Noncommutative Algebraic Geometry and Representations of Quantized Algebras

332 Pages·1995·6.71 MB·English

by  Alexander L. Rosenberg (auth.)

Noncommutative Algebraic Geometry and
Representations of Quantized Algebras
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centrefor Mathematics and Computer Science, Amsterdam, The Netherlands
Volume330
N oncommutative
Aigebraic Geometry and
Representations of
Quantized Aigebras
by
Alexander L. Rosenberg
Department 0/ Mathematics,
Indiana University,
Bloomington, Indiana, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA. B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4577-5 ISBN 978-94-015-8430-2 (eBook)
DOI 10.1007/978-94-015-8430-2
Printed on acid-free paper
All Rights Reserved
© 1995 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
Preface ......................................................... ix
Chapter I
Noncommutative affine schemes ........................................ 1
Introduction ...................................................... 1
O. Preliminaries: localizations and radical filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1. Left spectrum ................................................... 4
2. Localizations and the left spectrum ................................... 10
3. Morphisms of left spectra .......................................... 12
4. The left spectrum and the Levitzki radical .............................. 16
5. The Levitzki spectrum and the left spectrum ............................. 19
6. Structure presheaves. Reconstruction of modules . . . . . . . . . . . . . . . . . . . . . . . . .. 22
7. Affine and quasi-affine schemes. Projective spectra ......................... 31
Complementary facts and examples ........................ . . . . . . . . . . . .. 44
Cl. The left spectrum of a principal domain ............................... 44
C2. Left normal morphisms and quantum plane ............................ 46
Chapter 11
The left spectrum and irreducible representations of 'smali' quantized and
classical rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48
Introduction ..................................................... 48
O. Preliminaries: the spectrum of an abelian category . . . . . . . . . . . . . . . . . . . . . . . .. 50
1. The left spectrum of the ring of skew polynomials. Quantum plane ............ 51
2. Restricted skew polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
3. The left spectrum and the irreducible representations of hyperbolic rings. . . . . . . .. 59
4. Applications to basic examples ...................................... 73
Complementary facts and examples ........................ . . . . . . . . . . . .. 83
Cl. Hyperbolic rings of M(2)-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84
C2. Quantum Heisenberg algebra ...................................... 91
C3. Rings of U(sl(2))-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
C4. Other examples of hyperbolic rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
Chapter III
Noncommutative local algebra ....................................... 110
Introduction ........................................ . . . . . . . . . . .. 110
vi
1. The spectrum of an abelian category ................................. 111
2. The spectrum and exact localizations ................................. 112
3. Local abelian categories and localization at points of the spectrum . . . . . . . . . . . .. 117
4. The left spectrum of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119
5. Supports and localizing subcategories ..................... . . . . . . . . . . .. 121
6. Left closed subcategories and Zariski topology .......................... 124
7. Some other canonical topologies .................................... 133
8. Associated points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 136
9. Relative spectra ................................................ 139
Complementary facts and examples .......................... . . . . . . . . .. 141
Cl. The prime spectrum of an abelian category . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141
Chapter IV
Noncommutative local algebra and representations of certain rings of
mathematical physics .............................................. 142
Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 142
1. Quantized Weyl and Heisenberg algebras and hyperbolic rings ............... 143
2. Skew polynomial categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 145
3. The skew Laurent categories ....................................... 149
4. Apart of the spectrum of a skew polynomial category .................... 150
5. The hyperbolic categories ......................................... 153
6. The spectrum of a hyperbolic category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 159
Complementary facts and examples .................................... 176
Cl. Hyperbolic categories of higher rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 176
C2. Rings and categories of Heisenberg and Weyl types . . . . . . . . . . . . . . . . . . . . .. 178
Chapter V
Skew PBW monads and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 188
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 188
1. The spectrum of a quasi-exact category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190
2. From hyperbolic rings to PBW monads ............................... 192
3. Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 198
4. Graded monads and modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199
5. The spectrum of the category of X-graded modules ....................... 204
6. The spectrum of the category of modules over a PBW monad ............... 208
Complementary facts and examples .............................. . . . . .. 214
Cl. Morphisms of graded monads and the spectrum ........................ 214
C2. Quasi-holonomic modules and characters ................. . . . . . . . . . . .. 215
C3. Dualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 222
C4. Weyl algebras ................................................ 224
