Quadratic Algebra s

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University

LECTURE Series

Volume 3 7

Quadratic Algebra s Alexander Polishchu k

Leonid Positselsk i

American Mathematica l Societ y Providence, Rhode Islan d

EDITORIAL COMMITTE E Jerry L . Bon a (Chair ) Eri c M . Friedlande r Adr iano Gars i a Nige l J . Higso n

Peter Landwebe r

2000 Mathematics Subject Classification. P r imar y 16S37 , 16S15 , 16E05 , 16E30 , 16E45 , 16W50, 13P10 , 60G10 .

For addi t iona l informatio n an d upda te s o n thi s book , visi t www.ams.org/bookpages/ulect-37

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n Da t a

Polishchuk, Alexander , 1971 -Quadratic algebra s / Alexande r Polishchuk , Leoni d Positselski .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 37 ) Includes bibliographica l references . ISBN 0-8218-3834- 2 (acid-fre e paper ) 1. Quadrati c fields. 2 . Associativ e rings . 3 . Commutativ e rings . 4 . Stochasti c processes .

I. Positselski , Leonid , 1973 - II . Title . III . Universit y lectur e serie s (Providence , R.I. ) ; 37.

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Contents

Introduction vi i

Chapter 1 . Preliminarie s 1 0. Convention s an d notatio n 1 1. Ba r construction s 2 2. Quadrati c algebra s an d module s 6 3. Diagona l cohomolog y 7 4. Minima l resolution s 7 5. Low-dimensiona l cohomolog y 9 6. Lattice s an d distributivit y 1 1 7. Lattice s o f vector space s 1 5

Chapter 2 . Koszu l algebra s an d module s 1 9 1. Koszulnes s 1 9 2. Hilber t serie s 2 1 3. Koszu l complexe s 2 5 4. Distributivit y an d n-Koszulnes s 2 9 5. Homomorphism s o f algebra s an d Koszulness . I 3 2 6. Homomorphism s o f algebras an d Koszulness . I I 3 7 7. Koszu l algebra s i n algebrai c geometr y 4 0 8. Infinitesima l Hop f algebr a associate d wit h a Koszu l algebr a 4 5 9. Koszu l algebra s an d monoida l functor s 4 9 10. Relativ e Koszulnes s o f modules 5 3

Chapter 3 . Operation s o n grade d algebra s an d module s 5 5 1. Direc t sums , fre e product s an d tenso r product s 5 5 2. Segr e products an d Verones e powers . I 5 9 3. Segr e product s an d Verones e powers . I I 6 3 4. Interna l cohomomorphis m 6 8 5. Koszulnes s canno t b e checke d usin g Hilber t serie s 7 7

Chapter 4 . Poincare-Birkhoff-Wit t Base s 8 1 1. PBW-base s 8 1 2. PBW-theore m 8 2 3. PBW-base s an d Koszulnes s 8 4 4. PBW-base s an d operation s o n quadrati c algebra s 8 5 5. PBW-base s an d distributin g base s 8 6 6. Hilber t serie s o f PBW-algebra s 8 7 7. Filtration s o n quadrati c algebra s 8 8 8. Commutativ e PBW-base s 9 1

vi C O N T E N T S

9. Z-algebra s 9 5 10. Z-PBW-base s 9 6 11. Three-dimensiona l Sklyani n algebra s 9 8

Chapter 5 . Nonhomogeneou s Quadrati c Algebra s 10 1 1. Jacob i identit y 10 1 2. Nonhomogeneou s PBW-theore m 10 3 3. Nonhomogeneou s quadrati c module s 10 4 4. Nonhomogeneou s quadrati c dualit y 10 5 5. Example s 10 8 6. Nonhomogeneou s dualit y an d cohomolog y 11 1 7. Ba r constructio n fo r CDG-algebra s an d module s 11 2 8. Homolog y o f completed cobar-complexe s 11 7

Chapter 6 . Familie s o f quadrati c algebra s an d Hilber t serie s 11 9 1. Opennes s o f distributivit y 11 9 2. Deformation s o f Koszu l algebra s 12 0 3. Uppe r boun d fo r th e numbe r o f Koszul Hilber t serie s 12 2 4. Generi c quadrati c algebra s 12 3 5. Example s wit h smal l d im^ i an d dim^ 2 12 5 6. Koszulnes s i s no t constructibl e 12 7 7. Familie s o f quadrati c algebra s ove r scheme s 12 8

Chapter 7 . Hilber t serie s o f Koszu l algebra s an d one-dependen t processe s 13 3 1. Conjecture s o n Hilber t serie s o f Koszu l algebra s 13 3 2. Koszu l inequalitie s 13 5 3. Koszu l dualit y an d inequalitie s 13 8 4. One-dependen t processe s 13 9 5. PBW-algebra s an d two-block-facto r processe s 14 2 6. Operation s o n one-dependen t processe s 14 3 7. Hilber t spac e representation s o f one-dependent processe s 14 6 8. Hilber t serie s o f one-dependent processe s 14 7 9. Hermitia n constructio n o f one-dependen t processe s 14 9 10. Module s ove r one-dependen t processe s 15 1

Appendix A . DG-algebra s an d Masse y product s 15 3

Bibliography 155

Introduction

The goa l o f thi s boo k i s to introduc e th e reade r t o som e recen t development s in th e stud y o f associative algebra s define d b y quadrati c relations . Mor e precisely , we ar e intereste d i n (no t necessaril y commutative ) algebra s ove r a field tha t ca n be presente d usin g a finite numbe r o f generator s an d (possibl y nonhomogeneous ) quadratic relations . Thi s boo k i s devote d t o som e aspect s o f th e theor y o f suc h algebras, mostl y evolvin g aroun d th e notion s o f Koszu l algebr a an d Koszu l dual -ity. It s conten t i s a mixtur e o f know n result s wit h a fe w origina l result s tha t w e circulated sinc e 199 4 as a preprin t o f the sam e title .

One o f the origina l motivation s fo r th e stud y o f quadratic algebra s cam e fro m the theory o f quantum group s (se e [43 , 77]) . Namely , quadrati c algebra s provide a convenient framework fo r "noncommutativ e spaces " o n which quantu m group s ac t (see [78]) . On e o f th e basi c problem s tha t aros e i n thi s contex t i s ho w t o contro l the growt h o f a quadrati c algebr a (e.g. , measure d b y Hilber t series) . A relate d question i s whether there are generalizations o f the Poincare-Birkhoff-Wit t theore m (for universa l envelopin g algebras ) t o mor e genera l quadrati c algebras . Th e cor e of this boo k i s ou r attemp t t o presen t som e partia l solutions . I t turn s ou t tha t on e can shed som e light o n questions o f this kind usin g the remarkabl e notio n o f Koszul algebra introduce d b y S . Pridd y [104] . I n fact , th e stud y o f thi s notio n brough t some dramati c change s t o th e area . Loosel y speaking , ou r experienc e show s tha t general quadrati c algebra s behav e a s badl y a s possible , whil e fo r Koszu l algebra s the situatio n i s usually muc h nicer . A s w e hope t o convinc e th e reader , th e stud y of Hilber t serie s provide s a good illustratio n o f thi s principle .

