SStaircase algebras and graded nilpotent pairs
Magdalena BoosRuhr-Universität BochumFaculty of MathematicsD - 44780 [email protected]
Abstract
We consider a class of finite-dimensional algebras, the so-called “Staircasealgebras” parametrized by Young diagrams. We develop a complete classificationof representation types of these algebras and look into finite, tame (concealed) andwild cases in more detail. Our results are translated to the setup of graded nilpotentpairs for which we prove certain finiteness conditions.Keywords: Graded nilpotent pairs, finite-dimensional algebras, representation type
Graded nilpotent pairs naturally appear as a generalization of principal nilpotent pairsintroduced by Ginzburg [14] and studied by Panyushev [17], Yu [21] and others.Such pair is given as an element ( ϕ, ψ ) of the nilpotent commuting variety of a fixedbigraded vector space V , such that both nilpotent operators are compatible with thebigrading in a natural commuting way: ϕ respects the bigrading "horizontally" and ψ respects the bigrading "vertically". The non-zero components of the bigrading of V induce a partition λ and the graded nilpotent pair is called λ -shaped, thus.One of our main results gives an answer to a standard Lie-theoretic question: Arethere only finitely many λ -shaped graded nilpotent pairs up to base change by a Levirespecting the grading? Theorem (6.1) . The number of λ -shaped graded nilpotent pairs is finite (modulo basechange in the homogeneous components) if and only if1. λ ∈ { ( n ) , (1 k , n − k ) , (2 , n − , (1 n − , ) } for some k ≤ n,2. n ≤ and λ (cid:60) { (1 , , , (2 , ) , (1 , , , (1 , , } . Furthermore, we obtain (in)finiteness conditions for graded nilpotent pairs (moduloLevi-base change) on a fixed bigraded vector space V in Lemma 6.2 and Lemma 6.3.In order to obtain our results on graded nilpotent pairs, we introduce a class of finite-dimensional algebras, the so-called staircase algebras in Section 3. We show that every1 a r X i v : . [ m a t h . R T ] S e p raded nilpotent pair can be considered as a representation of such a staircase algebraand approach this class of algebras from the angle of representation theory of finite-dimensional algebras (for example with Auslander-Reiten techniques). Our results aretranslated to graded nilpotent pairs in Section 6.Staircase algebras, denoted by A ( λ ), are quite interesting themselves - they are parametrizedby Young diagrams Y ( λ ) (or, equivalently, partitions λ ) and are defined by quivers with(exclusively) commutativity relations. Our main aim concerning staircase algebras isto classify their representation types completely in Section 4. Theorem (4.5) . A staircase algebra A ( λ ) is • representation-finite if and only if the conditions of Theorem 6.1 hold true. • tame concealed if and only if λ comes up in the following list: (3 , , (1 , , , (1 , , , (2 , , (1 , , , (1 , , , (1 , ) , (1 , ) , (1 , , . • tame, but not tame concealed if and only if λ comes up in the following list: (4 , , (5 ) , (1 , ) , (2 , ) , (3 ) , (2 , , (1 , ) , (2 ) .Otherwise, A ( λ ) is of wild representation type. The orbit type of each staircase algebra is classified in Lemma 3.9 and we prove a cor-relation between representation types and orbit types for the class of staircase algebras.We obtain a complete hierarchy of algebras which makes clear the transitions betweenrepresentation types and is visualized in Appendix B.In Section 5, the finite, tame (concealed) and minimal wild cases are examined in moredetail. For example, all staircase algebras of finite type and most of the tame ones aretilted and (except for infinite families of representation-finite algebras) all Auslander-Reiten quivers of the former are attached in Appendix A.For each tame case, minimal nullroots which admit infinitely many isomorphism classesof representations are provided initiating the study of finiteness criteria for nilpotentgraded pairs of a fixed bi-graded vector space. For each case of wild representationtype, we construct a 2-parameter family of pairwise non-isomorphic representations.
Acknowledgments:
I am grateful to K. Bongartz and M. Reineke for various helpfuldiscussions and ideas. I would like to thank M. Bulois and J. Külshammer for helpfulremarks and suggestions. Furthermore, I am thankful to O. Kerner and L. Unger forhelping me with finding and getting to know literature and known results.
