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1 This article was downloaded by:[unicamp Universidade est Campinas] On: 17 June 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: On Moduli Spaces for Abelian Categories Vyacheslav Futorny a ; Marcos Jardim b ; Adriano A. Moura b a IME - USP, Department of Mathematics, S o Paulo-SP, Brazil b IMECC - UNICAMP, Department of Mathematics, Campinas-SP, Brazil Online Publication Date: 01 June 2008 To cite this Article: Futorny, Vyacheslav, Jardim, Marcos and Moura, Adriano A. (2008) 'On Moduli Spaces for Abelian Categories', Communications in Algebra, 36:6, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Algebra, 36: , 2008 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / Key Words: ON MODULI SPACES FOR ABELIAN CATEGORIES Vyacheslav Futorny 1, Marcos Jardim 2, and Adriano A. Moura 2 1 IME USP, Department of Mathematics, São Paulo-SP, Brazil 2 IMECC UNICAMP, Department of Mathematics, Campinas-SP, Brazil We show that if is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories. Abelian categories; Moduli spaces; Quivers; Stability Mathematics Subject Classification: Primary 14D20; Secondary 16G20, 17B INTRODUCTION Stability first arose as a geometric notion, within the context of Mumford s Geometric Invariant Theory. Roughly speaking, if G is an algebraic group acting on an algebraic variety X, a point x X is said to be stable with respect to the action of G if its orbit is closed. In the 1960s, Mumford and others used this geometric notion to construct moduli spaces of algebraic vector bundles over nonsingular algebraic curves, translating it into a notion of stability for vector bundles, the so-called Mumford Takemoto stability. This very successful theory has been greatly expanded in the past 40 years by several authors and today the construction of moduli spaces of sheaves over algebraic varieties is well understood alongside several different variations. It is also important to mention that moduli spaces of stable vector bundles have also been extremely useful in areas other than algebraic geometry, particularly in mathematical physics. King (1994) translated the geometric notion of stability into a concept of stability for finite dimensional modules over finite dimensional associative algebras and constructed moduli spaces of representations. This is especially useful in the study of wild algebras, since the task of describing the structure of indecomposable representations of such algebras is, from the purely algebraic point of view, hopeless. The best one can try to achieve is to describe the geometry of the moduli spaces of (semi)stable representations for a fixed dimension vector. A great deal was Received February 13, 2007; Revised April 26, Communicated by I. P. Shestakov. Address correspondence to Adriano Moura, Department of Mathematics, Institute of Mathematics, Statistics, and Computer Science IMECC UNICAMP, São Paulo , Brazil; Fax: ; aamoura@ime.unicamp.br 2171

3 2172 FUTORNY ET AL. accomplished by King (1994) and Schofield (2001) for the case of hereditary algebras. Rudakov (1997) introduced a purely categorical notion of stability for an object of an abelian category and observed that the Mumford Takemoto stability for algebraic vector bundles over curves and King s stability for modules were examples of his categorical notion. It was unclear however how Rudakov s categorical stability is related to geometric stability in other abelian categories. The goal of the present article is to apply King s and Rudakov s results to the realm of representation theory in the hope that these techniques will prove to be as useful as they were in other fields. In Section 2, we review Rudakov s categorical stability notion and use it to construct a special type of stability structure on a wide range of abelian categories. The central result (Theorem 2.11) states that if is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. In Section 3, we explore several examples of relevant abelian categories from representation theory to which we can apply Theorem The examples include highest weight categories with finite poset of simple objects (blocks of the BGG category for instance), the category of Harish Chandra bimodules, and the category of bimodules of a finite-dimensional Jordan algebra. We also give an example of a limit construction which enables us to define a (scheme theoretic) moduli space structure on a highest weight category whose underlying poset is infinite (Example 3.5). However, none of the examples above correspond to hereditary algebras, but rather, they correspond to quasi-hereditary or stratified algebras. This fact provides strong motivation to the study of the geometric structure of the moduli spaces of (semi)stable representations for such algebras. Some interesting examples of highest weight categories with finite poset coming from finite-dimensional representation theory of current algebras have just appeared in Chari and Greenstein (2007). We also mention article Grantcharov and Serganova (2006) where it was shown that the category of bounded modules over the symplectic Lie algebra sp 2n is equivalent to the category of weight modules for the nth Weyl algebra, whose blocks are just module categories of some quivers with relations (Bekkert et al., 2004). 2. CATEGORICAL STABILITY THEORY Let be an abelian category and denote by K 0 its Grothendieck group. We will write V meaning V Ob. The isomorphism class of V, as well as its image in K 0, will be denoted by V. Given K 0, set = V V =, and let be the additive closure of the full subcategory of whose objects consists of all of the subquotients of elements in. Clearly, is a full abelian subcategory of. Observe that if B is another abelian category and F B is an exact functor, then F induces an abelian group homomorphism K 0 B K 0 also denoted by F.

