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BEGIN:VEVENT
SUMMARY:Chelsea Walton (Rice University\, USA)
DTSTART:20210330T160000Z
DTEND:20210330T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/1
DESCRIPTION:Title: Universal Quantum Semigrupoids\nby Chelsea Walton (Rice Universit
y\, USA) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstr
act\nIn a recent paper (https://arxiv.org/abs/2008.00606)\, Hongdi Huang\,
Elizabeth Wicks\, Robert Won\, and I introduce the concept of a universal
quantum linear semigroupoid (UQSGd). This is a weak bialgebra that coacts
on a (not necessarily connected) graded algebra A. Our main result is tha
t when A is the path algebra kQ of a finite quiver Q each of the various U
QSGds introduced in our work is isomorphic to the face algebra attached to
Q (an important weak bialgebra due to Hayashi). Most of the talk will be
dedicated to setting up context and terminology towards the main result.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Gómez Torrecillas (Universidad de Granada\, Spain)
DTSTART:20210413T160000Z
DTEND:20210413T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/2
DESCRIPTION:Title: Biseparable extensions are not necessarily Frobenius\nby José G
ómez Torrecillas (Universidad de Granada\, Spain) as part of ONCAS Online
Noncommutative Algebra Seminar\n\n\nAbstract\nNecessary and sufficient co
nditions are given on an Ore extension A[x\; σ\, δ]\, where A is a finit
e dimensional algebra over a field F\, for being a Frobenius extension of
the ring of commutative polynomials F[x]. As a consequence\, as the title
of this talk highlights\, we provide a negative answer to a problem stated
by Caenepeel and Kadison. The involved ring-theoretical notions will be a
lso discussed. \n\n\nThe talk is based on the joint paper with F.J. Lobil
lo\, G. Navarro and P. Sánchez-Hernández\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Marsinkovsky (Northeastern University\, USA)
DTSTART:20210427T160000Z
DTEND:20210427T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/3
DESCRIPTION:Title: On Rings and Functors: Pictures at an Exhibition\nby Alexander Ma
rsinkovsky (Northeastern University\, USA) as part of ONCAS Online Noncomm
utative Algebra Seminar\n\n\nAbstract\nIt is banal to say that rings revea
l their nature through their module categories. However\, in the last few
years we have seen indications that a similar statement can be made about
stable module categories. Those are categories whose objects are the usual
modules\, but morphisms are formed by modding out maps factoring through
projectives (or injectives). Additive functors defined on such categories
are also said to be stable\, and the study of such functors adds even more
insights. The goal of this talk is to illustrate this thesis with simple
examples. We shall see how several familiar classes of rings can be charac
terized by properties of their stable functors. This will be an expository
talk. No prior knowledge of functor categories is assumed or needed.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivo Herzog (Ohio State University\, USA)
DTSTART:20210525T160000Z
DTEND:20210525T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/5
DESCRIPTION:Title: The Positselski-Štovíček Correspondence for the Recollements of Pu
rity\nby Ivo Herzog (Ohio State University\, USA) as part of ONCAS Onl
ine Noncommutative Algebra Seminar\n\n\nAbstract\nThe theory of purity for
modules over a ring R has been studied using the covariant\nas well as th
e contravariant functor categories. In their work on quiver Grassmanians\,
\nCrawley-Boevey and Sauter introduced a third such category\, the project
ive quotient \nfunctor category. We will explain how these three functor c
ategories are related to the three equivalent definitions of purity. Ironi
