SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS

Transcription

1 SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday ABSTRACT We study the multiplicative properties of the dual of Lusztig s semicanonical basis The elements of this basis are naturally indexed by the irreducible components of Lusztig s nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras We prove that the product of two dual semicanonical basis vectors and is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components and is again an irreducible component It follows that the semicanonical basis and the canonical basis coincide if and only if we are in Dynkin type with Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type We show that in this case the multiplicative properties of the dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type and by the corresponding elliptic root system of type "# %$ RÉSUMÉ Nous étudions les propriétés multiplicatives de la base duale de la base semi-canonique de Lusztig Les éléments de cette base sont naturellement paramétrés par les composantes irréductibles des variétés nilpotentes de Lusztig, qui peuvent être interprétées comme variétés de modules sur les algèbres préprojectives Nous démontrons que le produit de deux vecteurs et de la base semi-canonique duale est encore un vecteur de la base semi-canonique duale si la somme directe des composantes irréductibles et est encore une composante irréductible Il en résulte que les bases canonique et semi-canonique ne coïncident que pour le type de Dynkin avec ' Finalement, nous étudions en détail les variétés de modules sur l algèbre préprojective de type Nous montrons que dans ce cas les propriétés multiplicatives de la base semi-canonique duale sont controllées par la forme de Ringel d une algèbre tubulaire de type ())* et par le qui lui est associé système de racines elliptique de type #,$ CONTENTS 0 : 1 Introduction 2 2 Varieties of modules 3 Preprojective algebras 8 4 Constructible functions 9 Semicanonical bases 9 Comultiplication 12 7 Multiplicative properties of the dual semicanonical basis 14 8 Embedding of -/ into the shuffle algebra 1 9 A Galois covering of for type From Schur roots to indecomposable multisegments Cases 14, 17, 198 : the component graph 2 12 Cases 14, 17, 198 : the graph of prime elements of 2 Mathematics Subject Classification (2000): 14M99, 1D70, 1E20, 1G20, 1G70, 17B37, 20G42 C Geiss acknowledges support from DGAPA-UNAM for a sabbatical stay at the University of Leeds J Schröer was supported by a research fellowship from the DFG (Deutsche Forschungsgemeinschaft) He also thanks the Laboratoire LMNO (Caen) for an invitation in Spring 2003 during which this work was started B Leclerc is grateful to the GDR 2432 (Algèbre non commutative) and the GDR 2249 (Groupes, géométrie et représentations) for their support 1

2 2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 1 13 End of the proof of Theorem Case 1 : the tubular algebra and the weighted projective line 27 1 Case : the root system 30 1 Case 1 : parametrization of the indecomposable irreducible components 3 17 Case 1 : the component graph 3 18 Proof of Theorem Concluding remarks Pictures and tables 41 References 1 1 INTRODUCTION 11 Let be a simple Lie algebra of simply-laced type 1, and let be a maximal nilpotent subalgebra Let : be the canonical basis of the quantum enveloping algebra [29, 37] and : the basis dual to : When tends to 1, these two bases specialize to bases : and : of and, respectively Here stands for a maximal unipotent subgroup of a complex simple Lie group with Lie algebra Let denote an indexing set for the simple roots of Given a finite-dimensional -graded vector space with graded dimension " #", we denote by 0$ the corresponding nilpotent variety, see [38, % 12] This variety can be seen as the variety of modules over the preprojective algebra 0 attached to the Dynkin diagram of, with underlying vector space [4] let ')((*, be the set of irreducible components of Lusztig has shown that For a variety there are natural bijections '-((* 0$/ 01 :23" #"4 (resp : " #"4 ) 7 198;: (resp 8 : ) where < " #"4 (resp < " #"4 ) is the subset of : (resp : ) consisting of the elements of degree " #" Kashiwara and Saito [30] proved that the crystal basis of 3 can be constructed geometrically in terms of these irreducible components (this was a conjecture of Lusztig) This paper is motivated by several problems about the bases : and : and their relations with the varieties 0 $ and the preprojective algebra 0 12 One problem, which was first considered by Berenstein and Zelevinsky [4], is to study the multiplicative structure of the basis : Two elements 8 and 8 4 of : are called multiplicative if their product belongs to : up to a power of It was conjectured in [4] that 8 and 8 4 are multiplicative if and only if they -commute We refer to this as the BZ-conjecture The conjecture was proved for types 1 4 and 1 [4], and it also holds for 1 8 [1] More recently, Marsh and Reineke observed a strong relationship between the multiplicative structure of : and properties of the irreducible components of the varieties 0>$ They checked [42] that for of type 1 2?@BADCE, if the irreducible components GF 0$ and 4 F 0$H are the closures of the isomorphism classes of two indecomposable 0 -modules I and I 4, then 8 : and 8 :JH are multiplicative if and only if K2LNM O I I4P/SR This was verified by a case-by-case calculation, using the fact that for type 1 2,@#ATCE the preprojective algebra is of finite representation type, that is, it has only a finite number of isomorphism classes of indecomposable modules [1] They also calculated many examples in type 138 and conjectured that this property still holds in this case (note that 0 is again representation-finite for 1 8 ) But a conceptual explanation was still missing Let VU with Y \[ 4 denote the subset of 0W$ and Y 4 [ -X $ H consisting of all 0 -modules I isomorphic to Y ZU Y 4 4 It follows from a general decomposition theory for irreducible