4 MODULI SPACES FOR ABELIAN CATEGORIES Definition and Basic Properties The following definition was first proposed by Rudakov (see Rudakov, 1997 and the more recent article Gorodentscev et al., 2004), probably inspired by King (1994, Definition 1.1). Definition 2.1. A stability structure on consists of a preorder on the set of objects of satisfying the following properties: (i) Trichotomy: for any two nonzero objects A and B, either A B, orb A, or A B; (ii) Seesaw property: for each short exact sequence 0 A B C 0 of nonzero objects we have that: where: either or or (a) A B if A B and B A; (b) A B if A B but not A B; (c) A B if B A. A B B C A C A B B C A C A B B C A C Bridgeland has introduced the notion of stability on triangulated categories (Bridgeland, 2007). Stability on abelian categories is also discussed in Joyce (2007, Definition 4.1). Definition 2.2. A nonzero object B is said to be stable if every nontrivial subobject A B satisfies A B. Equivalently, B is stable if every nontrivial quotient-object B C 0 satisfies B C. A nonzero object B is said to be semistable if every nontrivial subobject A B satisfies A B. Equivalently, B is semistable if every nontrivial quotient-object B C 0 satisfies B C. It is not difficult to see that for any stability structure every simple object is stable, and every stable object is indecomposable. The converse is not true in general. Semistable objects may be either indecomposable or decomposable; a decomposable object A = j A j is semistable iff each A i is semistable and A i A j for all i j. We also have Schur s Lemma for stable objects: if A is a stable object and Hom A A is a finite dimensional vector space over an algebraically closed field, then Hom A A = (Rudakov, 1997, Theorem 1). Other general properties of stability structures and of (semi)stable objects can be found in Rudakov s original in Rudakov (1997). Every abelian category can be given a trivial stability structure, in which any two objects satisfy A B. For such structure, an object is stable if and only if it is simple, while every nonsimple object is semistable. It is unclear however whether every abelian category can be provided with a nontrivial stability structure.

5 2174 FUTORNY ET AL. Below, we show how nontrivial stability structures can be constructed on a large class of abelian categories, see Example 2.4 below Constructing Stability Structures Let R be a totally ordered -vector space satisfying a a > 0 r R r > 0 ar > 0 and r < 0 (2.1) Let also d K 0 be an additive function satisfying d A > 0if A 0 (A Obj ), and c K 0 R be a group homomorphism. We call the ratio A = c A d A the c d -slope of the object A. Given any two nonzero objects A and B, set A B A B It follows that A B A = B, and A B A < B. Note that: ( ) 1 d B c B A B = d A d B det d A c A where ( ) d B c B det = d B c A d A c B d A c A Thus A B if and only if the determinant above is smaller than 0, and A B if and only if the determinant above is the zero vector. We have (cf. Rudakov, 1997, Lemma 3.2) the following proposition. Proposition 2.3. on. The pre order defined above gives rise to a stability structure Proof. The first axiom of Definition 2.1 is easy to verify. To check the seesaw property, consider the exact sequence 0 A B C 0, so that B = A + C, and, hence, c B = c A + c C and d B = d A + d C. It follows that: ( ) ( ) d B c B d A + d C c A + c C det = det d A c A d A c A ( ) ( ) d C c C d C c C = det = det d A c A d A + d C c A + c C ( ) d C c C = det d B c B