cally\, the projective quotient functor category is closest in spirit to P
rüfer's original definition. \n\nEach of these functor categories may be
regarded as the middle term of a recollement of abelian categories whose l
ocalization/colocalization is given by the category R-Mod of R-modules. We
will describe the basic theory of recollements of functor categories and
indicate how it reveals the common features of the three functor categorie
s. Each of these functor categories is related to the other two by a trian
gle of Positselski-Štovíček correspondences\, which allows a detailed a
nalysis of its homological properties. \n\nThis is joint work with Xianhui
Fu.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Trlifaj (Charles University\, Prague)
DTSTART:20211012T160000Z
DTEND:20211012T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/6
DESCRIPTION:Title: Closure properties of $\\displaystyle \\lim_\\longrightarrow \\mathca
l C$\nby Jan Trlifaj (Charles University\, Prague) as part of ONCAS On
line Noncommutative Algebra Seminar\n\n\nAbstract\nLet $\\mathcal C$ be a
class of (right $R$-) modules closed under finite direct sums. If $\\mathc
al C$ consists of finitely presented modules\, then the class $\\displayst
yle \\lim_\\longrightarrow \\mathcal C$ of all direct limits of modules fr
om $\\mathcal C$ is well-known to enjoy a number of closure properties. Mo
reover\, if $R \\in \\mathcal C$\, $\\mathcal C$ consists of FP$_2$-module
s\, and $\\mathcal C$ is closed under extensions and direct summands\, the
n $\\displaystyle \\lim_\\longrightarrow \\mathcal C$ can be described hom
ologically: $\\displaystyle \\lim_\\longrightarrow \\mathcal C$ is the dou
ble perpendicular class of $\\mathcal C$ with respect to the Tor$_1^R$ bif
unctor [1]. \n\nThings change completely when $\\mathcal C$ is allowed to
contain infinitely generated modules: $\\displaystyle \\lim_\\longrightarr
ow \\mathcal C$ then need not even be closed under direct limits. After pr
esenting some positive general results (and their constraints)\, we will c
oncentrate on two particular cases: $\\mathcal C = add(M)$ and $\\mathcal
C = Add(M)$\, for an arbitrary module $M$. We will prove that if $S = \\En
d M$ and $\\mathcal F$ is the class of all flat right $S$-modules\, then $
\\displaystyle \\lim_\\longrightarrow add(M) = \\{ F \\otimes _S M \\mid F
\\in \\mathcal F \\}$. For $\\displaystyle \\lim_\\longrightarrow Add(M)$
\, we will have a similar formula\, involving the contratensor product $\\
odot _S$ and direct limits of projective right $S$-contramodules (for $S$
endowed with the finite topology). We will also show that for various clas
ses of modules $\\mathcal D$\, if $M \\in \\mathcal D$ then $\\displaystyl
e \\lim_\\longrightarrow add(M) = \\displaystyle \\lim_\\longrightarrow Ad
d(M)$. However\, the equality remains open in general\, even for (infinite
ly generated) projective modules. \n\nThe talk is based on my recent joint
work with Leonid Positselski [2].\n\n[1] L.Angeleri Hügel\, J. Trlifaj:
Direct limits of modules of finite projective dimension\, in Rings\, Modul
es\, Algebras\, and Abelian Groups\, LNPAM 236\, M.Dekker\, New York 2004\
, 27-44.\n\n[2] L.Positelski\, J.Trlifaj: Closure properties of $\\display
style \\lim_\\longrightarrow \\mathcal C$\, preprint.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:André Leroy (University of Artois)
DTSTART:20211019T160000Z
DTEND:20211019T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/7
DESCRIPTION:Title: Commutatively closed rings and their graphs\nby André Leroy (Uni
versity of Artois) as part of ONCAS Online Noncommutative Algebra Seminar\
n\n\nAbstract\nA subset S of a ring R is commutatively closed if for any e
lements a\, b in R\, the product ab is in S if and only if the product ba