3 @ A U X SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 3 components of varieties of modules developed in [14] that the condition K2LNM O I I4* R for some I I4* [ 4 is equivalent to U 4 being an irreducible component of 0 $ $H X In [31] counterexamples to the BZ-conjecture were found for all types other than 1 2 with In particular in type 1, for a certain of dimension 8 one can find an irreducible component of 0$ such that (1) 8 : 7 8 : 8 : where and are two irreducible components of 0 $ $, see also [2] This seems to be the smallest counterexample to the BZ-conjecture in type 1 Moreover, it also shows that the result of Marsh and Reineke does not generalize to 1W Note however that the BZ-conjecture was proved for large families of elements of : [8, 9, 10, 33] For example, in type 1 it holds for quantum flag minors, and the reformulation in terms of direct sums of irreducible components is also valid [49] So one would like to get a better understanding of the relationship between multiplicativity of elements of : and direct sum decompositions of irreducible components of varieties of 0 - modules 13 Another interesting problem concerns the singular supports of the simple perverse sheaves be a Dynkin quiver, which is obtained used by Lusztig [37] to define the canonical basis : Let from the Dynkin diagram of by choosing an orientation Let ( be the affine space of representations of with underlying finite-dimensional -graded vector space This is a finite union of isomorphism classes (or orbits) In Lusztig s geometric construction, the elements of :23" #"4 are given by the perverse extensions of the constant sheaves on the orbits In [38] Lusztig considered the singular supports of these sheaves and showed that they are unions of irreducible components of 0 $ (independent of the chosen orientation of the Dynkin diagram of ) He conjectured that in fact each is irreducible, equal to the closure 0 of the conormal bundle of Unexpectedly, Kashiwara and Saito [30] produced a counterexample to this conjecture They exhibited two orbits for type 1 such that 0 0 The corresponding vectors 8 and 8 of :2 have principal degree 1, and apparently this is the smallest counterexample in type 1 It turns out that this counterexample is dual to the counterexample above for :, in the sense that 0 and 0, see [31, Remark 1] One motivation for this paper was to find an explanation for this coincidence 14 What makes these problems difficult is that, although the canonical basis reflects by definition the geometry of the varieties (, we want to relate it to the geometry of some other varieties, namely the irreducible components of the nilpotent varieties 0 $ It is natural to think of an intermediate object, that is, a basis reasonably close to the canonical basis, but directly 0 $ : : " F 4 F U defined in terms of the varieties Lusztig [41] has constructed such a basis J:#" and called it the semicanonical basis This is a basis of (not of the -deformation 3 ) which gives rise, like, to a basis in each irreducible highest weight N -module Let $% denote the basis of dual to Our first main result is the following: Theorem 11 If 0$ and 0$WH are irreducible components such that is an irreducible component of 0 $ $WH, then %N: %J:JH '%J: X In other words, the dual semicanonical basis satisfies the multiplicative property which was expected to hold for the dual canonical basis :

4 4 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER An irreducible component [ ')((* 0 $ is called indecomposable if contains a dense subset of indecomposable 0 -modules By [14], every irreducible component of 0>$ has a canonical decomposition U 3U where the F 0$ are indecomposable irreducible components Our theorem implies that %J: '%J: %J: Hence has a natural description as a collection of families of monomials in the elements indexed by indecomposable irreducible components Such a description of resembles the description of : for type 1 obtained by Berenstein and Zelevinsky 1 So a natural question is how close are the bases and :? In type 1, Berenstein and Zelevinsky [4] proved that all minors of the triangular matrix of coordinate functions on belong to : We prove that they also belong to Hence using [33, 49], it follows that : contains all multiplicative products of flag minors However the two bases differ in general More precisely we have: Theorem 12 The bases and : coincide if and only if is of type 1 2 For example in type 1, we deduce from Equation (1) and Theorem 11 that (2) % : 8 : 8 : (where for simplicity we use the same notation 8 : and 8 : for the specializations at \ ) Nevertheless, since and : have lots of elements in common, we get an explanation why the BZ-conjecture (or rather its reformulation in terms of irreducible components of varieties of 0 - modules) holds for large families of elements of : Of course, by duality, these results also allow to compare the bases returning to the example of [30], we can check that (3) 8 O O and this is probably the smallest example in type 1 and : In particular, for which the canonical and semicanonical bases differ One may conjecture that, in general, the elements : occurring in the -expansion of 8 [ : are indexed by the irreducible components of, so that is irreducible if and only if 8 O (There is a similar conjecture of Lusztig [40] for the semicanonical basis of the group algebra of a Weyl group obtained from the irreducible components of the Steinberg variety) Assuming this conjecture we get an explanation of the relationship between the counterexamples to the conjectures of Berenstein-Zelevinsky and Lusztig 1 In the last part of the paper, we consider the first case which is not well understood, namely is representation-infinite, but it is still of tame type 1 In this case, the preprojective algebra 0 representation type [1] Motivated by our description of in terms of indecomposable irreducible components of varieties of 0 -modules, we give a classification of the indecomposable irreducible components for the case 1 We also give an explicit criterion to decide when the closure of the direct sum of two such components is again an irreducible component These results are deduced from [24], in which a general classification of irreducible components of varieties of modules over tubular algebras is developed They are naturally formulated in terms of the Ringel bilinear form 0 P0 of a convex subalgebra of a Galois covering of 0 The algebra is a tubular algebra of type JCJE and the corresponding 10-dimensional infinite root system is an elliptic root system of type 4 in the classification of Saito [0], with a 2-dimensional lattice of imaginary roots Note that the irreducible component of Equation (1) corresponds to a generator of this lattice (This is an a posteriori justification for calling 8: an imaginary vector in [31]) The Ringel form 0 P0 allows to define a distinguished Coxeter matrix of order acting on We prove the following:

5 1 SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS Theorem 13 There is a one-to-one correspondence between the set of Schur roots of and the set of indecomposable irreducible components of the nilpotent varieties of type 1> which do not contain an indecomposable projective 0 -module Moreover U 4P is an irreducible component if and only if the Schur roots and4 satisfy certain conditions which are all expressible in terms of 0 P0 and We also explain how to translate from the language of roots to the language of multisegments, which form a natural indexing set of canonical and semicanonical bases in type 1 17 The paper is organized as follows In Section 2 we recall the general theory of varieties of modules We explain a general decomposition theory for irreducible components of such varieties This is followed in Section 3 by a short introduction to preprojective algebras Then we recall the concept of a constructible function in Section 4 Following Lusztig [41], we review in Section the definition of the semicanonical basis of, which is obtained by realizing as an algebra - of constructible functions on the nilpotent varieties In order to study the dual semicanonical basis and its multiplicative properties we also need to describe the natural comultiplication of in terms of - This was not done in [41], so we provide this description in Section In and prove Theorem 11 Note Section 7 we introduce the dual semicanonical basis of - that for this theorem we do not restrict ourselves to types 1, and only assume that is the positive part of a symmetric Kac-Moody Lie algebra We end this section with the proof of the only if part of Theorem 12 In Section 8 we embed - into the shuffle algebra This gives a practical way of computing elements of We use this to prove that in type 1 all nonzero minors in the coordinate functions of belong to In the rest of the paper we focus on the Dynkin cases A In Section 9 we consider a Galois covering 0 of the algebra 0, with Galois group, and we use it to calculate the Auslander-Reiten quiver of 0 We also introduce an algebra whose repetitive algebra is isomorphic to 0 A, has finite representation type, while it is a tubular algebra of tubular type JCJE In Section 10 we recall from [24] that the indecomposable irreducible components of 0 are in one-to-one correspondence with the -orbits of Schur roots of 0 We also describe the map which associates to such a Schur root the multisegment indexing the corresponding indecomposable irreducible component The component graphs for the representation-finite cases 1 4, 1 and 18 are constructed in Section 11, and the corresponding graphs of prime elements of : are described in Section 12 In Section 13 we prove the if part of Theorem 12 All the remaining sections are devoted to the case 1 In Section 14 we relate the category of 0 -modules to the category of modules over the tubular algebra and to the category of coherent sheaves on a weighted projective line of type JCJE in the sense of Geigle and Lenzing [23] In Section 1 we consider the Grothendieck groups They are naturally endowed with a (non-symmetric) bilinear form 0/P0 (the Ringel form) and a Coxeter matrix This gives rise to 4 an elliptic root system of type We give an explicit description of its set of positive roots and of the subset of Schur roots In Section 1, we show that naturally parametrizes the -orbits of Schur roots of 0, hence also the indecomposable irreducible components of 0 Then Section 17 describes the component graph of 0 for type 1, thus making precise the statements of Theorem 13 Section 18 consists of the proof of Theorem 103 We conclude by noting the existence of similar results for type 38 and by pointing out some possible connections with the theory of cluster algebras of Fomin and Zelevinsky (Section 19) Section 20 contains a collection of pictures and tables to which we refer at various places in the text 18 Throughout, we use the following conventions If 1 4 and "# 4 1$ are maps, then the composition is denoted by " % 1 Similarly, if '( 1 and )# 1 C are arrows in a quiver, then the composition of ' and ) is denoted by )*' Modules are always assumed to be left modules

6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER All vector spaces are over the field of complex numbers We set * [ " R ", * [ " R " and " We also set [ " R " and T *R " 2 VARIETIES OF MODULES 21 A quiver is a quadruple where and are sets with non-empty, and E 1 are maps such that ) and ) are finite for all [ We call the set of vertices and the set of arrows of For an arrow ' [ one calls ' the starting vertex and ' the end vertex of ' A path of length in is a sequence ' ' 4 ' of arrows such that J ' ' for AA 0 Set JZ ' and Z ' Additionally, for each vertex [ let be a path of length 0 By we denote the path algebra of, with basis the set of all paths in and product given by concatenation A #"$%('*) for is a linear combination -, / where [ and the are paths of length at least two in with J 0J21* and 0 21 for all A43 A Thus, we can regard a relation as an element in An ideal in is admissible if it is generated by a set of relations for Note that this differs from the usual definition of an admissible ideal, where one also assumes that the factor algebra 78 is finite-dimensional 22 A map 9 #1 such that ;:<9 R is finite is called a dimension vector for We also write 9 instead of 9 ), and we often use the notation 9 9 > > By we denote the semigroup of dimension vectors for Let?A@CB J be the category of finite-dimensional -graded vector spaces Thus, the objects of?a@cbnn are of the form ED >F where the F are finite-dimensional vector spaces, and only finitely many of the F are nonzero We call " #"E ḦG F > the dimension vector or degree of The morphisms in?@cbnn are just linear maps respecting the grading Direct sums in?i@cb J are defined in the obvious way A representation of with underlying vector space [?J@CB J is an element I ILKMK *N [ ( PO K *N R FTS K FHU K For a representation I I K K N [ ( and a path ' ' 4 ' in set I2V TILK ILK H ILKW Then I satisfies a relation X #, if X #, I8V / R If is a set of relations for, then let ( be the set of all representations I [ ( which satisfy all relations in This is a closed subvariety of ( Let Y be the algebra Z8, where is the admissible ideal generated by Note that every element in ( can be naturally interpreted as an Y -module structure on, so we shall also write Y T( This is the affine variety of Y -modules with underlying vector space A dimension> vector for Y is by definition the same as a dimension vector for, that is, an element of For I [ YG we call [I\]BIV " #" the dimension vector of I