6 MODULI SPACES FOR ABELIAN CATEGORIES 2175 Therefore, A = B A = C B = C and A < B A < C B < C, which by definition imply the seesaw property. Example 2.4 (Jordan Hölder Categories). Let be an abelian category all of whose objects are of finite length, be the set of isomorphism classes of simple objects, and A be the multiplicity of in the object A. Fix an order on for which there exists a minimal element and consider the vector space R with basis equipped with the lexicographic order, which obviously satisfy (2.1). Given functions g with g > 0 for all, and f, set c A = f A and d A = g A Hence, every Jordan Hölder category can be equipped with a stability structure. Example 2.5 (Gieseker Stability for Torsion-Free Sheaves). Fix an n-dimensional projective variety X over, and let TF X be the quasi-abelian category of torsion-free coherent sheaves on X. Given a coherent sheaf E on X, its Hilbert polynomial is defined as p E t = E t = n 1 p dim H p X E t p=1 where as usual E t = E X t. This is a polynomial on t with rational coefficients of degree at most n, so-called Hilbert polynomial of E. It defines an additive function p K 0 TF X P n, where P n denotes the -vector space of polynomials of degree at most n = dim X. IfE is torsion-free, then p E t has degree exactly n whenever E is the nonzero sheaf; moreover, p E t = 0 if and only if E is the zero sheaf. We therefore can define an additive function r K 0 TF X, with r E given the leading coefficient of p E t, which is always positive for every nonzero sheaf. Providing P n with the lexicographic order of the coefficients, we get that the p r -slope slope function yields a stability condition on TF X, which is known as the Gieseker stability Harder Narasimhan Filtrations and Moduli Sets One of the main results for stability structures on abelian categories is the existence of a Harder Narasimhan filtration for any object of. Theorem 2.6 (Rudakov, 1997, Theorems 2 and 3). Let is a noetherian abelian category provided with a stability condition. IfA is a semistable object, then there exists a unique filtration A = F n+1 A F n A F 1 A F 0 A = 0 so-called Harder Narasimhan filtration, such that: (i) The factors Q k = F k 1 A/F k A are stable; (ii) Q 1 Q 2 Q n.

7 2176 FUTORNY ET AL. Definition 2.7. Two semistable objects in are said to be S-equivalent if their Harder Narasimhan filtrations have the same composition factors. It is easy to see that S-equivalence is indeed an equivalence relation, and that two stable objects are S-equivalent if and only if they are isomorphic. Definition 2.8. Let s be the collection of all semistable objects within. The moduli set of semistable objects of represented by the class up to S-equivalence is given by s = s /, where denotes S-equivalence. If the stability structure comes from a slope function as above, we shall write and instead of s and s. Under some circumstances, the moduli set s has the structure of an algebraic variety. To see this, we will need the concept of pullback for a stability structure. Let be an abelian category provided with a stability structure. Let B be an abelian category, and consider an exact functor F B. We can then pullback the stability structure in to one in B in the obvious way: given U W B we declare U W if and only if F U F W. The following proposition is easily verified. Proposition 2.9. Suppose B, and F are as above, K 0 B, and suppose that F B B F is an equivalence of categories. Then F induces a bijection F B s s F The Main Theorem We begin recalling the rephrasing of the notion of slope stability in terms of character stability. Given a totally ordered -vector space R satisfying (2.1), a c d -slope K 0 R, and K 0, let K 0 R be defined by V = c V + d V (2.2) Observe that if V, then V = 0 and V is stable (resp. semistable) iff U > 0 (resp. U 0) for all subobject U of V. Conversely, given abelian group homomorphisms K 0 R and d K 0, r R, and K 0 such that = 0 and d 0, set c V = V + d V d r and let be the corresponding slope. Then we see that, if V, then V is stable (resp. semistable) iff U > 0 (resp. U 0) for all subobject U of V. Hence, if we restrict ourselves to, for some K 0, slope stability can be defined by the choice of a character, i.e., an abelian group homomorphism K 0 R satisfying = 0. From now on we will assume that R is a totally ordered -vector space satisfying condition (2.1) and also that R is equipped with a norm such that r < r >0 such that r < r whenever r r (2.3)