C5. Aremark about relations between the spectrum of reductive and (quantized)
Kac-Moody Lie algebras and the spectrum of certain hyperbolic rings ...... . . . . .. 231
C6. Two-parameter deformations of M(2) and GL(2) ........................ 233
vii
Chapter VI
Six spectra and two dimensions of an abelian category . . . . . . . . . . . . . . . . . . . . . .. 238
Introduction .................................................... 238
1. The complete spectrum of an abelian category . . . . . . . . . . . . . . . . . . . . . . . . . .. 239
2. The flat spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 249
3. The Goldman spectrum .......................................... 255
4. The flat spectrum and injective objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 257
5. Injective spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 262
6. The Gabriel-Krull dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 266
Chapter VII
Noncommutative Projective Spectrum .......... . . . . . . . . . . . . . . . . . . . . . . .. 276
Introduction .................................................... 276
1. Quasi-schemes. Affine morphisms. Subschemes .......................... 276
2. Projective spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 282
3. Affine fibers and projective fibers ................................... 290
4. Blowing up and related constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 294
5. Generalizations. Quantized flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 296
Complementary facts and examples .................................... 298
Cl. Relative quasi schemes .......................................... 298
C2. Quasi-schemes with an ample auto-equivalence and Proj. .................. 299
C3. 'Schemes' with ample families of line bundles and generalized Proj. . . . . . . . . . .. 304
Reference ...................................................... 306
Index of Notations ............................................... 310
Index ......................................................... 314
PREFACE
This book is based on lectures delivered at Harvard in the Spring of 1991 and at the
University of Utah during the academic year 1992-93. Formally, the book assumes only general
algebraic knowledge (rings, modules, groups, Lie algebras, functors etc.). It is helpful, however,
to know some basics of algebraic geometry and representation theory.
Each chapter begins with its own introduction, and most sections even have a short
overview. The purpose of what follows is to explain the spirit of the book and how different
parts are linked together without entering into details.
The point of departure is the notion of the left spectrum of an associative ring, and the
first natural steps of general theory of noncommutative affine, quasi-affine, and projective
schemes. This material is presented in Chapter I.
Further developments originated from the requirements of several important examples
I tried to understand, to begin with the first Weyl algebra and the quantum plane. The book
reflects these developments as I worked them out in reallife and in my lectures.
In Chapter 11, we study the left spectrum and irreducible representations of a whole lot
of rings which are of interest for modern mathematical physics. The dasses of rings we
consider indude as special cases: quantum plane, algebra of q-differential operators, (quantum)
Heisenberg and Weyl algebras, (quantum) enveloping algebra ofthe Lie algebra sl(2) , coordinate
algebra of the quantum group SL(2), the twisted SL(2) of Woronowicz, so called dispin algebra
and many others.
We begin with an observation that most small rings of interest (e.g. all those listed
above) belong to a dass of rings particularly convenient for the study of the left spectrum and
irreducible representations which I call hyperbolic due to suggestive defining relations. Given
an automorphism (J and a central element ~ of a ring R, we define the hyperbolic ring over R
associated to this data as the ring generated by Rand by two elements, x and y, subject to the
relations:
xr = 8(r)x, ry = y8(r) for all rE Rj xy =~, yx = 8-1(~).
Note that in Chapter 11 we consider only hyperbolic rings over commutative noetherian
rmgs.
The treatment of special cases in Chapter 11 (and at the end of Chapter I) reveals that
the language of rings and ideals is not quite adequate to the task. Actually, we have to switch
from left ideals to left modules and reformulate the notion of the spectrum in purely
categorical terms from the very beginning. But this is not enough. With the exception of some
special cases, the obtained results provide an incomplete spectral picture. Besides, all the rings
considered in Chapter 11 are of GK-dimension lover some commutative rings. To overcome
x
this restriction, we need either much more sophisticated ring-theoretical investigations, or a
different point of view. This different point of view is presented in Chapt~r III, where the
spectrum of an abelian category is introduced and studied. We also study functorial properties
of the spectrum. This leads to the notion of the relative spectrum, i.e. the spectrum of a functor
from one category to another. The spectrum of a category is naturally isomorphie to the
spectrum of the corresponding identical functor.
After acquiring this new approach, we try it immediately - in Chapter IV - on
hyperbolic rings over an arbitrary noncommutative ring. The latter indude, for example, Weyl
algebras of any finite rank and their quantum analogues. Actually, the method we use enables
us to achieve more. Namely, we go from a hyperbolic ring over a ring R to a hyperbolic
category 'H. over a category A( - R·mod); this 'over' means that there is a natural functor ~
from'H. to A. Surprisingly, we get as areward a complete description of the spectrum of the
functor ~. Reflected back to rings, these results provide a complete description of the left
spectrum of any hyperbolic ring over a commutative noetherian ring, and much more.