7 [ S SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 7 23 Define $ > F This algebraic group acts on Y as follows For " D " *> [ $ and I I KMK *N [ YG define " I I K MK N where I K " U K ILK" K The $ -orbit of an Y -module I [ Y is denoted by IZ Two Y -modules I Y [ YG are isomorphic if and only if they lie in the same orbit For a dimension vector 9 for Y set > O > YG9J Y 9N $ Thus YG " #"4 Y for all J For this reason, we often do not distinguish between Y " #"4 and Y Any problems arising from this can be solved via the existence of an isomorphism between these two varieties which respects the group actions and the gradings 24 By abuse of notation, we identify [ with the dimension vector mapping to 1 and 3 to R If is an -graded vector space with " #"E, then the variety YG consists just of a single point and is denoted by The corresponding 1-dimensional Y -module is denoted by An element I [ Y is said to be nilpotent if there exists an [ such that for any path of length greater than we have I V R By YG we denote the closed subset of nilpotent elements in YG The nilpotent elements are exactly the Y -modules whose, [ composition series contains only factors isomorphic to the simple modules 2 An irreducible component [ '-((* YG is called indecomposable if it contains a dense subset of indecomposable Y -modules Let G ')((* Y be the set of indecomposable irreducible components of Y Let ')((*Y )$ '-((* YG9J be the set of all irreducible components of varieties of Y -modules, and set G V')(( Y/ $ G Z')((* YG9J 2 Let [ '-((* YG, A A be irreducible components of Y -modules, and set S U U Let U U be the set of modules in Y, which are isomorphic to I U U I with I for all The closure U U is irreducible, but in general it is not an irreducible component According to [14] any irreducible component [ ')((*Y/ has a decomposition U 3U for certain indecomposable irreducible components [ G ')((*Y/ Moreover the components are uniquely determined up to reordering This is called the canonical decomposition of For irreducible components and of Y -modules define ;LNM G ḦG \KLJM I I "NI I [ " This is the dimension of the extension group KLJM I I for all I I in a certain dense open subset of For irreducible components [ ')((*Y A A it is known that U U is an irreducible component if and only if ;LJM 1 7R for all 3, see [14]

8 K K 8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER The component graphy of Y is defined as follows The vertices ofy are the elements in G ')((*Y/ There is an edge between vertices and if and only if U is again an irreducible component, or equivalently if ;LJM ';LJM 7R 3 PREPROJECTIVE ALGEBRAS 31 Assume that S is a finite quiver without loops (A loop is an arrow ' with ' ', and is finite if is finite Note that this implies that is finite as well) The double quiver 3 of is obtained from by adding for each arrow ' [ an additional arrow '9 Define J ' / J ', ' / ', '9P ' and ' P J ' For any [ let *N S K, ' ' 0 *N U K, '*' be the Gelfand-Ponomarev relation associated to This is a relation for The preprojective algebra corresponding to is defined as where is generated by the relations Gelfand and Ponomarev, compare also [4] G7, [ These algebras were introduced and studied by 32 For a preprojective algebra 0\ and some [?J@CB J set Lusztig proved that 0W$ 0 $ G has pure dimension TG (, ie all irreducible components of 0$ have dimension ḦG ( Usually the varieties 0 $ are called nilpotent varieties If is a Dynkin quiver, then one might call them just preprojective varieties, since the nilpotency condition follows automatically in these cases, as shown by the next proposition Proposition 31 For a preprojective algebra G the following are equivalent: (a) is finite-dimensional; (b) G for all J ; (c) is a Dynkin quiver The equivalence of (a) and (c) is classical (see for instance [44]) The implication (c) proved by Lusztig [38, 142], and the converse by Crawley-Boevey [13] (b) is 33 The following remarkable property of preprojective algebras was proved in [12] Proposition 32 For finite-dimensional modules and over a preprojective algebra 0 we have ḦG KLJM O 2 TG \K2LNM O, Most preprojective algebras are of wild representation type The following proposition lists the exceptional cases We refer to [1] and [24] for further details Proposition 33 Let G be a preprojective algebra Then the following hold: (a) is of finite representation type if and only if is of Dynkin type 1 2 (b) is of tame representation type if and only if is of Dynkin type 1 or 78 ;

9 I I R R R I SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 9 4 CONSTRUCTIBLE FUNCTIONS 41 Let be an algebraic variety over, endowed with its Zariski topology A subset Y of is said to be constructible if it is a finite union of locally closed subsets It is easy to see that if is irreducible and if we have a partition Y Y into a finite number of constructible subsets, then there exists a unique Y containing a dense open subset of A function 1 is constructible if it is a finite -linear combination of characteristic functions for various constructible subsets Y Equivalently,2 is finite and is a constructible subset of for all [ The set of constructible functions on is denoted by S This is a -vector space 42 If is an irreducible component of and [ S, then is a finite partition into constructible subsets, hence there is a unique [ such that contains a dense open subset of In other words, a constructible function has to be constant on a dense open subset of each irreducible component of We denote by %J:,1 the linear form associating to its value on this dense open subset of 43 By Y we denote the Euler characteristic of a constructible subset Y If Y <, then WY < Y W < Hence can be regarded as a measure on the set of constructible subsets of For [, it is then natural to define 2IV2 W [ This is a linear form on, More generally, for a constructible subset Y of we write 2IV2 Y SEMICANONICAL BASES 1 In this section we assume that 9 is a finite quiver without loops, and as before for N let 0$ G be the corresponding nilpotent variety We denote by 0$/ the subspace of 0W$ consisting of the constructible functions which are constant on the orbits of /$ 2 For such that " " " "E " #", Lusztig [41] defines a bilinear map S 0 $ S 0 $ 01 0 $ as follows Let I [ 0W$ Define F to be the variety of all -graded subspaces of such that I and " " " " In other words, F is the variety of all 0 -submodules of I with dimension vector " " For such a let I [ 0 $ and I [ 0 be the elements induced by I, and let [ 0 $ and [ 0 $ be the elements obtained by transporting I and I via some isomorphisms 0 1 and 0 1 For [ 0 $ and [ 0 $ define F 1 by ; I Following [41] define ;IZ " ;