Perhaps on e o f th e importan t feature s o f th e theor y o f Koszu l algebra s i s du -ality: fo r eac h Koszu l algebr a ther e i s a dua l Koszu l algebr a (roughl y speaking , i t is obtaine d b y passin g t o th e dua l spac e o f generator s an d th e orthogona l spac e of quadrati c relations) . Thi s ofte n lead s t o remarkabl e connection s betwee n seem -ingly unrelate d problems . Fo r example , Koszu l dualit y o f th e symmetri c algebr a and th e exterio r algebr a underlie s th e famou s descriptio n o f coheren t sheave s o n projective space s in terms o f modules ove r the exterio r algebr a due to J . Bernstein , I. Gelfan d an d S . Gelfan d [27] . Mor e generally , i n a numbe r o f situations on e ca n prove a n equivalenc e o f derive d categorie s o f module s ove r Koszu l dua l algebra s (see [23 , 11 , 24, 51]) . Thi s topic i s beyond th e scop e of our boo k althoug h w e will discuss som e mor e elementar y aspect s o f Koszu l duality .

The notio n o f Koszulness als o proved t o b e a reall y impressiv e predictio n tool . In man y example s a fe w observation s ma y sugges t tha t som e quadrati c algebr a is Koszul . The n thi s conjectur e turn s ou t t o b e relate d t o som e importan t an d nontrivial feature s o f th e setting . I t i s als o quit e amazin g tha t man y importan t quadratic algebra s naturall y arisin g i n variou s fields o f mathematic s ar e Koszul . Examples know n t o u s aris e i n the followin g areas :

vii

viii I N T R O D U C T I O N

(i) algebrai c geometry—certai n homogeneou s coordinat e algebra s ar e Koszu l (se e [29, 37 , 39 , 67 , 50 , 72 , 73 , 89 , 96]) ; (ii) representation theory—certai n subcategorie s o f the category O for a semisimple complex Li e algebr a ar e governe d b y Koszu l algebra s (se e [19 , 24]) ; (iii) noncommutativ e geometry—th e Koszulnes s conditio n arise s naturall y i n th e theory o f exceptiona l collections ; th e algebra s describin g certai n noncommutativ e deformations o f projective space s ar e Koszu l (se e [30 , 31 , 117]) ; (iv) topology—Steenro d algebra , cohomolog y algebra s o f forma l rationa l K[K, 1]-spaces, holonom y algebra s o f supersolvabl e hyperplan e arrangements , a s wel l a s some algebra s relate d t o configuratio n space s o f surface s ar e Koszul ; th e categor y of perverse sheave s o n a triangulate d spac e i s equivalent t o module s ove r a Koszu l algebra (se e [104 , 88 , 113 , 28 , 97 , 127]) ; (v) numbe r theory—th e Milno r K-theor y rin g o f an y field (tensore d wit h Z/7 Z fo r a prim e I) i s conjecture d t o b e Koszul—thi s i s a strengthenin g o f th e Bloch-Kat o conjecture relatin g Milno r K-theor y wit h Galoi s cohomolog y (se e [103 , 102]) ; (vi) noncommutativ e algebra—th e universa l algebr a generate d b y pseudoroots o f a noncommutative polynomia l i s Koszul (se e [111 , 93]) .

Checking th e Koszu l propert y usuall y require s som e effor t an d th e method s o f proof var y fro m on e cas e t o another . Althoug h w e do no t tr y t o giv e a systemati c exposition o f thes e method s here , th e reade r wil l find a fe w sampl e technique s fo r checking Koszulnes s (mostl y i n chapte r 2) .

As we have already mentioned , on e of the centra l questions studied i n our boo k is ho w t o generaliz e th e Poincare-BirkhofT-Witt-theore m (PBW-theorem ) t o qua -dratic algebras . Recal l that th e classical PBW-theorem fo r the universal envelopin g algebra Ug o f a Li e algebr a g can b e formulate d i n tw o differen t ways . I n th e first formulation on e start s wit h a basi s o f g an d the n th e theore m state s tha t certai n standard monomial s in basis elements form a basis of Ug. Anothe r formulatio n sim -ply assert s tha t th e associate d grade d algebr a o f Ug wit h respec t t o th e standar d filtration coincide s with the symmetric algebr a Sg. Thus , the first wa y to generaliz e the PBW-theore m t o othe r algebra s i s to modif y th e notio n o f standar d monomi -als. Assum e tha t w e hav e a grade d quadrati c algebr a (i.e. , quadrati c relation s ar e homogeneous). The n usin g lexicographical orde r o n the se t o f all monomials in gen-erators one can define a certain se t o f standard monomial s (dependin g on quadrati c relations). Th e analogu e o f th e PBW-theore m i n thi s cas e state s tha t i f th e stan -dard monomial s for m a basi s i n th e gradin g componen t o f degre e 3 then th e sam e is also true fo r al l grading component s (s o that w e get a PBW-basis i n our algebra) . This theore m i s a particula r cas e o f th e so-calle d diamon d lemm a i n th e theor y of Grobne r base s develope d i n work s o n combinatoria l algebr a i n th e lat e 70 s (se e [26, 35 , 36]) . Not e tha t th e universa l envelopin g algebr a Ug ca n b e homogenize d by addin g a n extr a centra l generator , s o that th e classica l PBW-theore m woul d fit into thi s context .

Before statin g th e secon d generalizatio n o f the PBW-theore m le t u s sa y a fe w words abou t th e terminolog y adopte d i n th e book . W e us e th e ter m "quadrati c algebra" onl y i n referenc e t o algebra s define d b y homogeneou s quadrati c relation s (because with the exception o f chapter 5 we consider onl y such algebras) . Assignin g degree 1 to each generator one can view a quadratic algebra as a graded algebra A = ©n>o ^n s u c n t n a t ^ o i s the ground field and A i s the quotient o f the tensor algebr a of A\ b y a n idea l generated i n degree 2 . Not e tha t sometime s (e.g. , i n application s

I N T R O D U C T I O N i x

to representation theory ) i t i s necessary to consider more general quadratic algebra s such tha t AQ i s not necessaril y equa l t o the groun d field bu t rathe r i s a semisimpl e algebra. W e wil l briefly discus s algebra s o f thi s kin d i n sectio n 9 of chapte r 2 .