The main purpose of Chapter V is to extend results of Chapters II and IV on the
spectrum and representations of hyperbolic rings onto a larger dass of PBW-rings (pBW stands
for Poincare..Birkhoff-Witt) which indudes enveloping algebras of reductive Lie algebras and
Kac-Moody Lie algebras and the quantized enveloping algebras. In the case of a (quantized)
enveloping algebra, this approach describes, in a canonical way, the left spectrum and
irreducible representations of the enveloping algebra in terms of the left spectrum and
irreducible representations of its proper subalgebra. For 'generic' irreducible representations and
spectral points, this proper subalgebra happens to be the centralizer of the Cartan sub algebra
(we discuss this in CS).
Another topic of the chapter (related to PBW rings) is the study of the spectrum of the
category of graded modules over a graded ring. The obtained results together with the point
of view that the underlying topological space of a scheme is the spectrum of the category of
quasi-coherent sheaves on this scheme lead to a new notion of a graded scheme (and that of
a graded manifold).
As in Chapter IV, we work in a more general setting (much more convenient for our
task) which allows us to talk simultaneously about modules over rings and quasi-coherent
modules over sheaves of rings (e.g. D-modules on some manifold). So that, instead of graded
rings and PBW rings, we study graded monads and PBW monads in an abelian category.
Chapter VI is called "Six spectra and two dimensions of an abelian category". Below
follows an explanation.
The spectrum SpecA of an abelian category A is a minimal, in a certain sense,
extension of the set SimpleA of equivalence dasses of simple objects of A. the results of
Chapters I - VI show that SpecA is a quite satisfactory object except for one point: the
spectrum of a quotient category over an open set might be larger than this open set. In an
attempt to find a spectrum with better properties with respect to localizations, we introduce
the f/at spectrum Spec-A (which is the principal character of Chapter VI) and the naturally
coming with it complete spectrum, Spec' A. The three spectra are related as follows:
SpecA S; Spec-A S; Spec' A.
The remaining three spectra are: the Goldman's spectrum SpegA; the indecomposable
injective spectrum r SpecA; and the injective spectrum ISpecA.
The first two are well known objects which were intensively studied from respectively
the late sixties and late fifties (cf. [Gab] and [GoI2] and their bibliographies). The third notion
xi
is naturally suggested by that of SpecA.
The main reason to consider these spectra here is to see the connections between the
different approaches and acquire a new insight in both old an new spectral theories.
Since all the spectra listed above are ordered sets, each of them carries a notion of
dimension which can be defined the way it is defined for schemes (with a slight modification
if one wants to make a distinction between different infinities), so that we have six dimensions.
We choose, however, only the one which is related to the flat spectrum.
The second dimension mentioned in the heading of the Chapter is the Gabriel-Krull
dimension of an abelian category. We prove that if a category A has Gabriel dimension, then
five (of the six) spectra of A coincide. And two dimensions are equal to each other. In
particular, it is the case when A is the category of quasi-coherent sheaves over a noetherian
scheme, or A is the category of left modules over a left noetherian ring. This Fact is applied
(already in Chapter 11) to get a complete description of irreducible representations of the first
Weyl algebra over any field of zero characteristic.
In Chapter VII, we sketch some very basic constructions and facts of noncommutative
projective geometry over a 'scheme', where 'scheme' is represented by an abelian category seen
as the category of quasi-coherent sheaves its spectrum. A serious study of non-affine
phenomena requires cohomological machinery (nonabelian Cech resolutions, etc.) and a
detailed analysis of principal motivating examples (like flag varieties of quantized enveloping
algebras). It shall appear elsewhere.
I suggest reading the introductions to all Chapters. They are sufficiently detailed to give
an impression of the contents of the book.
Those who do not like categories and are interested only in 'low dimensional' examples
might confine themselves to the first two chapters.
Areader with experience in categorical thinking, but who is not particularly fond of
ring theory, may like to begin with Chapter III, then browse Section 1.3 and read Section 1.4
(which contains a nontrivial and important result -a description of the intersection of ideals
of the left spectrum - which is hard to obtain using categoricallanguage). Then go to the
second chapter for motivating examples, and after that read the remaining chapters in any
order.
It is worth mentioning that a considerable part of Chapter 11 consists of more or less
straightforward calculations; and Chapter IV (which is, in asense, a counterpart of Chapter 11
in categorical setting) is also rather technical.
Most Chapters have a section which is called "Complementary facts and examples". The
main body of the book does not depend on these complementary facts. But some of them
might be curious or useful to areader depending on his or her interests.
Acknowledgements.
The main notions and facts of affine geometry were prompted by the stimulating
interest of L.A. Bokut to whom I'm glad to express my thanks.
I thank Dmitry Leites who translated and published my notes on noncommutative
affine geometry (a much shortened version of which constitutes the 'affine' part of Chapter 1),
and also helped me in many other ways. This book might never have been written if not for
him.