Recent Developments in Representation Theory: Maurice Auslander Distinguished Lectures and International Conference, May 1-6, 2014, Woods Hole ... Woods Hole, Ma

258 Pages·2016·1.854 MB·English

by  Alex Martsinkovsky, Gordana Todorov, Kiyoshi Igusa

673
Recent Developments in
Representation Theory
Maurice Auslander Distinguished Lectures
and International Conference
May 1–6, 2014
Woods Hole Oceanographic Institute,
Woods Hole, MA
Alex Martsinkovsky
Gordana Todorov
Kiyoshi Igusa
Editors
AmericanMathematicalSociety
673
Recent Developments in
Representation Theory
Maurice Auslander Distinguished Lectures
and International Conference
May 1–6, 2014
Woods Hole Oceanographic Institute,
Woods Hole, MA
Alex Martsinkovsky
Gordana Todorov
Kiyoshi Igusa
Editors
AmericanMathematicalSociety
Providence,RhodeIsland
EDITORIAL COMMITTEE
Dennis DeTurck, Managing Editor
Michael Loss Kailash Misra Catherine Yan
2010 Mathematics Subject Classification. Primary 16G10,16G20, 16G70, 16T05, 14N15.
Library of Congress Cataloging-in-Publication Data
Names: Maurice Auslander Distinguished Lectures and International Conference (2014: Woods
Hole, Mass.) Martsinkovsky, A. (Alex), editor. — Todorov, G. (Gordana), editor. — Igusa,
Kiyoshi,1949-editor.
Title: Recentdevelopmentsinrepresentationtheory: MauriceAuslanderDistinguishedLectures
andInternationalConference: May1-6,2014,WoodsHoleOceanographicInstitute,WoodsHole,
Massachusetts/AlexMartsinkovsky,GordanaTodorov,KiyoshiIgusa,editors.
Description: Providence, Rhode Island: AmericanMathematicalSociety, [2016]—Series: Con-
temporarymathematics;volume673—Includesbibliographicalreferences.
Identifiers: LCCN2016001389—ISBN9781470419554(alk. paper)
Subjects: LCSH:Associativerings–Congresses. —Representationsofrings(Algebra)–Congresses.
— AMS: Associative rings and algebras – Representation theory of rings and algebras – Repre-
sentations of Artinian rings. msc — Associative rings and algebras – Representation theory of
rings and algebras – Representations of quivers and partially ordered sets. msc — Associative
rings and algebras – Representation theory of rings and algebras – Auslander-Reiten sequences
(almost split sequences) and Auslander-Reiten quivers. msc — Associative rings and algebras –
Hopf algebras, quantum groups and related topics – Hopf algebras and their applications. msc
— Algebraic geometry – Projective and enumerative geometry – Classical problems, Schubert
calculus. msc
Classification: LCCQA251.5.M282014—DDC515/.7223–dc23
LCrecordavailableathttp://lccn.loc.gov/2016001389
ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online)
DOI:http://dx.doi.org/10.1090/conm/673
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Contents
Preface v
Orders in Artinian rings, Goldie’s Theorem and the largest left quotient ring
of a ring
V. V. Bavula 1
Invariant theory of Artin-Schelter regular algebras: a survey
Ellen E. Kirkman 25
The Catalan combinatorics of the hereditary artin algebras
Claus Michael Ringel 51
Grassmannians, flag varieties, and Gelfand-Zetlin polytopes
Evgeny Smirnov 179
On the combinatorics of the set of tilting modules
Luise Unger 227
iii

Preface
This volume continues thenewly establishedtraditionofpublishing expository
lecturesfromtheMauriceAuslanderDistinguishedLecturesandInternationalCon-
ference. These events took place May 1-6, 2014 at the Woods Hole Oceanographic
Institute in Falmouth, MA.
The support from the National Science Foundation (Grant DMS-1162304) al-
lowedustoinvite59participantsfromAustria,Canada,CzechRepublic,Germany,
India, Mexico, Norway, Russia, Spain, Turkey, the UK, and the US. Twenty five of
them were either graduate students or postdocs.
The talks in the conference spread over a wide area of research. The same
appliestotheselectedexpositorylecturesinthisvolume. VladimirBavulapresents
a survey of old and new results on left orders in left Artinian rings. Ellen Kirkman
gives an introduction to invariant theory of Artin-Schelter regular algebras. Claus-
MichaelRingeldescribesthelatestresultsrelatingcombinatoricsandrepresentation
theory of hereditary artin algebras. Evgeny Smirnov's survey touches upon the
classical topic of Schubert calculus on Grassmannian and flag varieties and relates
it to the combinatorics of Gelfand-Zetlin polytopes. Luise Unger introduces the
reader to the combinatorics of tilting modules.
The list of the topics above, as spread out as it is, does not fully represent the
scientific breadth of the conference, even in the expository aspect of it. Because of
the time constraints, some of the presented expository lectures did not transition
into a written format. The editors hope very much that at least some of them will
eventually make it into print.
The support for the Conference from the NSF and the multi-year continued
support for the Maurice Auslander Distinguished Lecturesfrom BerniceAuslander
create a unique symbiotic relationship between the two events. The expository
component of the Conference has been attracting a large number of graduate stu-
dents and young researchers. The editors of this volume hope that its publication
will benefit many more mathematicians around the world.
The Editors
v