8 MODULI SPACES FOR ABELIAN CATEGORIES 2177 We say that an order satisfying conditions (2.1) and (2.3) is continuous or that R is a continuously ordered -vector space. Lemma Let R be a continuously ordered -vector space, be an abelian category, and K 0 be such that is a Jordan Hölder category. Then for any c d -slope function K 0 R, there exists a character K 0 such that: 1. = 0 and V for every simple object V ; 2. An object V is -semistable (resp. -stable) iff it is -semistable (resp. -stable). Proof. We can assume, without loss of generality, that =. Let be the character defined by (2.2). Since K 0 is a finite-dimensional subspace of R, we can also suppose that R = k equipped with the usual inner product. It suffices to obtain such that V for all simple V. Let I = 1 n be the index set of isomorphism classes of simple objects, i i I, be the corresponding images in K 0, and define i K 0 R by setting i j = ij i, so that = i i. Choose a basis 1 m 1 n for the -vector space generated by 1 n. Thus, we have i = j a ij j for some a ij. For each choice of 1 m k R, define i = j a ij j and extend by linearity. The linear independence of j immediately implies = 0. Since depends continuously on the choice of j and there are only finitely many K 0 which can be the class of a subobject of objects in, it follows that we can choose j so that an object V is -semistable (resp. -stable) iff it is -semistable (resp. -stable). Finally, for each choice of k, set =. As before we can choose so that we have 2. Now let B be a finite dimensional associative algebra over an algebraically closed field of characteristic zero, and let B = mod-b, the category of finite dimensional (left) B-modules. Given a class K 0 B and a group homomorphism K 0 B with = 0, it was shown in King (1994) that the categorical moduli set B (i.e., the set of -semistable objects up to S-equivalence) has the structure of a projective variety over. In light of this key example, we can conclude the following. Theorem Let be an abelian category, R be a continuously ordered -vector space, B = mod-b for some finite-dimensional algebra B, and K 0. Suppose is such that there exists an exact functor F B such that = F for some K 0 B and F B B is an equivalence of categories. Then, for any slope function K 0 R, the moduli set can be given a structure of projective -variety. Proof. By Lemma 2.10 we can find a character K 0 preserving (semi)stable objects in. Since F K 0 B K 0 is an isomorphism, we can regard as a character on B. From King (1994, Proposition 4.3) we know that B s, where the stability structure on B is the one pulled-back from, is a projective variety. Hence, by Proposition 2.9, this structure can be transported to.

9 2178 FUTORNY ET AL. For instance, if is a noetherian abelian category with finitely many nonisomorphic simple objects V 1 V n for which there are projective covers P i V i 0, then B = End P i is a finite dimensional associative algebra and is equivalent to mod-b. In this context, Theorem 2.11 applies, and we conclude that has the structure of a projective variety. 3. APPLICATIONS If B in Theorem 2.11 is a hereditary algebra, then more can be said about : it is irreducible, normal, and has dimension equal to dim Ext1 V V, for V being a generic object represented by ; the subset of isomorphism classes of stable objects is open (in the Zariski topology) and nonsingular King (1994). Furthermore, the birational type of these varieties is studied in Schofield (2001). However, many natural categories that arise in representation theory are equivalent to module categories of finite-dimensional algebras which are not hereditary, but say quasi-hereditary or standardly stratified. In this subsection we present some examples of such categories. It would be interesting to work out more detailed algebraic geometric properties of for them Highest Weight Categories and Quasi-Hereditary Algebras Let be an -linear category. Following Cline et al. (1988) we say that is locally artinian if it is closed under direct limit (union) and every object is a union of objects of finite length. We also assume that has enough injective modules and that V i U i = i V U i for any collection of objects V U i. An object W is said to be a composition factor of an object V if it is a composition factor of a finite-length subobject of V. In this case we denote by V W the supremum of the multiplicities of W in all such subobjects. A locally artinian category is called a highest weight category Cline et al. (1988) if there exists an interval-finite poset (i.e., the sets = z z are finite) such that: 1. Non isomorphic simple objects L in are parameterized by ; 2. For each there exists and a monomorphism L such that any composition factor L of /L satisfies < ; 3. For every, dim Hom and L are finite; 4. The injective envelope I of L has a filtration 0 = N 0 N 1 such that I = i N i, N 1, N n /N n 1 L for some = n > and given there exist only finitely many n for which = n. The elements of are called weights and condition 2 above explains the terminology highest weight category. Note that L is the socle of. Let f be the full (artinian) subcategory in consisting of objects of finite length. Theorem 3.1 (Cline et al., 1988, Theorem 3.5). If the poset is finite, then the category f is equivalent to a module category for some finite-dimensional algebra. In particular, since f is a Jordan Hölder category, it follows from Theorem 2.11 that for every K 0 f and any slope function with values on a

