Noetherian Semigroup Algebras (Algebra and Applications)

Noetherian Semigroup Algebras

Algebra and Applications Volume 7 Managing Editor: Alain Verschoren RUCA, Belgium Series Editors: Christoph Schweigert Hamburg University, Germany Ieke Moerdijk Utrecht University, The Netherlands John Greenlees Sheffield University, UK Mina Teicher Bar-llan University, Israel Eric Friedlander Northwestern University, USA Idun Reiten Norwegian University of Science and Technology, Norway Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C*algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science.

Noetherian Semigroup Algebras by

Eric Jespers Vrije Universiteit Brussel, Brussels, Belgium and

Jan Okniński Warsaw University, Warsaw, Poland

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-5809-8 (HB) 978-1-4020-5809-7 (HB) 1-4020-5810-1 (e-book) 978-1-4020-5810-3 (e-book)

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To Griet

Eric Jespers

To Iwona

Jan Okni´ nski

Contents 1 Introduction

1

2 Prerequisites on semigroup theory 2.1 Semigroups . . . . . . . . . . . . . 2.2 Uniform semigroups . . . . . . . . 2.3 Full linear semigroup . . . . . . . . 2.4 Structure of linear semigroups . . . 2.5 Closure . . . . . . . . . . . . . . . 2.6 Semigroups over a field . . . . . . .

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7 7 13 17 19 27 32

3 Prerequisites on ring theory 3.1 Noetherian rings and rings satisfying a polynomial identity 3.2 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Group algebras of polycyclic-by-finite groups . . . . . . . . 3.4 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Gelfand-Kirillov dimension . . . . . . . . . . . . . . . . . . 3.6 Maximal orders . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Principal ideal rings . . . . . . . . . . . . . . . . . . . . . .

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37 37 42 45 47 51 54 59

4 Algebras of submonoids of polycyclic-by-finite 4.1 Ascending chain condition . . . . . . . . . . . . 4.2 The unit group . . . . . . . . . . . . . . . . . . 4.3 Almost nilpotent case . . . . . . . . . . . . . . 4.4 Structure theorem . . . . . . . . . . . . . . . . 4.5 Prime ideals of K[S] . . . . . . . . . . . . . . . 4.6 Comments and problems . . . . . . . . . . . . .

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5 General Noetherian semigroup algebras 117 5.1 Finite generation of the monoid . . . . . . . . . . . . . . . . . 118 5.2 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . 127 vii

viii

Contents

5.3 5.4 5.5

Monomial semigroups and sufficient conditions . . . . . . . . 134 Gelfand-Kirillov dimension . . . . . . . . . . . . . . . . . . . 141 Comments and problems . . . . . . . . . . . . . . . . . . . . . 146

6 Principal ideal rings 6.1 Group algebras . . . . . 6.2 Matrix embedding . . . 6.3 Finite dimensional case 6.4 The general case . . . . 6.5 Comments and problems

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149 . 150 . 154 . 159 . 167 . 174

7 Maximal orders and Noetherian semigroup algebras 177 7.1 Maximal orders and monoids . . . . . . . . . . . . . . . . . . 178 7.2 Algebras of submonoids of abelian-by-finite groups . . . . . . 184 7.3 Comments and problems . . . . . . . . . . . . . . . . . . . . . 194 8 Monoids of I-type 8.1 A characterization . . . . . . . . 8.2 Structure of monoids of I-type . 8.3 Binomial monoids are of I-type . 8.4 Decomposable monoids of I-type 8.5 Algebras of monoids of I-type . . 8.6 Comments and problems . . . . .

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197 198 209 226 239 246 253

9 Monoids of skew type 9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 9.2 Monoids of skew type and the cyclic condition . . . 9.3 Non-degenerate monoids of skew type . . . . . . . 9.4 Algebras of non-degenerate monoids of skew type . 9.5 The cancellative congruence and the prime radical 9.6 Comments and problems . . . . . . . . . . . . . . .