Our secon d generalization o f the PBW-theorem deal s with a "nonhomogeneou s quadratic algebra" , i.e. , a n algebr a wit h a finite numbe r o f generator s an d non -homogeneous quadrati c definin g relations . I f A i s suc h a n algebr a the n on e ca n consider th e natura l filtration o n A determine d b y th e se t o f generators . Le t u s denote b y gr̂ 4 the associate d grade d algebra . O n the othe r hand , on e can truncat e the relation s i n A leavin g onl y thei r homogeneou s quadrati c parts . Le t A^ b e the obtaine d quadrati c algebra . Th e nonhomogeneou s PBW-theore m state s tha t the natura l ma p A^ — > giA i s an isomorphis m provide d A^ i s Koszul an d a cer -tain self-consistenc e conditio n i s satisfied (thi s resul t wa s proved independentl y b y A. Braverma n an d D . Gaitsgor y [33]) . Thi s self-consistenc y conditio n i s obtaine d by looking a t expression s o f degree 3 in generators . I n the cas e A = Ug i t coincide s with th e Jacob i identit y fo r th e Li e bracke t o n g.

It i s interestin g tha t th e notio n o f Koszulnes s appear s als o i n th e first gen -eralization o f th e PBW-theorem : quadrati c algebra s havin g a basi s o f standar d monomials, calle d PBW-algebras, ar e alway s Koszu l (thi s observatio n goe s bac k to S . Pridd y [104]) . However , th e convers e i s no t true : Koszu l algebra s ar e no t necessarily PB W (se e section 3 of chapter 4) . I n fact , th e clas s o f PBW-algebras i s substantially smalle r tha n tha t o f Koszul algebra s an d i s much easie r t o study . Fo r example, the set of PBW-algebras with a given number of generators is constructible in Zarisk i topolog y whil e th e se t o f Koszu l algebra s i s often no t constructibl e (se e section 3 o f chapte r 4 an d sectio n 6 o f chapte r 6) . O n th e othe r hand , ther e ar e many paralle l result s fo r bot h classe s o f algebras . Firstly , bot h propertie s ca n b e formulated i n term s o f distributivit y o f certai n lattice s o f vecto r spaces . Secondly , various natura l operation s wit h quadrati c algebras , suc h a s quadrati c duality , fre e product, tenso r product , Segr e product an d Verones e powers preserve bot h classes . The compariso n betwee n th e classe s o f Koszu l an d PBW-algebra s i s als o a n im -portant par t o f th e presen t work . I n ou r experienc e PBW-algebra s ofte n provid e a goo d testin g groun d fo r guessin g th e genera l patter n tha t migh t hol d fo r al l Koszul algebras . Usuall y ther e i s no problem wit h provin g tha t a pattern hold s fo r PBW-algebras; however , th e cas e of Koszu l algebra s i s often muc h harde r (i f a t al l accessible).

One o f the mos t strikin g propertie s o f Koszu l algebra s i s the following . Koszul Deformatio n Principl e (V . Drinfel d [43]) . If a formal family of graded quadratic algebras A(t) is flat in the grading components of degree ^ 3 and the algebra A(0) is Koszul then the family is flat in all degrees.

More precisely , a simila r statemen t hold s fo r loca l deformation s (i n Zarisk i topology) i f w e conside r onl y a finite numbe r o f gradin g component s (se e Theo -rem 2. 1 o f chapte r 6) . Th e secon d versio n o f th e PBW-theore m considere d abov e can b e easil y deduce d fro m thi s principle . Anothe r unexpecte d consequenc e tha t we derive from i t i s the finiteness o f the numbe r o f Hilbert serie s of Koszul algebra s with a fixed number o f generator s (th e analogou s statemen t fo r quadrati c algebra s is wrong). W e conjecture tha t Hilber t serie s of Koszul algebra s enjo y severa l inter -esting propertie s tha t ca n b e easil y checke d fo r PBW-algebra s (althoug h w e prov e that thes e two sets o f Hilbert serie s ar e different) . Fo r example , w e conjecture tha t the Hilber t serie s o f a Koszu l algebr a i s always rational .

x INTRODUCTIO N

The stud y o f Hilber t serie s o f Koszu l algebra s le d t o th e discover y i n [100 ] o f an interestin g connectio n wit h th e theor y o f discrete stochasti c processes . Namely , to ever y Koszu l algebr a A on e ca n associat e a one-dependent stationar y stochasti c sequence o f O' s an d l's . I t i s convenien t t o encod e probabilitie s o f variou s event s in suc h a proces s b y a linea r functiona l (j> : M{xo?#i} ~ > l o n th e fre e algebr a i n two variables , takin g nonnegativ e value s o n al l monomial s an d satisfyin g <j>(l) = 1 . Then th e conditio n o f one-dependence i s equivalent t o th e equatio n

<Kf-{x0 + x 1)-g) = <l>{f)<l>(g),

where / , g € R{a?o,#i} - Abusin g th e terminolog y w e cal l suc h a functiona l <p a one-dependent process. I t i s easy to see that <j) is uniquely determine d b y the value s (4>(xi)). No w th e one-dependen t proces s associate d wit h a Koszu l algebr a A i s defined b y

<£A W1 ) = a n/a%,

where a n = di m An. Nonnegativit y o f value s o f <j> o n al l monomial s i s equivalen t to a certai n syste m o f polynomia l inequalitie s fo r th e number s a n. Th e fac t tha t these inequalities ar e indeed satisfied fo r a Koszul algebra seems to be a remarkabl e coincidence. However , th e analog y betwee n th e tw o theorie s doe s no t en d here . I t turns out tha t unde r thi s correspondence the subclass of PBW-algebras maps to th e set o f so-called two-block-facto r processes . Th e relatio n betwee n al l one-dependen t processes an d th e subclas s o f two-block-factor s wa s intensivel y studie d i n th e 90 s after i t wa s prove d i n [2 ] that a one-dependen t proces s doe s no t hav e to b e a two-block facto r (se e [1 , 118, 122]) . Thi s topi c seem s to b e surprisingly simila r t o th e relation betwee n Koszu l an d PBW-algebras . Motivate d b y thi s analog y w e conjec -ture tha t th e Hilber t serie s associate d wit h ever y one-dependen t proces s admit s a meromorphic continuatio n t o th e entir e comple x plane . Rationalit y o f Hilber t se -ries o f Koszu l algebra s woul d follo w fro m thi s (b y a theore m o f E . Bore l [32]) . W e also observe that th e polynomia l inequalitie s satisfie d b y the number s (<f>(xi)) for m a subse t i n th e well-know n syste m o f inequalitie s definin g th e notio n o f a totally positive sequence (als o know n a s Poly a frequency sequence). I t i s know n tha t th e generating serie s o f a totally positiv e sequenc e admit s a meromorphic continuatio n (see [71]) . Thi s ca n b e considere d a s anothe r hin t i n favo r o f our conjecture .

Here i s the mor e detaile d outlin e o f the conten t o f the book . Chapter 1 contains some basic definitions an d results concerning cohomology of

graded algebras , quadrati c algebra s an d distributivit y o f lattices . I n particular , i n section 2 we define quadrati c dualit y fo r quadrati c algebra s an d quadrati c module s (we us e th e ter m "Koszu l duality " whe n referrin g t o thi s dualit y i n th e cas e o f Koszul algebra s an d Koszu l modules) .