10 MODULI SPACES FOR ABELIAN CATEGORIES 2179 continuously ordered -vector space, the moduli set can be given a structure of projective variety. Finite-dimensional algebras that correspond to highest weight categories are called quasi-hereditary. Such algebra B can be characterized by the existence of a hereditary chain of two-sided ideals 0 J 1 J 2 J t B where J s /J s 1 = B/J s 1 e s B/J s 1 for some idempotent e s of B/J s 1 for all 1 s t (Dlab, 1996). If is a highest weight category, then there are two canonical ways of constructing new highest weight categories. Let be the indexing poset for, a proper nonempty subset, = \. Denote by the full subcategory of consisting of objects M having composition factors isomorphic to some L,. Since is a Serre subcategory of, then the quotient category = / is defined. We say that is an ideal of if < and implies. The set is said to be finitely generated provided is the union of finitely many intervals < for some. We have the following recollement property of highest weight categories. Theorem 3.2 (Cline et al., 1988, Theorem 3.9). If is a finitely generated ideal of then is a highest weight category. Moreover, if is a finite coideal then is a highest weight category. Let be a finite poset and A be a quasi-hereditary algebra corresponding to the category f. Denote by e a complete sum of primitive idempotents representing. Then the category f is equivalent to the module category of eae if is an ideal, hence eae is quasi-hereditary. Moreover, A/eAe is quasi-hereditary if is coideal of. Example 3.3 (Poset Categories, Parshall and Scott, 1988, Example 6.9). Let P be a finite poset, P the geometric realization of the simplicial complex associated with P, P n+1 = P n \ maximal elements of P n, P 0 = P. Denote by Sh P the category of sheaves of -vector spaces on P which are locally constant on the natural strata P n \ P n+1. Then Sh P is a highest weight category which is isomorphic to the module category of the poset P. The next example is the key example that originated the theory of highest weight categories. Example 3.4 (Category ). Let be a complex finite-dimensional simple Lie algebra, be a fixed Cartan subalgebra, i be the fundamental weighs, P Q the weight and root lattices of, respectively. Let also Q + be the submonoid of Q generated by the positive roots and recall that can be equipped with the partial order iff Q +. The BGG category Bernstein et al. (1976) is the -linear category whose objects consists of -modules V satisfying: 1. V = V, where V = v V hv= h v h ;

11 2180 FUTORNY ET AL. 2. dim V is finite for all ; 3. There exist 1 m (depending on V ) such that V 0 implies j for some j = 1 m. The elements such that V 0 are called the weights of V. It turns out that every block of is equivalent to a block whose weights of its objects lie in P. Moreover, in such a block there exists a unique 0 P such that the weight of the objects in are bounded by 0 from above and there exists V such that V 0 0. is a highest weight category with poset given by = w 0 where w runs in the Weyl group of and the action of in is the so-called shifted action. In particular, is finite and, by Theorem 3.1, is equivalent to the module category of a (quasi hereditary) finite-dimensional algebra. If 0 is antidominant, then is a singleton and is a semisimple category with a unique simple object, but, in general is of wild representation type. The objects, are the so-called restricted duals of the Verma modules M. Let us look more closely at some natural stability structure on. Let R = be the -span of the fundamental weights and equip R with the lexicographic ordering determined by a choice of ordering the nodes of the Dynkyn diagram of and the usual order on. Given an object M in, fix x = x 0 x m 1 m, where m =, and define the slope j x j M L j j x M = j M L j This defines a stability structure on satisfying all the hypothesis of Theorem We now compute some examples of (semi) stable objects. We start with = 2, even though is always of finite representation type in this case. In fact, beside the semisimple block, there is only one more class of non equivalent blocks which is equivalent to mod-b, where B is the 5-dimensional algebra whose underlying quiver is 1 a 2 b ab = 0 The principal block (the one containing the trivial representation) is a representative of this class. We have m = 2 0 = 0, and 5 indecomposable objects: L 0 L 2 (the 2 irreducible objects), the Verma module M 0, its restricted dual M 0 = 0, and the big projective P 2. Here we have identified P with by sending 1 to 1. There are four nonsplit exact sequences: 0 L 2 M 0 L L 0 M 0 L L 2 P 2 M M 0 P 2 L 2 0