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257 257 261 264 275 284 287

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10 Examples 291 10.1 Monoids of skew type and the Gelfand-Kirillov dimension . . 292 10.2 Four generated monoids of skew type . . . . . . . . . . . . . . 295 10.3 Examples of Gelfand-Kirillov dimension 2 . . . . . . . . . . . 301 10.4 Non-degenerate monoids of skew type of Gelfand-Kirillov dimension one . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 10.5 Examples of maximal orders . . . . . . . . . . . . . . . . . . . 330 10.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Contents

ix

Bibliography

343

Notation

353

Index

357

Chapter 1 Introduction The first aim of this work is to present the main results and methods of the theory of Noetherian semigroup algebras. These general results are then applied and illustrated in the context of certain interesting and important concrete classes of algebras that arise in a variety of areas and have been recently intensively studied. One of the main motivations for this project has been the growing interest in the class of semigroup algebras (and their deformations) and in the application of semigroup theoretical methods. Two factors seem to be the cause for this. First, this field covers several important classes of algebras that recently arise in a variety of areas. Furthermore, it provides methods to construct a variety of examples and tools to control their structure and properties, that should be of interest to a broad audience in algebra and its applications. Namely, this is a rich resource of constructions not only for the noncommutative ring theorists (and not only restricted to Noetherian rings) but also to researchers in semigroup theory and certain aspects of group theory. Moreover, because of the role of new classes of Noetherian algebras in the algebraic approach in noncommutative geometry, algebras of low dimension (in terms of the homological or the Gelfand-Kirillov dimension) recently gained a lot of attention. Via the applications to the Yang-Baxter equation, the interest also widens into other fields, most notably into mathematical physics. A second factor for the growing interest in semigroup algebras is the natural relation to the extensively studied class of group algebras. However, one must realize that the scope of research on semigroup algebras must be in some sense more specialized. Namely, one has to study the “correct” classes of semigroups in order to get deep and applicable results. Our feeling is that this amounts to saying that again one 1

2

1. Introduction

should focus on important motivating constructions and examples that show up in other areas of algebra and its potential applications. It is also worth mentioning that, within the last decade, and at a rapidly increasing scale, semigroup theoretical methods have occurred naturally in many aspects of ring theory, algebraic combinatorics and representation theory. However, the knowledge of these methods is not so common among algebraists. The selection of the material is based on the principle that only those topics are dealt with for which a comprehensive general theory has been developed and that lead to nontrivial convincing results and applications. From the semigroup theoretic point of view, it turned out that the common feature for most of these topics is the required focus on a variety of methods based on cancellative semigroups and linear semigroups. The techniques built seem to be of independent interest, they form a core for the presented results that also contributes to the integrity of selected material. Some “isolated” topics, that often require a lot of prerequisites that would not fit in the scope of the methods developed in this book, are not included. However, all such exceptions are mentioned in the comments following every chapter. We denote by K[S] the semigroup algebra of a given field K and a semigroup S. So K[S] is a K-space with basis S and with multiplication extending, in a natural way,
the operation on S. A typical nonzero element α ∈ K[S] is of the form α = ni=1 ki si fore some nonzero ki ∈ K, si ∈ S and n ≥ 1. Then the support of α is defined by supp(α) = {s1 , . . . , sn }. For a semigroup S with zero θ, a related object, the contracted semigroup algebra K0 [S], is defined as the factor algebra K[S]/Kθ. So we identify the zero of S with the zero of the algebra. As, in this case, K[S] ∼ = K0 [S] ⊕ K, the two algebras are very strongly related and sometimes it is more convenient to deal with K0 [S]. So every
nonzero α ∈ K0 [S] can be considered as an element of the form α = ni=1 ki si with ki = 0 and si = θ, and in this case we define supp(α) as the set of all si . The following result ([54, Corollary 5.4]) is one of the starting points for the study of Noetherian semigroup algebras, both from the abstract point of view and from the point of view of applications: the semigroup algebra K[S] of an abelian monoid S is Noetherian if and only if S is finitely generated. It should be pointed out that the proof of the necessity is quite nontrivial. However, in the noncommutative case the situation is much more complicated and it is even hard to expect a complete characterization. Nevertheless, as in the group algebras case, one of the goals is to see whether the homogeneous information (that is, the information on S) reflects and determines the properties of K[S].