10 F, 10 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER F 0 $ ) be a $ -orbit (resp a $ -orbit) For I [ 0W$ let consisting of all 0 -submodules Y of I of isomor- and such that I Y has isomorphism type The above definition yields ;IZ IZ Note that in general the variety 0 $ has infinitely many orbits (Indeed, by Proposition 33 the 3 Let 0 $ (resp IZ be the constructible subset of F phism type algebra 0 has in general infinite representation type, hence, by the validity of the second Brauer- Thrall conjecture (see [2]), it has in general an infinite number of non-isomorphic representations of a given dimension) Therefore the support of the function may consist of an infinite number of orbits 4 Let - D $ S 0 $ where runs over the set of all > isomorphism classes of vector spaces N (For example, we can take [ P " 9 [ > " ) The operation defines the structure of an -graded associative -algebra on - For [, we recall that denotes the variety 0 $ where " #" ( is just a single point) Following [41] define - to be the subalgebra of - generated by the functions :, [ We set - $ - 0$/ For two distinct vertices 43 [, let 1 denote the negative of the number of arrows ' [ such that J ' ' " 43 " Set also [J Let be the symmetric Kac-Moody Lie algebra over with Cartan matrix 1 1 > Let be a maximal nilpotent subalgebra of, and let be its enveloping algebra We denote by [N the Chevalley generators of The defining relations of are 0 * 1/ where 0 > The algebra is -graded by assigning to the degree It is known that the number of irreducible components of Lusztig s nilpotent variety 0 $ is equal to the dimension of the homogeneous component of of degree " #" This was proved by Lusztig [38, 39] when is of finite or affine type and by Kashiwara and Saito [30] in general Lusztig has proved that there is an algebra isomorphism given by ( : 1-7RN To do this he constructed for every -graded vector space a -basis :#" [ ')((* 0$/ " of - $, naturally labelled by the irreducible components of 0 $ Using the notation of Section 42, it is characterized by (4) % : : 2 : : [ ')((* 0$ In other words, the function : is the unique element of - $ equal to on a dense open subset of and equal to R on a dense open subset of any other irreducible component of 0$, see [41, Lemma 2] The basis of obtained by transporting via : " [ ')((* 0 $ "E $ where the collection ranges over the set of all isomorphism classes of vector spaces J, is called the semicanonical basis of and is denoted by

11 SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 11 Example 1 Let be the quiver with two vertices and and one arrow ' 1 Thus is a Dynkin quiver of type 134 (a) Let F U F 4 with ḦG F ḦG F 4 Then 0$ PI 8 [ " N8 7R " The variety 0W$ has two irreducible components R " [ "E RN8; "8 [ " The group $ 7 7 acts on 0W$ with three orbits RNR "E 0 RNR "E 0 RNR " We have : :, : : 4 : : : 4 : (b) Let F U F 4 with ḦG F ḦG F 4* Then ( 4 4 and 0$ F ( has dimension The variety 0 $ has three irreducible components PI [ 0$D" ( ILKWA ( ILKP A "E PI " I K7R "E PI?" ILK 7R " We have: : : : : 4 : 4 : : : : : : 4 : 4 Note that : : and : [ - 2 : : : 4 : : : 4 : 4 : 4 : : 7 Next, we consider composition series of modules over preprojective algebras Let denote the set of pairs (\- where \Z is a sequence of elements of and [ *RN " The integer is called the length of (\- Given (\- [ such that X " #", we define a flag in of type (\- as a sequence T 7R of graded subspaces of for Thus ḦG such that " " is equal to R or So these are complete flags, with possible repetition of some subspaces (It will be convenient below to allow such flags with repeated subspaces) We denote by the variety of flags of type (\- When (flags without repetition), we simply write * Let I [ 0$ A flag is I -stable if I for all We denote by (resp ) the variety of I -stable flags of type (\- (resp of type \ ) Note that an I -stable flag is the same as a composition series of I regarded as a 0 -module 8 For (\- [ with X " #", define 9 : : [ - $ If for all, we simply write 9 instead of 9 In general, 9 9 where is the subword of \ consisting of the letters for which By definition, the functions 9, where runs over all words, span - The following lemma results immediately from the definition of the product of constructible functions Lemma 1 Let I [ 0W$ We have 9 IZ09*IZ W "

12 N R R R R $ 4 N N R R 12 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Example 2 Retain the notation of Example 1 (b) Let I 8 [ 0$ be given by the following matrices (with respect to some fixed bases of F and F 4 ) [ R R F F 4 8 [ R R F 4 F Let us calculate IV To construct a flag \ \ \ R [ we first have to choose a line in the 2-dimensional vector space F IV F We may take to be (a) the 1-dimensional image of I, or (b) any line except this one In case (a) the module I get In case (b), I R 8 [ induced by I in the quotient where 4 R F F 4* 8 R [ is also com- and at the next stage must be taken as the kernel of I pletely determined Thus, in case (b) we get So finally, > C is the semisimple module So we R F 4 F (no choice), and 9 In this section we assume that is a simple finite-dimensional Lie algebra Equivalently, associated to this quiver The image is a Dynkin quiver Then has a PBW-basis < < is easy to describe Let [?@CBJ The affine space ( can be regarded as a subset of 0W$ by identifying it to the set of I [ 0 $ with I KT R for every ' [ 0 Clearly this is an irreducible component of 0$ Moreover by our assumption it has finitely many $ -orbits Lemma 2 Let be a /$ -orbit in ( There exists a unique [ - $ whose restriction to ( is the characteristic function of The collection of all where runs through all $ -orbits in ( is equal to < - $ Proof By [38, 1019, 1212], the map from - $ to ( sending to its restriction to ( is a vector space isomorphism Moreover, the space N O ( (where ranges over all isoclasses of vector spaces in?i@cbnn ) endowed with the image of the product is the geometric realization of the Hall algebra of over due to Schofield (see [38, 1019]) In this setting, the PBW-basis is the basis of N given by the characteristic functions of the $ -orbits in ( Thus the lemma is proved COMULTIPLICATION The algebra is in fact a bialgebra, the comultiplication being defined 1@ [ In this section we describe the comultiplication of - obtained by transporting this comultiplication via

