set of rectangular matrices
r×s
over S of r rows and s columns; GL (S) := M (S)∗ is the general linear
r r×r
group of order r over S. If F is a matrix in M (S) := M (S), then its
r r×r
transpose is denoted by FT. Since we will mainly deal with skew PBW
extensions, we reserve the capital letter A to denote these types of rings.
Oswaldo Lezama
Bogot(cid:19)a, Colombia
June, 2020