1. Introduction

3

A substantial part of the book is devoted to algebras K[T ] of submonoids T of a group (in most cases, a polycyclic-by-finite group). One of the obvious reasons to study such algebras K[T ] is that they provide a natural class of orders and of Noetherian algebras, related to the extensively studied, quite well understood, and rich in beautiful results class of group algebras of polycyclic-by-finite groups [129, 130]. In this context, one should also point out the important role of such groups and of their algebras in several aspects of algebra, topology and of their applications. Not only the class of algebras K[T ] for subsemigroups T of such groups seems of a general interest, but such algebras arise in a natural way in a number of concrete problems. For example, in the problem of determining set theoretical solutions of the Yang-Baxter equation, which originated from the theory of quantum groups, as a tool for tackling certain problems in physics [33]. Another source of motivation comes from powerful homological methods developed for the study of Sklyanin algebras, a class of low dimensional algebras playing an important role in the attempt to build foundations for noncommutative geometry [150]. Surprisingly, these two problems have a meeting point, culminating in a fascinating class of semigroup algebras, presented in Chapter 8. A wider class, providing also several interesting examples of low dimensional algebras, is then considered in Chapter 9. Because of the general tendency to investigate concrete classes of algebras, we describe in full detail several examples in Chapter 10. It is a real satisfaction to see that the theory developed in the preceding chapters allows us to show that several ingredients of the structural description of these combinatorially defined algebras can be effectively computed. Looking from another perspective, the necessity for studying algebras K[T ] of cancellative semigroups T in the context of general Noetherian algebras is an obvious consequence of the structure theorem for linear semigroups [123], see Theorem 2.4.3. This results from the fact that a monoid S with K[S] right Noetherian has important homomorphic images that are linear, that is, subsemigroups of the full multiplicative monoid Mn (D) of n × n matrices over a division algebra D. Unexpectedly, we are able to show that the structure of S itself shares, in many cases, the flavor of the structure of a linear semigroup. As mentioned before, the main motivation comes from several important concrete classes of examples. Though, a general theory leads to surprising and very useful results of general character. For example, such are the main results of Chapter 4 and Chapter 5: Theorem 4.4.7 and its main consequences, Theorem 5.1.5, Theorem 5.1.6, as well as Theorem 5.2.3 and Theorem 5.3.7. All these results are heavily used in the next chapters of the

4

1. Introduction

book and they also illustrate the strategy of studying arbitrary Noetherian algebras K[S] via certain related algebras K[T ] of cancellative semigroups T. The material is organized as follows. Chapter 2 and Chapter 3 are included for the convenience of the reader. These chapters contain the main prerequisites of semigroup and ring theory used in the book. However, they are not intended as an overview of all relevant important classical results on semigroups and rings. We only state those results that are relevant for the main purpose of this monograph. In Chapter 4 we deal with algebras of submonoids of polycyclic-by-finite groups. Right Noetherian algebras of this type are completely characterized. It is shown that these algebras are finitely presented, they are Jacobson rings and information on their prime spectrum is obtained. Chapter 5 is devoted to the structure of arbitrary right Noetherian semigroup algebras. Certain necessary and sufficient conditions for K[S] to be Noetherian are proved, in terms of certain ideal chains of special type in S. Relations with the PI (polynomial identity) property and general results on the Gelfand-Kirillov dimension are also proved. The next two chapters deal with two special classes of algebras defined on the ring level: principal ideal rings and Noetherian maximal orders. In Chapter 6 a full description of principal ideal rings of the form K[S] is obtained. The structure of the underlying S is entirely determined, see Theorem 6.2.1, Theorem 6.3.6, Theorem 6.4.1 and Corollary 6.4.2. Noetherian PI domains K[S] of cancellative monoids S that are maximal orders are described in Chapter 7. The main results show that, also in this case, the information can be expressed in terms of S only, Theorem 7.2.5 and Theorem 7.2.7. A class of intriguing finitely presented algebras defined by quadratic relations, called algebras of skew type, is considered in Chapter 9. It provides us with many examples of Noetherian PI algebras. See Theorem 9.3.7, Theorem 9.4.2, and also, Theorem 10.1.2 and Theorem 10.1.3 for the main general results on these algebras. Before this, in Chapter 8, we study the special subclass arising independently from the set theoretical solutions of the Yang-Baxter equation, homological methods of Sklyanin algebras and from some considerations concerning Gr¨ obner bases techniques in quadratic algebras. The main accomplishments of the theory are stated in Theorem 8.1.4, Theorem 8.2.3, Theorem 8.3.10, Theorem 8.4.3, Theorem 8.5.1 and Theorem 8.5.6. These algebras are Noetherian PI domains, and also maximal orders of finite homological dimension. Their intriguing combinatorial properties lead to deep results of a structural nature. In particular, the underlying monoids can be characterized as certain submonoids of the semidirect product Fan
Symn of the free abelian group Fan of rank n and the symmetric group Symn .