13 I U I, $ U I $, H U I H U I U I SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 13 1 For J such that " " " " " #", define a linear map $ $ 0$/ 01 0 $ 0 $ as follows We have 01 U, so given I [ 0 $ and I [ 0 $ we can regard I as an element of 0W$ Here, I denotes the direct sum of I and I as endomorphisms of and, or equivalently as modules over the preprojective algebra 0 For [ $, I [ 0 $ and [ 0 $ set $ $ I I 2I This is clearly a constructible function on 0 $ 0 $ which is constant on $ $ -orbits 2 Let \ with X " #" Let U, I [ 0 $, I [ 0 $ and I TI [ 0$ Lemma 1 We have where the sum is over all pairs () / " Proof By definition we have $ $ $ 9 I I 9 I 29 I of sequences in *RN " such that " / " " R A A $ $ $ 9 I I 09 IV To [, we associate the pair of flags We regard as a flag in, and as a flag in by identifying with Clearly, we have [ and [ for some sequences, in *RN " satisfying the conditions () Let denote the set of pairs satisfying () For [, let be the subset of consisting of those for which [ Then clearly we have a finite partition where 4 consists of the pairs such that is nonempty Now for [ 4, the map ' 0 1 is a vector bundle, see [38, Lemma 44] Hence, and On the other hand, $ $ $ 9 I I 2 W W sending to 9 I M9 I 2 W W and it only remains to prove that 4 Clearly 4, so let [ Let [ and [ The assumption I 7I implies that is nonempty Indeed, the flag in whose th subspace is the direct sum of the th subspaces of and is I -stable and by construction [ So [ 4 and the lemma is proved

14 U I I I U $ U I U I I U I 14 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 3 By Lemma 1, the map by where the pairs $ $ induces a linear map from - $ to - $ - $, given satisfy () Taking direct sums, we obtain a linear map - $ 0 1 O - $ - $ $ $ and of -graded vector spaces such that " " " #" Taking direct sums over all isomorphism types, we get a linear map U I, is coassociative Since I the sum being over all isomorphism types " Since I cocommutative () Lemma 1 shows that is multiplicative on the elements 9, that is, for \Z where the product in - - is defined by 4P " " 4P " 4 "4* Proposition 2 Under the algebra isomorphism with the standard comultiplication of ", is - 0 1, the map gets identified Proof Equation () shows that is an algebra homomorphism Moreover the generators 9 : ( are clearly primitive The result follows 7 MULTIPLICATIVE PROPERTIES OF THE DUAL SEMICANONICAL BASIS > 71 The vector space - is -graded, with finite-dimensional homogeneous components Let - denote its graded dual Given an -graded space and an irreducible component of 0>$, we have defined a linear form %: on S 0$, see Section 42 We shall also denote by %: the element of - obtained by restricting % : to - $ and then extending by R on all - $ with " " " #" Note that by (4), the basis of - dual to the semicanonical basis : " is nothing but %J: " From now on %N: " will be called the dual semicanonical basis of - Lemma 71 For [ ')((* 0W$/ there exists an open dense /$ -stable subset : F such that for all [ - $ and all I [ : we have %N: 2IV Proof For a given, this follows from Section 42 Moreover, there exists such an open set simultaneously for all because - $ is finite-dimensional 72 For I [ 0W$ define the delta-function [ - by 22IZ [ - We then have (7) '%N: I [ : The next Lemma follows immediately Lemma 72 Let [ '-((* 0W$/ and suppose that the orbit of I [ Let denote the multiplication of - dual to the comultiplication of - Lemma 73 Let I [ 0$ and I 4 [ 0$H We have H H X Proof For [ -, one has H V 2 H V ;I I 4*22I U I 4P is open dense Then % : X H

15 Lemma 74 Suppose that SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 1 U 4 is an irreducible component of 0 $ Then there exists I [ : of the form I TI U I 4 with I [ >: and I 4 [ >:JH Proof The direct sum U 4 is the image of the morphism $ 4/01 defined by "I I4P" I U I 4P Since : (resp :JH ) is open dense in (resp in 4 ), the subset $ : :JH is open dense in $ 4 Now, since is a dominant morphism between irreducible varieties, the image under of /$ : : H contains a dense open subset of, hence it has a nonempty intersection with : Since both $ : >:JH and : are $ -stable we can find I in their intersection of the form I TI U I 4 with I [ : and I4 [ :JH 73 We can now give the proof of Theorem 11 yield Proof of Theorem 11 Choose I I I4 as in Lemma 74 Then Lemma 71 and Lemma 73 U U be the canonical decomposition of the irreducible com- of 0 $ The dual semicanonical basis vector % : factorizes as % : '% : % : Corollary 7 Let ponent %J: %N: H H X H '%N: Proof For this follows from Theorem 11 Assume that By [14] U 3U is an irreducible component Moreover U so by Theorem 11 we get % : '% : %N: The result follows by induction on The factorization given by Corollary 7 will be called the canonical factorization of %V: 74 We shall now deduce from Theorem 11 the proof of the only if part of Theorem 12 Theorem 7 Let be of type 1 do not coincide 72@ or Then, the bases : and Proof Assume first that is of type 1W or 8 Then the preprojective algebra 0 is of tame representation type In this case, we have ;LNM O R for any irreducible component of 0$, see [24, 2] Therefore by Theorem 11 and [14] the square of any vector of belongs to On the other hand, it was shown in [31] that for the cases 1 and 8 there exist elements of : whose square is not in : They are called imaginary vectors of : This shows that : and are different in these cases Now if is not of type 1 2 type 1 or 8, and the result follows from the cases 1W and 8, then the Dynkin diagram of contains a subdiagram of In the next sections we shall prepare some material for the proof of the if part of Theorem 12, to be given in Section 13 8 EMBEDDING OF -/ INTO THE SHUFFLE ALGEBRA We describe a natural embedding of - a certain family of elements of in type 12 into the shuffle algebra This is then used to describe