12 MODULI SPACES FOR ABELIAN CATEGORIES 2181 and the right regular representation of A has the following structure = A A where 1 stands for L 2 and 2 stands for L 0. Now we compute the slopes L = 0 L 2 = 2x 2 (3.1) M 0 = M 0 = x 2 P 2 = 4 3 x 2 (3.2) If x 2 > 0, then M 0 is stable, while M 0 is stable if and only if x 2 < 0. On the other hand, P 2 is semistable if and only if x 2 = 0, in which case the stability structure is trivial, i.e., V W for all V W. Now let us look at = 3. This time the principal block is wild so we do not attempt to characterize the semistable objects completely. If is the principal block then 0 = 0 and = = 0 5 It is not difficult, but a bit tedious, to see that for all k, there always exists a choice of x j j = 1 5, such that M k is stable. We also remark that the blocks of the following subcategory of are also highest weight categories. Given a parabolic subalgebra, let be the subcategory of whose objects split into a sum of finite-dimensional modules of. Such categories were studied in Rocha-Caridi (1980). The highest weight category of the next example is not equivalent to the module category of a finite-dimensional algebra. However, it is the limit of some truncated highest weight subcategories to which Theorem 3.1 applies. Example 3.5 (A Limit Construction). Let be an affine Kac Moody algebra with a Cartan subalgebra, Q the root lattice and Q + the submonoid of Q generated by the positive roots, and = 1 n be a set of simple roots. Define the height of = n i=1 k i i Q + by = i k i. Categories of truncated -modules were studied in Rocha-Caridi and Wallach (1982). For k +, set Q + k = Q + >k.if = n i=1 k i i Q, set + = j k j + k j j and, given, k +, denote by = k the set of all such that + Q + \Q + k. Let also k be the full category of the category of all -modules consisting of those modules V such that: 1. V = V, where V = v V hv= h v h ; 2. dim V is finite for all ; 3. V = 0 for all \. Clearly, any simple object of the category k is a quotient of the corresponding Verma module. We will denote an irreducible module with highest

13 2182 FUTORNY ET AL. weight by L. It follows from the results of Rocha-Caridi and Wallach (1982) that k is a highest weight category as well as its full subcategory f k consisting of finite-length objects. The later category has infinitely many simple objects and hence is not a module category for a finite dimensional algebra. Nevertheless, this category can be approximated by certain finite-dimensional algebras. Indeed, consider the following stratification of the category f k. Given a positive integer m denote by f k m the full subcategory in f k consisting of those objects whose composition factors are isomorphic to L for some with m. Let be the set of all with m (we set =0 if < ). Then clearly is a coideal in and hence, f k m is a highest weight category by Theorem 3.2. Moreover, since is finite then f k m is equivalent to the module category for some quasi-hereditary algebra. We have an embedding of full subcategories and f k 1 f k m f k = lim f k m To shorten the notation, set = f k and m = f k m. Fix a slope function K 0 R, where R is an -vector space with a continuous total order and let K 0. Then = k L for some non-negative integers k, all but finitely many equal to zero. Let m 0 = max k 0 It follows that = m 0 has a structure of quasi-projective variety. The results of this section are valid for a larger class of algebras introduced in Rocha-Caridi and Wallach (1982), which includes symmetrizable Kac Moody algebras and the Witt algebra Stratified Categories and Algebras In this section we consider examples of categories equivalent to module categories of finite-dimensional algebras which are a certain generalization of quasi-hereditary algebras. Let A be a finite-dimensional algebra and be a poset parameterizing the isomorphism classes of simple A-modules. Then A is called -filtered (Agoston et al., 1998; Dlab, 1996, 2000) with respect to the right (respectively, left) module structure if the following conditions are satisfied: (i) There exists a collection of right (resp. left) A-modules, such that has a unique irreducible quotient L corresponding to and all other composition factors L satisfy ; (ii) The right (resp. left) regular representation of A is filtered by s, i.e., for each, is a homomorphic image of the projective cover P of L and the kernel has a finite filtration with factors with <.