1. Introduction

5

Several examples are studied in detail in Chapter 10. Each of the chapters, except for the chapters with prerequisites, contains bibliographic comments and a list of problems related to the main results presented. Most of the presented material is due to the authors. Some results have been obtained in joint work with F.Cedo and T.Gateva-Ivanova. In Chapter 8, we also incorporate some of the latest developments on set theoretic solutions of the Yang-Baxter equation and on algebras of I-type, due to T.Gateva-Ivanova and M.Van den Bergh, P.Etingof, T.Schedler and A.Soloviev, T.Gateva-Ivanova, and W.Rump. For the previous work on semigroup algebras we refer to the monographs [54] and [119]. Clearly, we strongly rely on results on group algebras [129] and, more generally, on graded rings [130]. Some of our results can be proved in the more general setting of certain quantized deformations of the considered algebras, such as for example twisted semigroup algebras. However, this is beyond the scope of the monograph. Some material related to the present work was presented in the survey paper [82]. This project was supported by the Flemish-Polish bilateral agreements BIL 01/31 and BIL2005/VUB/06, by the KBN research grant 2P03A 033 25 (Poland), Onderzoeksraad of Vrije Universiteit Brussel and by Fonds voor Wetenschappelijk Onderzoek (Belgium).

Chapter 2 Prerequisites on semigroup theory This chapter is devoted to the necessary background on semigroups and especially on semigroups of matrices over a division algebra, referred to as linear semigroups. The material is a mixture of very classical results of semigroup theory, basic structural results of the theory of linear semigroups and certain rather special technical results that provide us with important tools used in the book. Unless explicit references are given, the results can be found in the monograph [123]. The reader should be warned however that the latter is mostly devoted to linear semigroups over a field. So, a few proofs are given in Section 2.5, as they cannot be found in the required form in the existing literature and also since they might put more light onto the topic for readers with little experience in semigroup theory.

2.1

Semigroups

In this section we recall some basic facts about semigroups and we introduce notation and terminology. Most of the material can be found in [27] and [66]. If not stated otherwise, multiplicative notation will be always used for the operation in a semigroup S. If A is a subset of a semigroup S, then by ≺A we denote the subsemigroup of S generated by A. If S is a monoid, then A denotes the submonoid generated by A. If A = {a1 , . . . , ak }, then we also write ≺a1 , . . . , ak  and a1 , . . . , ak for the corresponding subsemigroup or submonoid of S. In particular, ≺a is the cyclic semigroup generated by a ∈ S. Then a is called a periodic element if ≺a is finite. In this case an is an idempotent for some 7

8

2. Prerequisites on semigroup theory

n ≥ 1 and e≺a is a an ideal and a subgroup of ≺a. If G is a group and A ⊆ G, then gr(A) stands for the subgroup of G generated by A. If S is a semigroup that is not a monoid, then by S 1 we mean the monoid S ∪ {1} obtained by adjoining an identity element to S. Otherwise, we put S 1 = S. Moreover U(S) denotes the set of all invertible elements of S, called the unit group of S. If S has a zero element, we often write θ for the zero of S. By S 0 we mean the semigroup S with zero adjoined. Let e = e2 ∈ S. Then eSe is a monoid and hence U(eSe) is a subgroup of S with identity e. Subgroups of this type are exactly the maximal subgroups of S. For example, if S = Mn (K) the multiplicative semigroup of n × n matrices over a field K, then every e Mn (K)e is isomorphic to Mj (K), where j is the rank of e. Hence, the maximal subgroup of Mn (K) containing e is isomorphic to GLj (K). A nonempty subset I of S is a right (left) ideal of S if IS ⊆ I (SI ⊆ I, respectively). Green’s relations on S reflect the ideal structure. They are defined as follows: • aRb if aS 1 = bS 1 • aLb if S 1 a = S 1 b • aHb if aRb and aLb • aJ b if S 1 aS 1 = S 1 bS 1 . An equivalence relation ρ on S is a right congruence if sρt implies suρtu for every u ∈ S. A left congruence is defined symmetrically. A left and right congruence is simply called a congruence. For convenience sake we sometimes write sρt as (s, t) ∈ ρ. Clearly R is an example of a left congruence, and L is a right congruence. If ρ is a congruence on S then we denote by S/ρ the set of ρ-classes which has a natural semigroup structure inherited from S. If K is a field then the kernel of the natural epimorphism K[S] −→ K[S/ρ] is denoted by I(ρ) and it is the K-span of the set consisting of all elements s − t with sρt. If ρ = S × S then I(ρ) is called the augmentation ideal of K[S] and is usually denoted by ω(K[S]). If I is a (two-sided) ideal of S, then one can form the Rees factor S/I, which is the semigroup with zero (S \ I) ∪ {θ} with operation defined by a · b = ab if ab ∈ S \ I and ab = θ otherwise. Clearly I corresponds to a congruence τI on S determined by sτI t if and only if either s = t or s, t ∈ I. It is often convenient to define S/I = S for I = ∅. If a ∈ S, then the set Ia of non-generators of the ideal S 1 aS 1 is an ideal of S, if nonempty. The Rees factor S 1 aS 1 /Ia is called the principal factor