Let K = K be an algebraically closed field and let GL n (cid:66) GL n ( K ) be the general lineargroup for a fixed integer n ∈ N regarded as an a ffi ne variety. We begin by defining thenotion of a graded nilpotent pair before including some facts about the representationtheory of finite-dimensional algebras. We refer to [1] for a thorough treatment of thelatter. 2 .1 Graded nilpotent pairs Let V = (cid:76) s , t ∈ Z ≥ V s , t be an N -dimensional bigraded K -vector space; we formally set V x , y : = x , y ) (cid:60) Z ≥ .Denote by N ( V ) the nilpotent cone of nilpotent operators on V . The nilpotent commut-ing variety of V is defined by C ( V ) : = { ( ϕ, ψ ) ∈ N ( V ) × N ( V ) | [ ϕ, ψ ] = } , its elements are called commuting nilpotent pairs .Such pair is called graded nilpotent pair , if ϕ restrict to each V s , t "horizontally" via ϕ s , t : = ϕ | V s , t : V s , t → V s − , t and ψ restrict to each V s , t "vertically" via ψ s , t : = ψ | V s , t : V s , t → V s , t − . Note that the study of this setup is quite natural regarding the context of principalnilpotent pairs [14].We define the shape of V bysh( V ) : = { ( s , t ) | ∃ p , q ∈ Z ≥ , p ≥ s , q ≥ t : V p , q (cid:44) } which defines a Young diagram corresponding to a partition λ ( V ). In more detail, thelatter is given by λ ( V ) i = (cid:93) { ( s , h ( V ) − i ) ∈ sh( V ) | s ∈ Z ≥ } if we define h ( V ) : = max { t | V , t (cid:44) } + ϕ, ψ ) is called a λ -graded nilpotent pair . Note that these definitions de-pend on a fixed chosen grading of V , but not on the nilpotent pairs compatible withit. Example 2.1.
Let V : = (cid:76) s , t ∈ Z ≥ V s , t be a bigraded K-vector space, such that V , = V , = V , = K, V , = V , = K , V , = K and V s , t = , otherwise. Let ( ϕ, ψ ) be agraded nilpotent pair on V. Then we can illustrate the latter by V , V , V , V , KV , V , V , V , = K K V , V , V , V , K K K ϕ , ψ , ψ , ϕ , ϕ , The Young diagram is accentuated on the right hand side, it is given byY ( λ ) = (4,1)(3,1)(2,1) (2,2)(1,1) (1,2) (1,3) . Thus, λ ( V ) = (1 , , , and ( ϕ, ψ ) is a λ ( V ) -graded nilpotent pair. .2 Basics of Representation Theory A finite quiver Q is a directed graph Q = ( Q , Q , s , t ), given by a finite set of vertices Q , a finite set of arrows Q and two maps s , t : Q → Q defining the source s ( α ) andthe target t ( α ) of an arrow α . A path is a sequence of arrows ω = α s . . . α , such that t ( α k ) = s ( α k + ) for all k ∈ { , . . . , s − } .We define the path algebra K Q as the K -vector space with a basis consisting of allpaths in Q , formally included is a path ε i of length zero for each i ∈ Q starting andending in i . The multiplication is defined by concatenation of paths, if possible, andequals 0, otherwise.Let us define the arrow ideal R Q of K Q as the (two-sided) ideal which is generated (asan ideal) by all arrows in Q . In particular, the arrow ideal of Q equals the radical of K Q if Q does not contain oriented cycles. An arbitrary ideal I ⊆ K Q is called admissible if there exists an integer s , such that R s Q ⊂ I ⊂ R Q . Given such admissible ideal I , wedenote by I ( i , j ) the set of paths in I starting in i and ending in j .Given such admissible ideal I , the path algebra of Q , bound by I is defined as A : = K Q / I ; it is a basic and finite-dimensional K -algebra [1]; the elements of I are the so-called relations of A . Then A is called triangular , if Q does not contain an orientedcycle.Let rep K A be the abelian K -linear category of all finite-dimensional A -representationswhich is equivalent to the category of K-representations of Q , which are bound by I , de-fined as follows: The objects are given by tuples (( M i ) i ∈Q , ( M α ) α ∈Q ), where the M i are K -vector spaces, and the M α : M s ( α ) → M t ( α ) are K -linear maps. For each path ω in Q asabove, we denote M ω = M α s · . . . · M α and ask a representation M to fulfill (cid:80) ω λ ω M ω = (cid:80) ω λ ω ω ∈ I . A morphism of representations M = (( M i ) i ∈Q , ( M α ) α ∈Q ) and M (cid:48) = (( M (cid:48) i ) i ∈Q , ( M (cid:48) α ) α ∈Q ) consists of a tuple of K -linear maps ( f i : M i → M (cid:48) i ) i ∈Q , suchthat f j M α = M (cid:48) α f i for every arrow α : i → j in Q .The dimension vector of an A -representation M is defined by dim M ∈ N Q ; in moredetail (dim M ) i = dim K M i for i ∈ Q . Let us fix such a dimension vector d ∈ N Q anddenote by rep K A ( d ) the full subcategory of rep K A which consists of all representa-tions of dimension vector d .By defining the a ffi ne space R d K Q : = (cid:76) α : i → j Hom K ( K d i , K d j ), one realizes that itspoints m naturally correspond to representations M ∈ rep K K Q ( d ) with M i = K d i for i ∈ Q . Via this correspondence, the set of such representations bound by I correspondsto a closed subvariety R d A ⊂ R d K Q .The algebraic group GL d = (cid:81) i ∈Q GL d i acts on R d K Q and on R d A via base change, fur-thermore the GL d -orbits O M of this action are in bijection to the isomorphism classesof representations M in rep K A ( d ).Due to Krull, Remak and Schmidt, every finite-dimensional A -representation decom-poses into a direct sum of indecomposables (which by definition do not decomposefurther). For certain classes of finite-dimensional algebras, a convenient tool for theclassification of these indecomposable representations is the Auslander-Reiten quiver Γ A = Γ ( Q , I ) of rep K ( Q , I ). Its vertices [ M ] are given by the isomorphism classes of4ndecomposable representations of rep K ( Q , I ); the arrows between two such vertices[ M ] and [ M (cid:48) ] are parametrized by a basis of the space of so-called irreducible mapsf : M → M (cid:48) . One standard technique to calculate Γ A is the knitting process (see, forexample, [1, IV.4]).A component Π A of Γ A is called preprojective if it does not contain oriented cycles andif every module in Π A lies in the τ -orbit of some projective. Its corresponding orbitquiver Υ A is defined as the quiver whose vertices are given by the τ -orbits [ X ] of Π A .The number of arrows [ X ] → [ X (cid:48) ] in Υ A coincides with the maximal number of arrows M → M (cid:48) in Π A , where M ∈ [ X ] , M (cid:48) ∈ [ X (cid:48) ]. The orbit type of A is defined to be thetype of Υ A . A representation M is called sincere , if every simple indecomposable of A comes up in a composition series of M .We say that an algebra B = K Q (cid:48) / I (cid:48) is a convex subcategory of A = K Q / I , if Q (cid:48) is a convex subquiver of Q (that is, if two vertices i , j are contained in Q (cid:48) , then every pathof Q from i to j is completely contained in Q (cid:48) ) and I (cid:48) (cid:66) (cid:104) I ( i , j ) | i , j ∈ Q (cid:48) (cid:105) .An indecomposable projective P has a so-called separated radical, if for arbitrary twonon-isomorphic direct summands of its radical, their supports are contained in di ff erentcomponents of the subquiver Q obtained by deleting all starting points of paths endingin i . We say that A fulfills the separation condition , if every projective indecomposablehas a separated radical. If this condition is fulfilled, A admits a preprojective compo-nent, see [3]. In general, the definition of an algebra to be strongly simply connected algebra is quite involved. In case of a triangular algebra A , there is an equivalentdescription, though : A is strongly simply connected if and only if every convex sub-category of A satisfies the separation condition [19]. Consider a finite-dimensional basic K -algebra A : = K Q / I . It is called of finite repre-sentation type , if there are only finitely many isomorphism classes of indecomposablerepresentations. If it is not of finite representation type, the algebra is of infinite rep-resentation type . These infinite types split up into two disjoint cases; we say that thealgebra A has • tame representation type (or is tame ) if for every integer n there is an integer m n and there are finitely generated K [ x ]- A -bimodules M , . . . , M m n which are freeover K [ x ], such that for all but finitely many isomorphism classes of indecom-posable right A -modules M of dimension n , there are elements i ∈ { , . . . , m n } and λ ∈ K , such that M (cid:27) K [ x ] / ( x − λ ) ⊗ K [ x ] M i . • wild representation type (or is wild ) if there is a finitely generated K (cid:104) X , Y (cid:105) - A -bimodule M which is free over K (cid:104) X , Y (cid:105) and sends non-isomorphic finite-dimensional indecomposable K (cid:104) X , Y (cid:105) -modules via the functor _ ⊗ K (cid:104) X , Y (cid:105) M tonon-isomorphic indecomposable A -modules.In 1979, Drozd proved the following dichotomy statement [11]. Theorem 2.2.