14 MODULI SPACES FOR ABELIAN CATEGORIES 2183 If A has either right or left -filtration, then it is also called standardly stratified (Futorny et al., 2002). Note that strandardly stratified algebras are special cases of stratified algebras introduced in Cline et al. (1996). If A has both left and right filtrations then it is called properly stratified. Finally, a properly stratified algebra is quasi-hereditary if and only if the choice of s is the same for right and left regular representations. One way to obtain new standardly stratified algebras out of a given projectively stratified (or even quasi-hereditary) algebra A is by taking eae for some idempotent e. Notice that even if A is quasi-hereditary, eae may not be so. Here we just list some examples, for details, see Mazorchuk (2002). Example 3.6 (Harish Chandra Bimodules). Let be a complex finite-dimensional simple Lie algebra and a central character. The category H of Harish Chandra bimodules consists of finitely generated algebraic U -bimodules. Denote by H the full subcategory of modules with central character (with respect to the right action of the center). Then H is equivalent to a certain subcategory of Bernstein and Gelfand (1980) which in turn is equivalent to the category mod-eae for some quasi-hereditary algebra A and some idempotent e. Example 3.7 (Parabolic Category ). A parabolic generalization of the category which contains nonhighest weight irreducible modules was defined in Coleman and Futorny (1994) and studied extensively in Futorny et al. (2000a,b, 2002). This category corresponds to a fixed parabolic subalgebra of a simple finite-dimensional Lie algebra. The role of Verma modules is played by the generalized Verma modules. In Futorny et al. (2001) this theory was extended to the case of infinite-dimensional Lie algebras with triangular decomposition. Note that suitable blocks of the categories mentioned in these two examples are equivalent. Some other equivalences of these categories were exploited in Koenig and Mazorchuk (2002) Jordan Algebras Now we present an example of a category equivalent to the module category of a finite-dimensional algebra which is neither hereditary, nor quasi-hereditary, nor stratified. Recall, that a Jordan -algebra is an -vector space J with a binary operation compatible with the vector space structure such that any a b J a b = b a a a b a = a a b a (3.3) It is known that the category of finite-dimensional J-bimodules is equivalent to the category of left finite-dimensional modules over some associative algebra U J which is called the universal multiplicative envelope of J. If J is finite-dimensional, then U J is finite-dimensional as well and, hence, the whole stability machinery can be applied to the category of J-bimodules. If J is a semisimple Jordan algebra, then

15 2184 FUTORNY ET AL. U J is semisimple and the variety of semistable objects is trivial. On the other hand, if J is not semisimple but Rad 2 J = 0, then the category of bimodules might even be of wild representation type and, thus, have very nontrivial moduli sets. We point out that the algebra U J decomposes into a product of subalgebras U J = U 0 U 1 2 U 1, where U 0 = and U 0 U 1 2 is the special universal envelope of J. Then the module category U J -mod is equivalent to a direct sum U 0 -mod U 1 2 -mod U 1 -mod and the problem is reduced to the study of these module categories. For detailed study of the category U 0 U 1 2 -mod, see Kashuba et al. (Preprint) Finite-Dimensional Representations of Loop Algebras We end the section with an example of an abelian category which is a very active research topic and is not equivalent to a module category of a finite-dimensional algebra. Since it is a Jordan Hölder category, we can construct the moduli sets. However, nothing can be said about geometric properties of these moduli sets at the moment. This is the category of finite-dimensional representations of the loop algebras = t t 1, where is a complex finite-dimensional simple Lie algebra. We remark that, although this category is not a highest weight category, it is a very close cousin of the highest weight category studied in Chari and Greenstein (2007). Since every finite-dimensional -module is in category when regarded as a -module, we can define slope functions in exactly the same manner as we did in Example 3.4. However, here we have other natural choices for the function d. Instead of using -length as in Example 3.4, we can use -length or even dimension. However, we do not expect such choice to significantly change the structure of the generic moduli sets. REFERENCES Agoston, I., Dlab, V., Lukacs, E. (1998). Stratified algebras. C. R. Math. Rep. Acad. Sci. Canada 20: Bekkert, V., Benkart, G., Futorny, V. (2004). Weight modules for Weyl algebras. In: Kac-Moody Lie Algebras and Related Topics. Contemp. Math 343 Amer. Math. Soc., Providence, RI, pp Bernstein, I., Gelfand, I. (1980). Tensor product of finite and infinite-dimensional representations of semisimple Lie algebras. Compositio Math. 41(2): Bernstein, I. N., Gel fand, I. M., Gel fand, S. I. (1976). Category of modules. Funct. Anal. Appl. 10: Bridgeland, T. (2007). Stability conditions on triangulated categories. Annals of Math. 166(2): Chari, V., Greenstein, J. (2007). Current algebras, highest weight categories and quivers, Adv. Math. 216(2): Cline, E., Parshall, B., Scott, L. (1988). Finite dimensional algebras and highest weight categories. J. Reine Agne. Math. 391: Cline, E., Parshall, B., Scott, L. (1996). Stratifying endomorphism algebras. Mem. Amer. Math. Soc. 124:591.

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