In chapte r 2 we describe various equivalen t definition s o f Koszulness, includin g Backelin's criterio n i n term s o f distributivity o f lattices (se e [15]) . W e give simila r equivalent definition s fo r a relate d notio n o f n-Koszulnes s tha t ha s a n advantag e of bein g define d b y a finite numbe r o f conditions . W e als o show tha t man y result s about quadrati c an d Koszu l algebra s hav e natura l analogue s fo r quadrati c an d Koszul modules. I n section 5 we consider th e problem o f preservation o f Koszulnes s under homomorphism s o f various types between graded algebras , generalizing som e results o f Backeli n an d Frober g [20] . I n sectio n 7 we giv e example s o f projectiv e varieties with Koszul homogeneous coordinate algebras . I n section 8 we explain how to associat e t o a Koszul algebra A a (graded ) infinitesimal bialgebra (or e-bialgebra)

INTRODUCTION x i

VA- Thi s constructio n ca n b e viewe d a s a categorificatio n o f th e one-dependen t process 4>A associated wit h A, becaus e th e value s o f (\>A o n monomial s ar e give n by dimension s o f certain multigradin g component s o f VA • In sectio n 9 we conside r some generalization s o f th e notio n o f Koszulnes s includin g a n importan t cas e o f graded algebra s A — © n > 0 A n suc h tha t AQ i s a semisimple algebr a (i n the res t o f the boo k w e assume tha t AQ is the groun d field). W e also give a n interpretatio n o f Koszul algebra s i n terms o f monoidal functor s fro m a certain universa l (nonunital ) monoidal category .

In chapte r 3 we conside r severa l natura l operation s o n quadrati c algebra s an d modules tha t preserv e Koszulnes s an d discus s th e behavio r o f Hilber t serie s unde r these operations . Followin g [20 ] w e conside r fre e sums , fre e products , alon g wit h several types of tensor products , th e Segr e product AoB, th e dua l operation "blac k circle product " A • B an d Verones e power s A^ n\ Th e operatio n A • B i s als o closely relate d t o th e interna l cohomomorphis m operatio n introduce d b y Mani n (see [77 , 79]) . W e prov e tha t i f on e o f th e algebra s i s Koszu l the n th e Hilber t series o f A • B ca n b e compute d i n term s o f thos e o f A an d B an d sho w tha t thi s is impossibl e i f bot h algebra s ar e no t Koszul . A n interestin g applicatio n o f thes e operations i s given i n section 5 , where we show, following D . Piontkovskii [92] , that Koszulness o f a quadrati c algebr a A canno t b e determine d fro m th e knowledg e o f the Hilber t serie s o f A an d A 1.

Chapter 4 i s devote d t o PBW-algebras . W e star t b y givin g a proo f o f th e PBW-theorem fo r quadrati c algebra s tha t give s a criterio n fo r th e existenc e o f a PBW-basi s (a s w e hav e mentione d before , thi s i s reall y a particula r cas e o f th e diamond lemma) . The n we prove that PBW-algebra s ar e Koszul and give a criterion of the PBW-propert y i n terms o f distributivity o f lattices i n the spiri t o f Backelin' s criterion o f Koszulness . W e als o chec k tha t th e clas s o f PBW-algebra s i s stabl e under quadrati c dualit y an d unde r al l operation s considere d i n chapte r 3 . The n we discus s Hilber t serie s o f PBW-algebras . W e sho w tha t th e Hilber t serie s o f a PBW-algebra i s a generatin g functio n fo r th e numbe r o f path s i n a finite oriente d graph an d henc e i s rational . I n sectio n 7 we prov e a generalizatio n o f th e PBW -theorem involvin g filtrations wit h value s i n a n ordere d semigroup . I n sectio n 8 we consider commutativ e PBW-algebras . W e prove that the y ar e Koszu l an d comput e their Hilber t series . W e also present som e examples showing that th e sets of Hilber t series o f PBW-algebra s an d Koszu l algebra s ar e different . I n sectio n 9 we discus s a generalizatio n o f th e classe s o f Koszu l an d PBW-algebra s fro m grade d algebra s to Z-algebras . I n sectio n 1 1 we conside r 3-dimensiona l ellipti c Sklyani n algebras . We prov e tha t the y ar e Koszu l bu t d o no t admi t a PBW-basi s eve n viewe d a s Z-algebras.

In chapte r 5 we conside r nonhomogeneou s quadrati c algebras . Fo r thes e alge -bras w e prove i n sectio n 2 the PBW-theore m involvin g a n analogu e o f th e Jacob i identity an d Koszulnes s o f the correspondin g homogeneou s quadrati c algebra . W e also prov e i n sectio n 3 a versio n o f thi s theore m fo r nonhomogeneou s quadrati c modules. I n sectio n 4 we conside r a n analogu e o f quadrati c dualit y fo r th e nonho -mogeneous case . I t turn s ou t tha t th e dua l objec t t o a nonhomogeneou s quadrati c algebra i s a so-called CDG-algebr a (curve d DG-algebra) . I n section 5 we give some examples o f nonhomogeneou s quadrati c algebra s an d modules . I n particular , w e list al l solution s o f th e analogu e o f th e Jacob i identit y i n th e cas e o f th e quadra -tic relation s correspondin g t o a fre e commutatativ e superalgebra , an d conside r a n

xii I N T R O D U C T I O N

example relate d t o th e PBW-theore m fo r quantu m universa l envelopin g algebra s (Example 6) . Th e remainde r o f thi s chapte r i s devote d t o variou s cohomologica l calculations relate d t o nonhomogeneou s quadrati c duality .

Chapter 6 is devoted to the Koszul Deformation Principl e for quadratic algebra s and som e o f it s consequences , suc h a s finiteness o f the numbe r o f Hilber t serie s of Koszul algebra s wit h a fixed numbe r o f generators . Furthermore , i n sectio n 3 we give an explici t boun d o n this number an d i n section 7 we prove that th e number of such Hilber t serie s i s finite eve n i f the groun d field i s allowed t o vary . I n sectio n 4 we discuss som e result s o n generi c algebra s amon g quadrati c algebra s wit h a give n number o f generator s an d relations . I n sectio n 5 we consider example s o f possibl e Hilbert series for algebras with a small number of generators and relations. Sectio n 6 contains counterexample s fro m [56 ] showing tha t th e se t o f Koszu l algebra s i s no t constructible an d tha t th e se t o f Hilber t serie s o f quadrati c algebra s wit h a give n number o f generator s i s infinite .