Every finite-dimensional algebra is either tame or wild. A yields that there are at most one-parameter familiesof pairwise non-isomorphic indecomposable A -modules; in the wild case there areparameter families of arbitrary many parameters.A finite-dimensional K -algebra is called of finite growth , if there is a natural number m , such that the indecomposable finite-dimensional modules occur in each dimension d ≥ m one-parameter families. For a triangular algebra A = K Q / I , the Tits form q A : Z Q → Z is the integral quadraticform defined by q A ( v ) = (cid:88) i ∈Q v i − (cid:88) α : i → j ∈Q v i v j + (cid:88) i , j ∈Q r ( i , j ) v i v j ;here r ( i , j ) equals the number of elements in R ∩ I ( i , j ) whenever R is a minimal setof generators of I , such that R ⊆ (cid:83) i , j ∈Q I ( i , j ). The corresponding symmetric bilinearform is denoted b A (_ , _) and fulfills the condition q A ( v + w ) = q A ( v ) + b A ( v , w ) + q A ( w ).Any non-zero vector v ∈ N Q is called positive . The quadratic form q A is called weaklypositive , if q A ( v ) > v ∈ N Q ; and (weakly) non-negative , if q A ( v ) ≥ v ∈ Z Q (or v ∈ N Q , respectively).For a non-negative form q A , the radical of q A is rad q A : = { u ∈ Z Q | q A ( u ) = } ,we call its elements nullroots . In a similar manner, we define the set of isotropic roots as rad q A : = { u ∈ N Q | q A ( u ) = } and the set of rational isotropic roots to berad Q q A : = { u ∈ Q + Q | q A ( u ) = } (here, Q + is the set of non-negative rationalnumbers). The maximal dimension of a connected halfspace in rad Q q A is the isotropiccorank corank q A of q A .The definiteness of the Tits form is closely related to the representation type of A , andthere are connections between roots and certain dimension vectors of representations.Many results are, for example, summarized by De la Peña and Skowro´nski in [9] whereall definitions can be found, too. It is well known that q A is weakly positive if A isrepresentation-finite and q A is weakly non-negative if A is tame. In certain cases, theopposite directions are true, as well. The following criterion for finite representationtype is due to Bongartz [3]. Theorem 2.3.
Let A = K Q / I be a triangular algebra, which admits a preprojectivecomponent. Then A is representation-finite if and only if the Tits form q A is weaklypositive. If the equivalent conditions hold true, then the dimension vector function X (cid:55)→ dim X induces a bijection between the set of isomorphism classes of indecomposable A -modules and the set of positive roots of q A . Assume that Γ A has a preprojective component. The algebra A is called critical if q A is not weakly positive, but every proper restriction of q A is weakly positive. The term"critical" is actually intuitive, since the conditions of Theorem 2.3 are equivalent to A not having a convex subcategories which is critical; a classification of the critical6lgebras can be found in [4, 15].The algebra A is called hypercritical if q A is not weakly non-negative while everyproper restriction of q A is weakly non-negative. The hypercritical algebras have beenclassified in Unger’s list [20]. For strongly simply connected algebras, they turn outto be the minimal wild algebras as the following classification of tame types due to ofBrüstle, De la Peña and Skowro´nski yields [7]. Theorem 2.4.