16 S 1 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 81 Let be the free associative algebra over generated by A monomial in is called a word This is nothing else than a sequence \V in Let be the surjective algebra homomorphism given by ) :, and more generally by (\ 9 Let denote the graded dual of We thus get an embedding of vector spaces - 01 Let \ " denote the basis of dual to the basis \ " of words in Let [ - ;(\ \ By Lemma 1, we obtain in particular (8) (\ \ W \ 9 \ We have 82 Denote by the multiplication on - obtained by pushing with, that is, for Z [ - set Lemma 81 The product is the restriction to - P of the classical shuffle product on defined by where the sum runs over the permutations [ 2 S 2 such that S J * and * Proof This follows easily from Lemma 1 and the duality of and Indeed, ;9 \ ; 9 \ 9 9 \ where the pairs are as in Lemma 1 Now it is clear that the coefficient of \ in this last sum is the same as the coefficient of \ in the shuffle product and the lemma is proved Suppose that is of type 132 Then -, where is the group of * -matrices 2

17 K 1 SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS Let us construct an explicit isomorphism ' E 1 - Let 1 denote the coordinate function assigning [ its 1 Then " A 3 A\@ It is known that in the isomorphism, the natural basis of consisting of monomials in the 1 gets identified to the dual of the PBW-basis of associated to the quiver KEH K 2 " the dual in - of the PBW-basis " in -, 1 to the element (see for example [33, 3]) The $ -orbits of ( 2 are naturally labelled by the multisegments of degree " #", and if we denote by then more precisely the above isomorphism maps the monomial indexed by the multisegment For A 3, let I 43 denote an indecomposable representation of 2 with socle and top H1 (up to isomorphism there is exactly one such representation) Then the orbit of I 43 is open dense so 1 belongs to - by Lemma 72 On the other hand, by Lemma 2 1 if 43) 2 Hence 1 1 otherwise and ' is the algebra homomorphism determined by ' If we regard the functions 1 as entries of a * matrix we may consider some special elements of 4 given by the minors of this matrix Proposition 82 The images under ' of all nonzero minors of the matrix belong to Proof We shall use the embedding First note that since I 43 has a unique composition series, ' Let 1 be the -minor taken on the sequence of rows and the sequence of columns 3 #3 3" Since is a unitriangular matrix with algebraically independent entries 1 above the diagonal, the function 1 is nonzero if and only if A 3 for every We shall assume that this condition is satisfied Let #3" 43" 43 0 * 0 * Then is a skew Young diagram We identify it with the following subset of 8 " AB8 A A " Each pair 8 is called a cell of Let Y be a standard Young tableau of shape, that is, a total ordering of the cells of which is increasing both on the rows and on the columns We associate to Y the element Y V 0 8 0?8 of, where 8 A A Before we continue with the proof of Proposition 82 we need the following lemma Lemma 83 ' Y where Y runs over the set of standard Young tableaux of shape 1 1

18 S R S 18 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Proof Set We shall prove that 1 'W 1 and Y 1 $ by induction on the number of cells of If, then 1 Z for some, and the statement is clear So suppose define [ K R by if 2 It is immediate to check that otherwise is a derivation with respect to the shuffle product, ie " "N " [ "N It is also clear that " if and only if rows and columns 3 of the matrix W " for every Note that 0 0 R R R 1 is the minor on where in the expansion of the determinant the shuffle product of the entries is used It follows that, if 3 [ 3 and 3 [ 3 then 1 and otherwise R On the other hand, 1 1E where is obtained from 3 by replacing 3 by 3, where ranges over the Young tableaux whose shape is a skew Young diagram L obtained from by removing an outer cell 8 with 0B8 3 It is easy to check that there is one such diagram only if 3 above So, by induction in this case, and 1 proof of Lemma 83 [ 3 and 3 [ 3, and that this diagram then corresponds to the pair 1E 1E 1 R 1 1, otherwise Therefore " 0 We continue with the proof of Proposition 82 Let 1 This finishes the be the multisegment corresponding to the pair 43 (Here we leave out 43 0 in case 3 Following [47] this parametrizes a laminated 0 -module I, that is, a direct sum of indecomposable subquotients of projective 0 -modules Let be the underlying -graded vector space of I It is known that the $ -orbit of I is open dense in its irreducible component, hence the function belongs to the dual semicanonical basis Now it is easy to see that the types \ of composition series of I are in one-to-one correspondence with the standard Young tableaux of shape, and that for each tableau, the corresponding flag variety is reduced to a point Therefore, comparing with Lemma 83 we see that 2 'W 1 Hence ' 1 belongs to the dual semicanonical basis This finishes the proof of Proposition 82 )