In chapte r 7 w e explai n th e connectio n betwee n Koszu l algebra s an d one -dependent processes . W e start b y formulating severa l conjecture s o n Hilber t serie s of Koszu l algebras , suc h a s th e rationalit y conjecture . The n w e derive a syste m o f polynomial inequalitie s satisfie d b y the number s a n = di m An fo r a Koszu l algebr a A. Th e polynomial s o f a n appearin g i n thes e inequalitie s expres s th e dimension s of multigradin g component s o f th e e-bialgebr a VA- The n w e sho w tha t thes e in -equalities allo w one to associat e a one-dependent proces s to th e sequenc e (a n). W e show that Koszu l duality correspond s to the natura l dualit y on one-dependent pro -cesses and als o introduce analogues of some other operation s on Koszul algebras fo r one-dependent processes . I n sectio n 5 we show tha t th e one-dependen t proces s as -sociated wit h a PBW-algebra i s a two-block-factor an d tha t ever y two-block-facto r can b e approximate d b y thos e obtaine d fro m PBW-algebras . I n sectio n 7 we re -view th e notio n o f a Hilber t spac e representatio n o f a one-dependen t proces s du e to V . d e Val k [121] . I n sectio n 8 we discuss th e conjectur e tha t th e Hilber t serie s of a one-dependent proces s can be extended meromorphicall y t o the entire comple x plane. W e show tha t thi s serie s alway s admit s a meromorphi c continuatio n t o th e disk \z\ < 2 (i t converge s fo r \z\ < 1 ) and prov e th e conjectur e fo r two-block-facto r processes. I n section 9 we give a construction du e to B. Tsirelson of a one-dependen t process associate d wit h a n arbitrar y quadrati c algebr a an d a Hermitia n for m o n the spac e o f generators . I n sectio n 1 0 we conside r a n analogu e fo r Koszu l modules of the constructio n o f a one-dependen t proces s fro m a Koszu l algebra .

In th e Appendi x w e recall some definition s concernin g DG-algebras , DG-mod -ules an d Masse y products .

Acknowledgments. First , w e would lik e to than k A . Vaintrob whose question abou t possible generalization s o f th e PBW-theore m t o quadrati c algebra s starte d thi s work i n 1991 . Also , w e ar e gratefu l t o J . Backelin , A . Braverman , J . Bernstein , P. Etingof, V. Ginzburg, V. Ostrik, D. Piontkovskii, J.-E . Roos, A. Schwarz, B. Shel-ton, B . Tsirelson , an d S . Yuzvinsk y fo r man y interestin g discussion s an d sugges -tions. Specia l thanks ar e due to J . Backeli n fo r pointin g ou t severa l mistakes i n th e manuscript. Finally , w e are gratefu l t o th e refere e fo r man y usefu l suggestions .

Bibliography

J. Aaronson , D . Gilat, M . Keane . On the structure o f 1-dependent Marko v chains . J. Theoret. Probab. 5 , # 3 , 545-561 , 1992 . J. Aaronson , D . Gilat , M . Keane , V . d e Valk . A n algebrai c constructio n o f a clas s o f one -dependent processes . Ann. Probab. 17 , # 1 , 128-143 , 1989 . M. Aguiar . Infinitesima l Hop f algebras . New trends in Hopf algebra theory (La Falda, 1999), 1-29. Contemp . Math . 267 , AMS , Providence , RI , 2000 . M. Aguiar . Infinitesima l bialgebras , pre-Li e an d dendrifor m algebras . Hopf Algebras, 1-33 . Lect. Note s i n Pur e an d Appl . Math . 237 , 2004 . R. Aharoni . A proble m i n rearrangement s o f (0 , 1 ) matrices . Discrete Math. 30 , # 3 , 191 -201, 1980 . R. Ahlswede , G . O . H . Katona . Graph s wit h maxima l numbe r o f adjacen t pair s o f edges . Acta Math. Acad. Sci. Hungar. 32 , #1-2 , 97-120 , 1978 . D. J . Anick . Noncommutativ e grade d algebra s an d thei r Hilber t series . J. Algebra 78 , # 1 , 120-140, 1982 . D. J . Anick . Generi c algebra s an d CW-complexes . Algebra, topology and algebraic K-theory (Princeton, NJ, 1983), 247-321 , Ann . o f Math . Stud . 113 , Princeto n Univ . Press , 1987 . D. J . Anick . Diophantin e equations , Hilber t series , an d undecidabl e spaces . Annals of Math. 122, # 1 , 87-112 , 1985 . E. Arbarello , M . Cornalba , P . Griffiths , J . Harris . Geometr y o f algebrai c curve s I , Springer -Verlag, Ne w York/Berlin , 1985 . S. Arkhipov . Koszu l dualit y fo r quadrati c algebra s ove r a complex . Fund. Anal. Appl. 28 , # 3 , 202-204 , 1994 . M. Artin , W . Schelter . Grade d algebra s o f globa l dimensio n 3 . Advances in Math. 6 , # 2 , 171-216, 1987 . M. Artin , J . Tate , M . Va n de n Bergh . Som e algebra s associate d t o automorphism s o f ellipti c curves. The Grothendieck Festschrift, vol . 1 , 33-85 , 1990 . L. Avramov , D . Eisenbud . Regularit y o f module s ove r a Koszu l algebra . J. Algebra 153 , # 1 , 85-90 , 1992 . J. Backelin . A distributivenes s propert y o f augmente d algebra s an d som e relate d homological results . Ph . D . Thesis , Stockholm , 1981 . Availabl e a t ht tp: / /www . matematik.su. se/~joeb/avh / J. Backelin . O n th e rate s o f growth o f the homologie s o f Veronese subrings . Algebra, topology and their interactions (Stockholm, 1983), 79-100 , Lectur e Note s i n Math . 1183 , Springer , Berlin, 1986 . J. Backelin . Som e homologica l propertie s o f "high " Verones e subrings . J. Algebra 146 , # 1 , 1-17, 1992 . J. Backelin . Privat e communication , 1991 . J. Backelin . Koszu l dualit y fo r paraboli c an d singula r categor y O. Represent. Theory 3 , 139-152, 1999 . J. Backelin , R . Froberg . Koszu l algebras , Verones e subrings an d ring s with linea r resolutions . Revue Roumaine Math. Pures Appl. 30 , # 2 , 85-97 , 1985 . D. Bayer , M . Stillman . A criterio n fo r detectin g m- regularity. Inventiones Math. 87 , # 1 , 1-11, 1987 . D. Bayer , M . Stillman . O n th e complexit y o f computin g syzygies . J. Symbolic Comput. 6 , # 2 - 3 , 135-147 , 1988 . A. Beilinson , V . Ginzburg , V . Schechtman . Koszu l duality . J. Geom. Phys. 5 , # 3 , 317-350 , 1988.

155

156 BIBLIOGRAPH Y

[24] A . Beilinson , V . Ginzburg , W . Soergel . Koszu l dualit y pattern s i n representatio n theory . J . Arner. Math. Soc. 9 , # 2 , 473-527 , 1996 .

[25] A . Beilinson , R . MacPherson , V . Schechtman . Note s o n motivi c cohomology . Duke Math, J, 54 , # 2 , 679-710 , 1987 .