is in S. This concept was introduced in a recent paper and intended to hav
e another perspective on Dedekind finite\, reversible\, semicommutative\,
... rings. A topology was attached to this concept and in the present work
we attach a graph and are able to compute the diameter of this graph for
semisimple algebras. This answers some questions left open. \n\nThis is jo
int work with Mona Abdi.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Reyes (University of California)
DTSTART:20211102T160000Z
DTEND:20211102T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/8
DESCRIPTION:Title: Dual coalgebras as a quantized maximal spectrum\nby Manuel Reyes
(University of California) as part of ONCAS Online Noncommutative Algebra
Seminar\n\n\nAbstract\nIf an algebra A has “many” finite-dimensional r
epresentations\, we argue that its Sweedler dual coalgebra is a reasonable
functorial quantization of the maximal spectrum of A. Many such algebras
arise as twisted tensor products of commutative algebras\, including Ore e
xtensions and smash products. This leads to the problem of understanding t
he dual coalgebra of a twisted tensor product. We will discuss when the Sw
eedler dual of a twisted tensor product can be computed as a cross product
coalgebra\, a result that is achieved using methods of topological algebr
a.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Van C Nguyen (United States Naval Academy)
DTSTART:20211116T170000Z
DTEND:20211116T180000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/9
DESCRIPTION:Title: Twisting of graded quantum groups and solutions to the quantum Yang-B
axter equation\nby Van C Nguyen (United States Naval Academy) as part
of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\nLet $H$ be a
Hopf algebra over a field $k$ such that $H$ is $\\mathbb Z$-graded as an
algebra. In this talk\, we introduce the notion of a twisting pair for $H$
and show that the Zhang twist of $H$ by such a pair can be realized as a
2-cocycle twist. We use twisting pairs to describe twists of Manin's unive
rsal quantum groups associated to quadratic algebras. Furthermore\, we dis
cuss a strategy to twist a solution to the quantum Yang-Baxter equation vi
a the Faddeev-Reshetikhin-Takhtajan construction. If time permits\, we ill
ustrate this result for the quantized coordinate rings of $GL_n(k)$.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Musson (University of Wisconsin Milwaukee)
DTSTART:20211130T170000Z
DTEND:20211130T180000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/10
DESCRIPTION:Title: How to construct the lattice of submodules of a multiplicity free mo
dule from partial information\nby Ian Musson (University of Wisconsin
Milwaukee) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbs
tract\nIn general it is a difficult problem to construct the lattice of su
bmodules $L(M )$ given module $M$. In [Sta12] a method is outlined for con
stucting a distributive lattice from a knowledge of its join irreducibles.
However it is not an easy task to identify all join irreducible submodule
s of a given module. In the case of a multiplicity free module M we presen
t an alternative method based on the composition factors. As input we requ
ire a set of submodules $A_1\,\\ldots\, A_n$ whose submodule lattice is kn
own\, which contain all composition factors of $M$\, and for which all int
ersections $A_i \\cap A_j$ are known. From this we can reconstruct $L(M)$.
We illustrate the process for a Verma module M for the Lie superalgebra $
\\mathfrak{osp}(3\, 2)$. In this case\, $L(M)$ is isomorphic to the (exten
ded) free distributive lattice of rank 3. This is well-known\, but quite c
omplicated lattice. Indeed $M$\nhas 20 submodules and 8 composition factor
s\, each with multiplicity one.\n\n$\\mathbf{Bibliography}$\n\n[Sta12] R.