2.1. Semigroups

9

of a in S. It is clear that, if a = θ, then the set of nonzero elements of S 1 aS 1 /Ia can be identified with the J -class of a in S. Principal factors are used to study the structure of S ‘locally’. This approach is especially useful in the class of finite semigroups. A semigroup S is called 0-simple if S = S 2 and S has no ideals other than S and possibly {θ} if S has a zero element θ. If additionally S has a primitive idempotent, then S is completely 0-simple. (Notice that, for practical reasons, we include the class of completely simple semigroups into the class of completely 0-simple semigroups, which is not a standard convention.) It is known that every principal factor of a semigroup S is either completely 0-simple or a null semigroup (that is, a semigroup with zero multiplication). Let X, Y be nonempty sets, G a group and P = (pyx ) a Y ×X matrix with entries in G0 . (Clearly, by this we mean a function (y, x) → pyx from Y × X to G0 .) By S = M(G, X, Y ; P ) we denote the set of all triples (g, x, y) ∈ G0 ×X ×Y with (g, x, y, )·(g
x
, y
) = (gpyx
g
, x, y
), where all triples (θ, x, y) are identified with the zero of S. In particular, θ also denotes the zero of S. S is called a semigroup of matrix type over the group G with sandwich matrix P . Then the sets S(x) = {(g, x, y) ∈ S | g ∈ G0 , y ∈ Y }, x ∈ X, are called the rows of S, while S (y) = {(g, x, y) ∈ S | g ∈ G0 , x ∈ X}, y ∈ Y , are the columns of S. If |X| = k < ∞, then we also write S = M(G, k, Y ; P ), identifying X with the set {1, . . . , k}. The same convention applies to the set Y of columns of S.  For any subsets A ⊆ X, B ⊆ Y we put S(A) = x∈A S(x) , S (B) =  (y) , S (y) = S (y) and (x) ∩ S y∈B S (x)  (B) (y) S(x) . S(A) = S(A) ∩ S (B) = x∈A,y∈B

(B)

Clearly, S(A) = M(G, A, B; PBA ) is a semigroup of matrix type, where PBA is the B × A submatrix of P . S can be represented as the set of X × Y matrices over G ∪ {0} with at most one nonzero entry (under the identification of every (g, x, y) with the matrix with g in position (x, y) and zeros elsewhere) with a · b = aP b, where the latter is the usual multiplication of matrices. It is known that a semigroup S with zero is completely 0-simple if and only if it is isomorphic to a semigroup of matrix type M(G, X, Y ; P ) such that every row and every column of the matrix P has a nonzero entry. If S is a semigroup with no zero element, then it is completely 0-simple if and only if S 0 is of the latter type (in particular, the sandwich matrix P has no

2. Prerequisites on semigroup theory

10

zero entries). The basic information on the properties of semigroups of the above type is given below. Lemma 2.1.1. Let S = M(G,X,Y\P) Then

be a semigroup of matrix type.

1. for every nonempty subsets A C X,B CY, S^ and S^ is a left ideal of S,

is a right ideal of S

2. E = {(g,x,y) | pyx € G,g = pyx1} is the set of nonzero idempotents of S. If e = (g,x,y) € E, then SW — G° via the map (h,x,y)»-> hpyx, 3. if Pyx = 0, then SW is a semigroup with zero multiplication, 4. ifa,b,c€ S and ab ^ 9,bc^ 9, then abc ^ 9\ if a ^9, then any of the conditions ab = a,ba = a implies that b is an idempotent, 5. if ax = bx ^ 9 for some a,ft£ S^y\ then a = ft, 6. if S is completely Q-simple, then for every nonzero a = (g, x,y) € 5 there exist e,feE such that a = ea = af and there exist b,c € S such that bac lies in a maximal subgroup of S. Moreover, aS = 5(x) andSa = S(y). Note that the last assertion identifies the K, C and 7i-classes of a completely 0-simple 5. In particular, we get an egg-box pattern on 5 \ {9} (the boxes being the nonzero W-classes S^l \ {9}), which will be crucial for the general structural approach presented in Section 2.4. Let S = M(G, X, Y; P) be a semigroup of matrix type over a group G. If H is a normal subgroup of G, then