[26] G . Bergman . Th e diamon d lemm a fo r rin g theory . Advances in Math. 29 , # 2 , 178-218 , 1978.

[27] I . Bernstein , I . Gelfand , S . Gelfand . Algebrai c vecto r bundle s o n P n an d problem s o f linea r algebra. (Russian ) Funktsional . Anal , i Prilozhen . 12 , # 3 , 66-67 , 1978 .

[28] R . Bezrukavnikov . Koszu l DG-algebra s arisin g fro m configuratio n spaces . Geom. and Funct. Anal 4 , # 2 , 119-135 , 1994 .

[29] R . Bezrukavnikov . Koszu l propert y an d Frobeniu s splittin g o f Schuber t varieties . Preprin t alg-geom/9502021.

[30] A . I . Bondal . Helices , representations o f quivers and Koszu l algebras . Helices and vector bun-dles, 75-95 , Londo n Math . Soc . Lecture Not e Ser. , 148 , Cambridge Univ . Press , Cambridge , 1990.

[31] A . I . Bondal , A . E . Polishchuk . Homologica l propertie s o f associativ e algebras : th e metho d of helices . Russian Acad. Sci. Izvestiya Math. 42 , # 2 , 219-260 , 1994 .

[32] E . Borel . Su r un e applicatio n d'u n theorem e d e M . Hadamard . Bull. Sc. math., 2 e serie , t.18, 22-25 , 1894 .

[33] A . Braverman , D . Gaitsgory . Th e Poincare—Birkhoff-Wit t theore m fo r quadrati c algebra s of Koszu l type . J. Algebra 181 , # 2 , 315-328 , 1996 .

[34] W . Bruns , J . Gubeladze , Ng o Vie t Trung , Norma l polytopes , triangulations , an d Koszu l algebras. J. Reine Angew. Math. 485 , 123-160 , 1997 .

[35] B . Buchberger . Grobne r bases : A n algorithmi c metho d i n polynomia l idea l theory . Multidi-mensional Systems Theory (N . K . Bos e ed.) 5 184-232 , Reidel , Dordrecht , 1985 .

[36] B . Buchberger , F . Winkler , eds. , Grobne r base s an d applications , Cambridg e Univ . Press , 1998.

[37] D . C . Butler . Norma l generatio n o f vecto r bundle s ove r a curve . J. Diff. Geom, 39 , # 1 , 1-34, 1994 .

[38] A . Conca . Grobne r base s fo r space s o f quadric s o f lo w codimension . Adv. in Appl. Math. 24, # 2 , 111-124 , 2000 .

[39] A . Conca , M . E . Rossi , G . Valla . Grobne r flags an d Gorenstei n algebras . Compositio Math. 129, # 1 , 95-121 , 2001 .

[40] A . Conca , N . V . Trung , G . Valla . Koszu l propert y fo r point s i n projectiv e spaces . Math. Scand. 89 , # 2 , 201-216 , 2001 .

[41] A . Connes . Noncommutativ e differentia l geometry . Publ. Math. IHES 62 , 257-360 , 1985 . [42] A . A . Davydov . Totall y positiv e sequence s an d i?-matri c quadrati c algebras . Journal of

Math. Sciences 100 , # 1 , 1871-1876 , 2000 . [43] V . G . Drinfeld . O n quadrati c quasi-commutationa l relation s i n quasi-classica l limit . Mat.

Fizika, Funkc. Analiz, 25-34 , "Naukov a Dumka" , Kiev , 1986 . English translatio n in : Selecta Math. Sovietica 11 , # 4 , 317-326 , 1992 .

[44] L . Ein , R . Lazarsfeld . Syzygie s an d Koszu l cohomolog y o f smoot h projectiv e varietie s o f arbitrary dimension . Inventiones Math. I l l , # 1 , 51-67 , 1993 .

[45] D . Eisenbud , S . Goto . Linea r free resolution s an d minima l multiplicity . J. Algebra 88 , # 1 , 89-133, 1984 .

[46] D . Eisenbud , A . Reeves , B . Totaro . Initia l ideals , Verones e subrings , an d rate s o f algebras . Advances in Math. 109 , # 2 , 168-187 , 1994 .

[47] P . Etingof, V . Ostrik . Modul e categorie s ove r representation s o f SL q(2) an d graphs . Preprin t math.QA/0302130.

[48] B . Feigin , A . Odesskii . Sklyanin' s ellipti c algebras . Funct. Anal. Appl. 23 , # 3 , 207-214 , 1990.

[49] B . Feigin , B . Tsygan . Cycli c homolog y o f algebra s wit h quadrati c relations , universa l en -veloping algebras , an d grou p algebras . K-theory, arithmetic and geometry (Moscow, 1984^ 1986), 210-239 , Lectur e Note s i n Math . 1289 , Springer , Berlin , 1987 .

[50] M . Finkelberg , A . Vishik . Th e coordinat e rin g o f genera l curv e o f genu s ^ 5 i s Koszul . J. Algebra 162 , 535-539 , 1993 .

[51] G . Floystad . Koszu l dualit y an d equivalence s o f categories , preprin t math.RA/0012264 . [52] R . Froberg . Determinatio n o f a clas s o f Poincar e series . Math. Scand. 37 , # 1 , 29-39 , 1975 .

BIBLIOGRAPHY 157

[53] R . Froberg . A stud y o f grade d extrema l ring s an d o f monomia l rings . Math. Scand. 51 , # 1 , 22-34, 1982 .

[54] R . Froberg . Ring s wit h monomia l relation s havin g linea r resolutions . J. Pure Appl. Algebra 38, # 2 - 3 , 235-241 , 1985 .

[55] R . Froberg . Koszu l algebras . Advances in commutative ring theory (Fez, 1997), 337-350 . Dekker, Ne w York , 1999 .

[56] R . Froberg , T . Gulliksen , C . Lofwall . Fla t one-paramete r famil y o f Artinian algebra s with in -finite numbe r o f Poincar e series . Algebra, topology and their interactions (Stockholm, 1983), 170-191, Lectur e Note s i n Math . 1183 , Springer , Berlin , 1986 .

[57] R . Froberg , C . Lofwall . Koszu l homolog y an d Li e algebras wit h appplicatio n t o generi c form s and points . Homology Homotopy Appl. 4 , no . 2 , 227-258 , 2002 .

[58] W . Fulton , Intersection theory, Springer , 1998 . [59] A . Gandolfi , M . Keane , V . d e Valk . Extrema l two-correlation s o f two-value d stationar y

one-dependent processes . Probab. Theory Related Fields 80 , # 3 , 475-480 , 1989 . [60] I . M. Gelfand , V . A . Ponomarev . Mode l algebra s an d representation s o f graphs. Fund. Anal.