19 ' 2 2 K K ' 1 H SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 19 9 A GALOIS COVERING OF 0 FOR TYPE 12 In order to prove the if part of Theorem 12 we need to study the canonical decomposition of [ ')((* 0 for type 172@ A Our main tool for this will be the Auslander-Reiten quiver of 0, which we will calculate by using a Galois covering 0 of 0 This covering is in fact important for and it will also play an essential rôle in our investigation of type 1 in the last sections of the paper So we shall work in type 132 for in the next two sections, and we shall specify which results are only valid We will also exclude the trivial case 1 and assume 91 let again 2 be the quiver K H of Dynkin type 12 Let 032 2N be the preprojective algebra corresponding to 2 Thus where the double quiver 2 of 2 is K K KEH and the ideal 2 is generated by ' ' ' '2 ' 0#' ' AWA\@ 0 E 92 Next, let where 2 is the quiver with vertices 1 " A AT@243 [ " and arrows 1 *1 1 1 ' 1 1>1 *1 AWA\@ 0 43 [ and the ideal 2 is generated by ' 1 ' 1 ' 1 ' 2 1 ' 1 ' 10#' 1 ' 1 A A\@ 0 J43 [ we illustrate these definitions in Figure 1 Denote by 2 the full (and convex) subquiver of 2 which has as vertices the set 1 [ 2 " C A C3 AB8 "E and denote by 2 the restriction of 0 2 to 2, see 910 for an example 93 The group acts on 032 by -linear automorphisms via 1 1 ' 1 ' 1 ' 1 ' 1 This induces an action 032N where denotes the 032 -module obtained from by twisting the action with Roughly speaking, is the same 0 2 -module as, but translated one level upwards If we consider 032 and 032 as locally bounded categories we have a functor 032B defined by 1 1 ' 1 ' ' 1 ' This is a Galois covering of 0 2 in the sense of [22, % 31], with Galois group It provides us with the push-down functor [22, % 32] 032N 0 1 K K 032N [ 0 2 be a 0 2 -module with which we also denote by It is defined as follows Let I underlying graded vector space D F Then IZ has the same underlying vector space 1

20 O O K K H H 4 K K 9 4 H K H 8 K K K 8 H 20 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER K H K KEH H K 4H K H H K H K K K H K 8 K K K K KEH K 4 K 8 # H K # # K K H KEH K K K FIGURE 1 The Galois covering with the grading D F where F D 1 F, and IV has maps IV K IZ K D 1 I K The next lemma follows from [22, % 3] and from [1] % 3] Lemma 91 Let A\@?A D 1 I K and, as noted in [1, Any finite-dimensional indecomposable 0 2 -module is isomorphic to IZ for some indecomposable 032 -module I, which is unique up to a translation I 1 I by the Galois group 94 For a dimension vector [, We have a morphism of varieties 9N 032[2 1 9 for 0 2 define where N where the push-down morphism is defined by " D" D For [ let [ be the th shift of [, that is, 9 AWA\@243 [ Thus, if [ is the dimension vector of a 032 -module, then is the dimension vector of the shifted In this case we write [ This defines an equivalence relation To simplify module our notation, we shall not always distinguish between [ and its equivalence class However, it will always be clear from the context which one is meant To display a dimension vector (or its equivalence class) for 0 2 we just write down the relevant entries, all other entries are assumed to be 0

21 2 2 4 SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS 21 9 Recall that to any finite-dimensional algebra Y over a field (and more generally to any locally bounded -algebra or locally bounded category, see eg [48, p 4]) is associated a translation quiver called the Auslander-Reiten quiver of Y It contains a lot of information about is representation-finite and standard, one can recover the category Y In particular if Y Y/ from The vertices of consist of the isomorphism classes of indecomposable finite-dimensional Y -modules If and are two such modules, then there are ḦG, arrows from to in, where denotes the radical of the category Y (compare [4, Chapter 2]) This means that there is an arrow from to if there exists a nonzero irreducible homomorphism from to Here, by abuse of notation, we do not distinguish between a module and its isomorphism class The quiver is endowed with an injective map, the translation, defined on the subset of vertices corresponding to non-projective modules If is indecomposable and non-projective then where R R is the Auslander-Reiten sequence (or almost split sequence) ending in The stable Auslander-Reiten quiver of Y is obtained from by removing all translates [ of the projective vertices and the injective vertices as well as the arrows involving these vertices Thus the translation induces a permutation on the vertices of We refer the reader to [1, Chapter VII] for more details on Auslander-Reiten theory, or to [21] and [4, Chapter 2] for algebraically closed as we assume here Note however that these papers use slightly different conventions 9 In Section 201 we display the Auslander-Reiten quivers of 0 4, 0 and 038 To calculate them one first determines the Auslander-Reiten quivers of their coverings 0 4, 03 and 038 Indeed the algebras 032 are directed, that is, there is no sequence of indecomposable 032 -modules of the form with all homomorphisms being nonzero and non-invertible It follows that the Auslander-Reiten quiver of 032 can be calculated by a combinatorial procedure known as the knitting procedure, see for example [21, % ] By applying the push-down functor to this quiver, one then obtains the Auslander-Reiten quiver of 0 2 [22, % 3] In our pictures, each indecomposable 0 2 -module is represented by the dimension vector [ of a 032 -module such that > One has to identify each vertex in the extreme left column with the vertex in the extreme right column represented by the same dimension vector up to a shift by the Galois group The Auslander-Reiten quiver of 0 is shaped like a Moebius band, and for 0 4 and 038 one gets a cylinder In particular, we see that 0 4 has 4 isoclasses of indecomposable modules, 0 has 12, and 038 has 40 We should point out that module via the push-down functor there are 0 2 -modules which are not obtained from a For an algebra Y let Y be the stable category of finite-dimensional Y -modules ([48, p ]) By definition, the objects of Y/ are the same as the objects of Y, and the morphism space is defined as modulo the morphisms factoring through projective modules The isomorphism classes of indecomposable objects in Y/ correspond naturally to the isomorphism classes of non-projective indecomposable Y -modules The stable category Y is no longer abelian, but if Y is a Frobenius algebra then Y has the structure of a triangulated category ([2, % I2]) with translation functor, the inverse of Heller s loop functor