ContemporaryMathematics
Volume673,2016
http://dx.doi.org/10.1090/conm/673/13488
Orders in Artinian rings, Goldie’s Theorem and the largest
left quotient ring of a ring
V. V. Bavula
Abstract. Thisshortsurveyisaboutsomeoldandnewresultsonleftorders
inleftArtinianrings,newcriteriaforaringtohaveasemisimpleleftquotient
ring,newconcepts(e.g.,thelargestleftquotientringofaring).
Contents
(1) Introduction.
(2) New criteria for a ring to have a semisimple left quotient ring.
(3) Old criteria for a ring to have a left Artinian left quotient ring.
(4) Necessary and sufficient conditions for a ring to have a left Artinian left
quotient ring.
(5) A criterion via associated graded ring.
(6) Criteria similar to Robson’s Criterion.
(7) A left quotient ring of a factor ring.
(8) The largest denominator sets and the largest left quotient ring of a ring.
(9) The maximal left quotient rings of a ring.
(10) Examples.
1. Introduction
Inthispaper,modulemeansaleftmodule,allringsareassociativewith1. The
present paper comprises three parts.
Part I, ‘New Criteria for a Ring to have a Semisimple Left Quotient Ring’
(Section 2). Goldie’s Theorem (1958, 1960) is an old and up to 2013 was the only
example of such criteria. Four new criteria will be given that are independent of
Goldie’s Theorem and are based on completely new ideas and approach.
Part II, ‘Left Orders in Left Artinian Rings’ (Sections 3–7), deals with old and
new criteria for a ring to have a left Artinian left quotient ring.
2010 Mathematics Subject Classification. Primary 16U20, 16P40, 16S32, 13N10, 16P20,
16U20,16P60.
Key words and phrases. Goldie’s Theorem, orders, left Artinian ring, the left quotient ring
ofaring,thelargestleftquotientringofaring,thelargestregularleftOreset.
(cid:2)c2016 American Mathematical Society
1
2 V.V.BAVULA
Part III, ‘The Largest Left Quotient Ring of a Ring’ (Sections 8–10), is about
new recent concepts and results obtained in order to answer the old question:
Why does the classical left quotient ring of a ring not always exist?
A positive step in this direction is the fact that for an arbitrary ring R there
alwaysexiststhelargestleftquotientringQ (R),[4],whichcoincideswiththeclas-
l
sicalleftquotientringifthelatterexists. Anothernewconcept/factistheexistence
of the maximal left quotient rings (for an arbitrary ring R). Their existence gives
an affirmative answer to the following question: given a ring R, replace the ring R
byitsleftlocalizationS−1Rataleftdenominatorset(thatnotnecessarilyconsists
1
of regular elements). Then repeat the step again and again (infinitely many times
of arbitrary cardinality, if necessary): S−1(S−1R), S−1(S−1(S−1R)),...; will this
2 1 3 2 1
process stop? (i.e. do we reach the moment we cannot invert anything new?) The
answer is yes and the rings we obtain are called the maximal left quotient rings of
a ring and any such a ring can be written as S−1R for some left denominator set
S of the ring R, [4].
Goldie’s Theorem (1960), which is one of the most important results in Ring
Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of
thepaperistogivefournewcriteria(usingacompletelydifferentapproachandnew
ideas). The first one is based on the recently discovered fact that for an arbitrary
ring R the set M of maximal left denominator sets of R is a non-empty set [4].
Theorem (The Firs(cid:2)t Criterion). A ring R has a semisimple left quotient ring
Q iff M is a finite set, ass(S) = 0 and, for each S ∈ M, the ring S−1R is
S∈M (cid:3)
a simple left Artinian ring. In this case, Q(cid:4) S−1R.
S∈M
The Second Criterion is given via the minimal primes of R and goes further
than the First one in the sense that it describes explicitly the maximal left denom-
inator sets S via the minimal primes of R. The Third Criterion is close to Goldie’s
Criterionbutitiseasiertocheckinapplications(basically,itreducesGoldie’sThe-
oremtotheprimecase). TheFourthCriterionisgivenviacertainleftdenominator
sets.
The conditions in old criteria for a ring R to have a left Artinian left quotient