Appl. 13 , 157-166 , 1979 . [61] V . Ginzburg , M . Kapranov . Koszu l dualit y fo r operads . Duke Math. J. 76 , # 1 , 203-272 ,

1994. Erratu m i n Duke Math. J. 80 , # 1 , 293 , 1995 . [62] E . S . Golod . Standar d base s an d homology . Algebra —some current trends (Varna, 1986),

88-95, Lectur e Note s i n Math . 1352 , Springer , Berlin , 1988 . [63] E . S . Golod , I . R . Shafarevich . O n th e towe r o f clas s fields . Izvestiya Akad. Nauk. SSSR,

Ser. Mat. 28 , # 2 , 261-272 , 1964 (i n Russian) . [64] G . Gratzer . Genera l lattic e theory . Birkhauser , Basel , 1978 . [65] M . Green , R . Lazarsfeld . A simpl e proo f o f Petri' s theore m o n canonica l curves , Geometry

Today (Rome, 1984), 129-142 , Progres s i n Math . 60 , Birkhauser , Boston , 1985 . [66] R . Hartshorne . Algebrai c geometry . Springer , Ne w York , 1977 . [67] S . P . Inamdar , V . B . Mehta . Frobeniu s splittin g o f Schuber t varietie s an d linea r syzygies .

Amer. Journ. of Math. 116 , # 6 , 1569-1586 , 1994 . [68] S . A . Joni , G . C . Rota . Coalgebra s an d bialgebra s i n combinatorics . Stud. Appl. Math. 61 ,

# 2 , 93-139 , 1979 . [69] B . Jonsson . Distributiv e sublattice s o f a modula r lattice . Proc. Amer. Math. Soc. 6 , # 5 ,

682-688, 1955 . [70] P . Jorgensen , Linea r fre e resolution s ove r non-commutativ e algebras . Compositio Math. 140 ,

# 4 , 1054-1058 , 2004 . [71] S . Karlin . Tota l positivity . Stanfor d Univ . Press , 1968 . [72] G . Kempf . Som e wonderfu l ring s i n algebrai c geometry . J. Algebra 134 , # 1 , 222-224 , 1990 . [73] G . Kempf . Syzygie s fo r point s i n projectiv e space . J. Algebra 145 , # 1 , 219-223 , 1992 . [74] G . Kempf . Projectiv e coordinat e ring s o f abelia n varieties . Algebra analysis, geometry, and

number theory (Baltimore, MD, 1988), 225-235 , John s Hopkin s Univ . Press , Baltimore , MD, 1989 .

[75] C . Lofwall . O n th e subagebr a generate d b y th e one-dimensiona l element s i n th e Yoned a Ext-algebra. Algebra, algebraic topology and their interactions (Stockholm, 1983), Lectur e Notes i n Math . 1183 , 291-338 , Springer , Berlin , 1986 .

[76] A . Malkin , V . Ostrik , M . Vybornov , Quiver varieties and Lusztig's algebra, preprin t math.RT/0403222.

[77] Yu . I . Manin . Som e remark s o n Koszu l algebra s an d quantu m groups . Ann. Inst. Fourier 37, # 4 , 191-205 , 1987 .

[78] Yu . I . Manin . Quantu m group s an d non-commutativ e geometry . CRM , Universit e d e Montreal, 1988 .

[79] Yu . I . Manin . Topic s i n noncommutativ e geometry . Princeto n Univ . Press , Princeton , 1991 . [80] Yu . I . Manin . Note s o n quantu m group s an d quantu m d e Rha m complexes . Theoret. and

Math. Physics 92 , # 3 , 997-1019 , 1992 . [81] R . Martinez-Villa . Application s o f Koszu l algebras : th e preprojectiv e algebra . Representa-

tion Theory of Algebras (Cocoyoc, 1994), 487-504 , CM S Conf . P r o c , vol . 18 , AMS , Provi -dence, RI , 1996 .

[82] J . P . May . Th e cohomolog y o f augmente d algebra s an d generalize d Masse y product s fo r DGA-algebras. Trans. Amer. Math. Soc. 122 , 334-340 , 1966 .

158 BIBLIOGRAPHY

[83] T . Mora . A n introductio n t o commutativ e an d noncommutativ e Grobne r bases . Theoret. Comput. Science 134 , # 1 , 131-173 , 1994 .

[84] D . Mumford . Lecture s o n curve s o n a n algebrai c surface . Princeto n Univ . Press , Princeton , 1966. D. Mumford . Varietie s define d b y quadrati c relations , Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), 29-100 , Edizion i Cremonese , Rome , 1970 . R. Musti , E . Buttafuoco . Su i subreticol i distributiv i de i reticol i modular! . Boll. Unione Mat. Ital (3 ) 11 , # 4 , 584-587 , 1956 . P. Orlik , H . Terao . Arrangement s o f hyperplanes . Springer-Verlag , Berlin , 1992 . S. Papadima , S . Yuzvinsky . O n rationa l K{w,l] space s an d Koszu l algebras . J. Pure Appl. Algebra 144 , # 2 , 157-167 , 1999 . G. Pareschi . Koszu l algebra s associate d t o adjunctio n bundles . J. Algebra 157 , # 1 , 161-169 , 1993. G. Pareschi , B . Purnaprajna . Canonica l rin g o f a curv e i s Koszul : A simpl e proof . Illinois J. of Math. 4 1 , # 2 , 266-271 , 1997 . I. Peeva , V . Reiner , B . Sturmfels . Ho w t o shel l a monoid . Math. Ann. 310 , # 2 , 379-393 , 1998. D. Piontkovskii . O n th e Hilber t serie s o f Koszu l algebras . Fund. Anal. Appl. 35 , # 2 , 133 -137, 2001. D. Piontkovskii , Noncommutativ e Koszu l nitrations . Preprin t math.RA/0301233 . D. Piontkovskii . Set s o f Hilber t serie s an d thei r applications . Preprin t math.RA/0502149 . Fundam. Prikl . Mat . 10 , 143-156 , 2004 . D. Piontkovskii . Algebra s associate d t o pseudo-root s o f noncommutativ e polynomial s ar e Koszul. Preprin t math.RA/0405375 . A. Polishchuk . O n th e Koszu l propert y o f th e homogeneou s coordinat e rin g o f a curve . J. Algebra 178 , # 1 , 122-135 , 1995 . A. Polishchuk . Pervers e sheave s o n a triangulate d space . Math. Res. Letters 4 , #2 -3 , 191 -199, 1997 . A. Polishchuk . Koszu l configuration s o f point s i n projectiv e spaces . Preprin t math.AG / 0412441. L. Positselski . Nonhomogeneou s quadrati c dualit y an d curvature . Funct. Anal. Appl. 27 , # 3 , 197-204 , 1993 . L. Positselski . Koszu l inequalitie s an d stochasti c sequences . M . A . Thesis , Mosco w Stat e University, 1993 . L. Positselski . Th e correspondenc e betwee n th e Hilber t serie s o f dua l quadrati c algebra s does no t impl y thei r havin g th e Koszu l property . Funct Anal. Appl. 29 , # 3 , 1995 . L. Positselski . Koszu l propert y an d Bogomolo v conjecture . Harvar d Ph . D . Thesis , 1998 , available a t http://www.math.uiuc.edu/K-theory/0296 . L. Positselski , A . Vishik . Koszu l dualit y an d Galoi s cohomology . Math. Research Letters 2 , # 6 , 771-781 , 1995 . S. Priddy . Koszu l resolutions . Trans. Amer. Math. Soc. 152 , 39-60 , 1970 . J.-E. Roos . Homolog y o f free loo p spaces , cycli c homology , an d non-rationa l Poncare-Bett i series i n commutativ e algebra . Algebra—some current trends (Varna, 1986), 173-189 , Lec -ture Note s i n Math . 1352 , Springer , Berlin , 1988 . J.-E. Roos . O n th e characterizatio n o f Koszu l algebras . Fou r counter-examples . Comptes Rendus Acad. Sci Paris, ser . I , 321 , # 1 , 15-20 , 1995 . J.-E. Roos . A descriptio n o f th e homologica l behavio r o f familie s o f quadrati c form s i n fou r variables. Syzygies and Geometry (Boston, 1995), 86-95 , Northeaster n Univ. , Boston , MA , 1995. M. Rosso . A n analogu e o f P.B.W . theore m an d th e universa l K-matri x fo r Uh,sl(N + 1) . Commun. Math. Phys. 124 , 307-318 , 1989 . P. Schenzel . Ube r di e freie n Auflosunge n extremale r Cohen-Macauley-Ringe . J. Algebra 64 , 93-101, 1980 . A. Schwarz . Noncommutativ e supergeometr y an d duality . Lett. Math. Phys. 50 , # 4 , 309 -321, 1999 . S. Serconek , R . Wilson . Th e quadrati c algebra s associate d wit h pseudo-root s o f noncommu -tative polynomial s ar e Koszu l algebras . J. Algebra 278 , 473-493 , 2004 .