P. Stanley\, Enumerative combinatorics. Volume 1\, 2nd ed.\, Cambridge Stu
dies in Advanced Mathematics\, vol. 49\, Cambridge University Press\, Camb
ridge\, 2012. MR2868112\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Gaddis (Miami University\, Ohio)
DTSTART:20211214T170000Z
DTEND:20211214T180000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/11
DESCRIPTION:Title: Reflexive hull discriminants and applications\nby Jason Gaddis (
Miami University\, Ohio) as part of ONCAS Online Noncommutative Algebra Se
minar\n\n\nAbstract\nIn algebraic number theory\, the discriminant is an i
mportant invariant of a Galois field extension. There is a notion of the d
iscriminant for noncommutative algebras that are finite modules over their
centers. This has been used to solve several challenging problems\, such
as to classify the automorphism groups of certain families of noncommutati
ve algebras. But the discriminant is notoriously difficult to compute in l
arge rank. In this talk\, I will review some of the history behind the dis
criminant invariant and introduce a new notion\, the reflexive hull discri
minant. This modification has a geometric interpretation and\, moreover\,
is well-suited for algebras that are finitely generated but not necessaril
y free over their centers. As an application\, I will show how this invari
ant can be used to determine the automorphism groups for certain quantum g
eneralized Weyl algebras.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Posur (Munster University\, Germany)
DTSTART:20220215T170000Z
DTEND:20220215T180000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/13
DESCRIPTION:Title: On free abelian categories for theorem proving\nby Sebastian Pos
ur (Munster University\, Germany) as part of ONCAS Online Noncommutative A
lgebra Seminar\n\n\nAbstract\nComputing explicitly within a free mathemati
cal object can be interpreted as theorem proving. In this talk\, we discus
s the constructiveness of free abelian categories. A very concrete descrip
tion of free abelian categories was given by Murray Adelman\, and we demon
strate how his description can be employed to validate homological lemmata
like the Snake lemma computationally.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miodrag Iovanov (University of Iowa\, USA)
DTSTART:20220308T170000Z
DTEND:20220308T180000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/14
DESCRIPTION:Title: Quantum groups of finite representation type\nby Miodrag Iovanov
(University of Iowa\, USA) as part of ONCAS Online Noncommutative Algebra
Seminar\n\n\nAbstract\nAlgebras of finite representation type - that is\,
those who have only finitely many indecomposable finite dimensional repre
sentations up to isomorphism - have been of central interest in representa
tion theory. Classically\, they appeared from work in modular representati
ons\; a (finite) group has finite representation type iff its p-Sylow subg
roup is cyclic. On the quantum side\, results of Farnsteiner describe the
structure of finite group schemes (finite dimensional co-commutative Hopf
algebras). Among the first examples of non-commutative and non-cocommutati
ve quantum groups (Hopf algebras) are the Sweedler algebra and Taft algebr
as. These are pointed (their simple modules are 1-dimensional - they are p
oints)\, and they are also of finite type.\n\nTo study this in the general
ity of infinite dimensional quantum groups (which includes gl_n\, quantum
sl\, etc.)\, one defines an algebra to be of finite type if given any dime
nsion vector\, there are only finitely many indecomposables of this dimens
ion vector\; by the well known Brower-Thrall problems\, this is equivalent
to the above for finite dimensional algebras. We give an overview of vari
ous examples of infinte quantum groups of finite type\, and give a complet
e classification of the pointed quantum groups of finite representation ty
pe. We re-obtain results known for the finite dimensional case (including
Taft algebras and their generalizations)\, and show that these include sev
eral interesting Hopf algebras\, such as those whose categories of comodul
es form the category of chain complexes or the category of double chain co
mplexes\, and in general\, the list includes these and certain kind of twi
sts and deformations of theirs.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pace Nielsen (Brigham Young University\, USA)
DTSTART:20220315T160000Z
DTEND:20220315T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/15
DESCRIPTION:Title: Nilpotent polynomials with non-nilpotent coefficients\nby Pace N
ielsen (Brigham Young University\, USA) as part of ONCAS Online Noncommuta
tive Algebra Seminar\n\n\nAbstract\nIt is well known that the coefficients
of nilpotent polynomials over noncommutative rings generally are not all
nilpotent. We show that this remains true even under extremely strong res