ring Q are ‘strong’ (like ‘the ring R is a left Goldie ring’) and given in terms of
the ring R itself and its ideals. The conditions of the new criteria are ‘weak’ (like
‘the ring R is a left Goldie ring’ where R := R/n and n is the prime radical of R)
and given in terms of the ring R (rather than R) and of its finitely many explicit
modules.
Goldie’s Theorem [12] characterizes left orders in semi-simple rings, it is a
criterion of when the left quotient ring of a ring is semi-simple (earlier, character-
izations were given, by Goldie [11] and Lesieur and Croisot [16], of left orders in
a simple Artinian ring). Talintyre [26] and Feller and Swokowski [10] have given
conditionswhicharesufficientforaleftNoetherianringtohavealeftquotientring.
Further, foraleftNoetherianringwhichhasaleftquotientring, Talintyre[27]has
established necessary and sufficient conditions for the left quotient ring to be left
Artinian. Small [21,22], Robson [20], and latter Tachikawa [25] and Hajarnavis
[13]havegivendifferentcriteriaforaringtohavealeftArtinianleftquotientring.
In this paper, three more new criteria are given (Theorem 4.1, Theorem 5.1 and
Theorem 6.1).
ORDERS IN ARTINIAN RINGS 3
Theorem 7.1 gives an affirmative answer to the question: Let R be a ring with
a left Artinian left quotient ring Q and I be a C-closed ideal of R such that I ⊆n.
Is the left quotient ring Q(R/I) of R/I a left Artinian ring?
The set C of regular elements of a ring R is not always a left (or right) Ore
set in R (hence, the left quotient ring C−1R or the right quotient ring RC−1 does
not always exist) but there always exists the largest regular left Ore set S and
l,0
the largest regular right Ore set S in C of the ring R, [4]. In general, S (cid:6)=S ,
r,0 l,0 r,0
[3]. In [4], the largest left quotient ring Q (R) := S−1R and the largest right
l l,0
quotient ring Qr(R(cid:4)) := RSr−,01 are introduced. In [3], these rings are found for the
ring I = K(cid:7)x,∂, (cid:8) of polynomial integro-differential operators over a field K of
1
characteristic zero, S (I )(cid:6)=S (I ) and Q (I )(cid:6)(cid:4)Q (I ).
l,0 1 r,0 1 l 1 r 1
Part 1. New criteria for a ring to have a semisimple left quotient ring
In the paper, the following notation is fixed:
• Risaringwith1, R∗ isitsgroupofunits, n=n isitsprimeradicaland
R
Min(R) is the set of minimal primes of R;
• C = C is the set of regular elements of the ring R (i.e. C is the set of
R
non-zero-divisors of the ring R);
• Q = Q (R) := C−1R is the left quotient ring (the classical left ring of
l,cl
fractions) of the ring R (if it exists) and Q∗ is the group of units of Q;
• Den (R,a)isthesetofleftdenominatorsetsS ofRwithass(S)=awhere
l
a is an ideal of R and ass(S):={r ∈R|sr =0 for some s∈S};
• max.Den (R)isthesetofmaximalleftdenominatorsetsofR(itisalways
l
a non-empty set, [4]);
• Ore (R):={S|S is a left Ore set in R};
l
• Den (R):={S|S is a left denominator set in R};
l
• Loc (R):={S−1R|S ∈Den (R)};
l l
• Ass (R):={ass(S)|S ∈Den(R)};
l l
• S = S (R) = S (R) is the largest element of the poset (Den(R,a),⊆)
a a l,a l
and Q (R):=Q (R):=S−1R is the largest left quotient ring associated
a l,a a
to a, S exists (Theorem 2.1, [4]);
a
• In particular, S = S (R) = S (R) is the largest element of the poset
0 0 l,0
(Den(R,0),⊆), i.e. the largest regular left Ore set of R, and Q (R) :=
l l
S−1R is the largest left quotient ring of R [4];
0
• Loc (R) := {[S−1R]|S ∈ Den (R)} where [S−1R] is an R-isomorphism
l l
class of the ring S−1R (a ring isomorphism σ : S−1R → S(cid:5)−1R is called
an R-isomorphism if σ(r) = r for all elements r ∈ R); we usually write
1 1
S−1R instead of [S−1R] if this does not lead to confusion;
• Loc (R,a):={[S−1R]|S ∈Den (R,a)}.
l l
The largestleftquotientringof aring. LetRbearing. Amultiplicatively
closed subset S of R (i.e. a multiplicative sub-semigroup of (R,·) such that 1 ∈ S
and 0(cid:6)∈S) is said to be a left Ore set if it satisfies the left Ore condition: for each
r ∈R and s∈S,
(cid:5)
Sr Rs(cid:6)=∅.