BIBLIOGRAPHY 159

B. Shelton , C . Tingey . O n Koszu l algebra s an d a ne w constructio n o f Artin-Schelte r regula r algebras. J. Algebra 241 , 789-798 , 2001. B. Shelton , S . Yuzvinsky . Koszu l algebra s fro m graph s an d hyperplan e arrangements . J . London Math. Soc. (2) 56 , # 3 , 477-490 , 1997 . J. Stasheff . Homotop y associativit y o f if-spaces , II . Trans. Amer. Math. Soc. 108 , 293-312 , 1963. B. Sturmfels . Grobne r base s an d conve x polytopes . Universit y Lectur e Series , 8 . AMS, Prov -idence, RI , 1996 . B. Sturmfels . Fou r counterexample s i n combinatoria l algebrai c geometry . J. Algebra 230 , # 1 , 282-294 , 2000 . J. Tate , M . va n de n Bergh . Homologica l propertie s o f Sklyani n algebras . Inventiones Math. 124, # 1 - 3 , 619-647 , 1996 . B. Tsirelson . A new framework fo r ol d Bel l inequalities . Helv. Phys. Acta 66 , #7-8 , 858—874, 1993. V. A . Ufnarovski . Combinatoria l an d asymptoti c method s i n algebra . Algebra, VI, 1-196 . Encyclopaedia Math . Sci . 57 , Springer , Berlin , 1995 . V. d e Valk . A proble m o n 0- 1 matrices . Compositio Math. 7 1 , # 2 , 139-179 , 1989 . V. d e Valk . Hilber t spac e representation s o f m-dependen t processes . Ann. Probab. 21 , # 3 , 1550-1570, 1993 . V. d e Valk. One-dependen t processes : two-bloc k factor s an d non-two-bloc k factors . Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1994 . M. Va n de n Bergh . Noncommutativ e homolog y o f som e three-dimensiona l quantu m spaces . K-theory, 8 , 213-230 , 1994 . M. Va n de n Bergh . A relation betwee n Hochschil d homolog y an d cohomolog y fo r Gorenstei n rings. Proc. AMS, 126 . # 4 , 1345-1348 , 1998 . A. M . Vershik . Algebra s wit h quadrati c relations . Spektr. teoriya operatorov i beskonech-nomern. analiz, 32-57 , Akad . Nau k Ukrain . SSR , Inst . Mat. , Kiev , 1984 . English translatio n in: Selecta Math. Sovietica 11 , # 4 , 293-316 , 1992 .

[126] M . Vybornov . Mixe d algebra s an d quiver s relate d t o cel l complexe s an d Koszu l duality , Math. Res. Letters 5 , 675-683 , 1998 .

[127] M . Vybornov . Sheave s o n triangulate d space s an d Koszu l duality . Preprin t math.AT / 9910150.

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Titles i n Thi s Serie s

37 Alexande r Polishchu k an d Leoni d Positselski , Quadrati c algebras , 200 5

36 Mati ld e Marcolli , Arithmeti c noncommutativ e geometry , 200 5

35 Lue a Capogna , Carlo s E . Kenig , an d Loredan a Lanzani , Harmoni c measure :

Geometric an d analyti c point s o f view , 200 5

34 E . B . Dynkin , Superdiffusion s an d positiv e solution s o f nonlinea r partia l differentia l

equations, 200 4

33 Kristia n Seip , Interpolatio n an d samplin g i n space s o f analyti c functions , 200 4

32 Pau l B . Larson , Th e stationar y tower : Note s o n a cours e b y W . Hug h Woodin , 200 4

31 Joh n Roe , Lecture s o n coars e geometry , 200 3

30 Anato l e Katok , Combinatoria l construction s i n ergodi c theor y an d dynamics , 200 3

29 Thoma s H . Wolf f (Izabell a Lab a an d Caro l Shubin , editors) , Lecture s o n harmoni c

analysis, 200 3

28 Ski p Garibaldi , Alexande r Merkurjev , an d Jean-Pierr e Serre , Cohomologica l

invariants i n Galoi s cohomology , 200 3

27 Sun-Yun g A . Chang , Pau l C . Yang , Karste n Grove , an d Jo n G . Wolfson ,

Conformal, Riemannia n an d Lagrangia n geometry , Th e 200 0 Barret t Lectures , 200 2

26 Susum u Ariki , Representation s o f quantu m algebra s an d combinatoric s o f Youn g

tableaux, 200 2

25 Wil l ia m T . Ros s an d Harol d S . Shapiro , Generalize d analyti c continuation , 200 2

24 Victo r M . Buchstabe r an d Tara s E . Panov , Toru s action s an d thei r application s i n

topology an d combinatories , 200 2

23 Lui s Barreir a an d Yako v B . Pes in , Lyapuno v exponent s an d smoot h ergodi c theory ,

2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations ,

2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Frobeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 Stephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf f an d Dietma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra , 1993

TITLES I N THI S SERIE S

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o low-dimensional topology , 198 9