trictions on the set of nilpotents in the coefficient ring. If $R$ is a r
ing and its set of nilpotents\, ${\\rm Nil}(R)$\, satisfies ${\\rm Nil}(R)
^2=0$\, then in general ${\\rm Nil}(R[x])\\not \\subseteq {\\rm Nil}(R)[x]
$. This is proven by constructing an explicit polynomial example. The sm
allest possible degree of such a polynomial is seven. Related problems ar
e raised\, as well as connections to Kothe's conjecture and work of Agata
Smoktunowicz.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahn Majid (Queen Mary University of London\, UK)
DTSTART:20220329T160000Z
DTEND:20220329T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/16
DESCRIPTION:Title: Quantum jet bundles\nby Shahn Majid (Queen Mary University of Lo
ndon\, UK) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbs
tract\nWe formulate a notion of jet bundles over a possibly noncommutative
algebra \nA equipped with a torsion free connection. Among the conditions
needed for 3rd-order jets and above is that the connection also be flat a
nd its `generalised braiding tensor' σ:Ω1⊗AΩ1→Ω1⊗AΩ1 obey the Y
ang-Baxter equation or braid relations. We also cover the case of jet bund
les of a given `vector bundle' over A in the form of a bimodule E with fla
t bimodule connection with its braiding σEobeying the coloured braid rela
tions. Examples include the permutation group S3 with its 2-cycles calculu
s\, M2(ℂ) and the bicrossproduct model quantum spacetime in two dimensio
ns.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henrik Holm (University of Copenhagen\, Denmark)
DTSTART:20220426T160000Z
DTEND:20220426T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/17
DESCRIPTION:Title: The Q-shaped derived category of a ring\nby Henrik Holm (Univers
ity of Copenhagen\, Denmark) as part of ONCAS Online Noncommutative Algebr
a Seminar\n\n\nAbstract\nThe derived category D(A) of the category Mod(A)
of modules over a ring A is an important example of a triangulated categor
y in algebra. It can be obtained as the homotopy category of the category
Ch(A) of chain complexes of A-modules equipped with its standard model str
ucture. One can view Ch(A) as the category Fun(Q\,Mod(A)) of additive func
tors from a certain small preadditive category Q to Mod(A). The model stru
cture on Ch(A) = Fun(Q\,Mod(A)) is not inherited from a model structure on
Mod(A) but arises instead from the "self-injectivity" of the special cate
gory Q. We will show that the functor category Fun(Q\,Mod(A)) has two inte
resting model structures for many other self-injective small preadditive c
ategories Q. These two model structures have the same weak equivalences\,
and the associated homotopy category is what we call the Q-shaped derived
category of A. We will also show that it is possible to generalize the hom
ology functors on Ch(A) to homology functors on Fun(Q\,Mod(A)) for most se
lf-injective small preadditive categories Q. The talk is based on a joint
paper with Peter Jørgensen (arXiv:2101.06176)\, which has the same title
as the talk.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Rothmaler (CUNY)
DTSTART:20220510T160000Z
DTEND:20220510T170000Z
DTSTAMP:20260404T110914Z
UID:ONCAS/18
DESCRIPTION:Title: Every module has an Ulm length\nby Philipp Rothmaler (CUNY) as p
art of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\nTo make
sense of the statement in the title\, I introduce a concept of Ulm submodu
le that grew out of discussions with Alex Martsinkovsky and applies to any
module over any associative ring with 1. In abelian groups it coincides w
ith the classical notion\, and so does the rest of the investigation. \n\n
As usual\, a module is said to have Ulm length 0 if it coincides with its
own Ulm submodule. These modules form a definable subcategory\, which\, ov
er domains\, coincides with that of divisible modules. The subcategory is
equal to the entire category if and only if the ring is absolutely pure (o
n the same side). Over RD-domains\, like Prüfer domains or the first Weyl
algebra over a field of characteristic 0\, Ulm length 0 modules are injec
tive.\n\nTaking the Ulm submodule constitutes a functor\, which\, by itera
tion\, leads to higher Ulm functors as usual and in turn to Ulm sequences
and the notion of Ulm length for any module. One of the main results is th
at the (first) Ulm submodule of a pure-injective has length 0. In other wo
rds\, pure-injectives have Ulm length at most 1\, just as over the intege
rs. As a consequence\, every module is a pure (even elementary) submodule
of a module of Ulm length at most 1.\n\nAs another consequence one obtains
\, for any pure-injective over an RD domain\, a direct decomposition into
a largest injective submodule (= the first Ulm submodule)\, and a reduced
module\, that is\, a module with zero Ulm submodule.\n
LOCATION:https://stable.researchseminars.org/talk/ONCAS/18/
END:VEVENT
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