Let A be strongly simply connnected. Then the following are equivalent:1. A is tame;2. q A is weakly non-negative;3. A does not contain a full convex subcategory which is hypercritical. Theorem 2.4 and Theorem 2.2 yield a su ffi cient criterion for wildness. Corollary 2.5.
Let A be strongly simply connnected. Whenever there exists v ∈ N Q ,such that q A ( v ) ≤ − , then A is of wild representation type. By De la Peña [8], the following description of finite growth is available.
Proposition 2.6.
Let A be a strongly simply connected algebra. Then A is of finitegrowth if and only if q A is weakly non-negative and corank q A ≤ . Some of our algebras A do appear as tilted algebras, that is, there is a hereditary algebra B and a B -tilting module T , such that End B ( T ) = A . In general, there is a su ffi cientcriterion for an algebra to be tilted which reads as follows [18]: Lemma 2.7. If A has a preprojective component which contains an indecomposablesincere representation, then A is a tilted algebra. The following lemma provides a classification for certain cases.
Lemma 2.8.
If there is a preprojective component of A containing all indecomposableprojective A -modules, but no injective module, then the following are equivalent:1. A is tilted of Euclidean type;2. A is tame;3. q A is non-negative. Especially the notion of a tame concealed algebra comes up in our setup. These arethe Euclidean tilted algebras, such that the tilting module is preprojective; also shownto be the minimal representation-infinite, that is, critical algebras [18]. The so-calledBongartz-Happel-Vossieck list (BHV-list) lists the tame concealed algebras [4, 15].7
Staircase algebras
Let n ∈ N and let λ (cid:96) n be an increasing partition of n . Let Y ( λ ) be the Young diagramof λ , that is, the box-diagram of which the i -th row contains λ i boxes. We denote by λ T the transposed increasing partition given by the columns of the Young diagram (fromright to left). Starting with (1 ,
1) in the bottom left corner, we number the boxes of λ by ( i , j ) increasing i from bottom to top and j from left to right.Corresponding to Y ( λ ), let us define the quiver Q ( λ ): its vertices are given by the tuples( i , j ) appearing in Y ( λ ); the arrows are given by all horizontal arrows α i , j : ( i , j ) → ( i − , j ) and all vertical arrows β i , j : ( i , j ) → ( i , j − Q ( λ ) can be easily visualized by turning the Young diagram 90 o anti-clockwise and drawing arrows from left to right and from top to bottom.Let us define the path algebra A ( λ ) = K Q ( λ ) / I with relations given by I : = I ( λ ), whichis the 2-sided admissible ideal generated by all commutativity relations in the appearingsquares in Q ( λ ) which are, if defined, of the form β i , j α i , j + − α i − , j + β i , j + . Since I isadmissible and Q ( λ ) is connected, A ( λ ) is a basic, connected, finite-dimensional K -algebra.We call n the size of A ( λ ) and l : = l ( λ ) the length of A ( λ ), if λ = ( λ , ..., λ l ). For easiernotation, we merge similar entries of λ by potencies, for example (1 , , , , , , , = :(1 , , , ). Then the length is given by the number of entries in original notation; thenumber of entries in potency-notation is called the number of steps s : = s ( λ ) of A ( λ ).Since the quivers Q ( λ ) look like staircases, the following definition is reasonable. Definition 3.1.
Let A be a finite-dimensional K-algebra. It is called • n-staircase algebra, if there is an increasing partition λ (cid:96) n, such that A (cid:27) A ( λ ) . • staircase algebra, if there exists a natural number n, such that A is an n-staircasealgebra. • m-step, if the number of steps equals m. We denote the Tits quadratic form by q λ : = q A ( λ ) and the Auslander-Reiten quiver by Γ ( λ ) : = Γ A ( λ ) .Let us consider an example to illustrate these definitions before discussing staircasealgebras and their properties in detail. Example 3.2.
Let n = and λ = (1 , , , and its Young diagram drawn in Example2.1. Its quiver Q ( λ ) is given by (1 , • (2 ,
2) (1 ,
2) ˆ = • • (4 ,
1) (3 ,
1) (2 ,
1) (1 , • • • • α , β , β , β , α , α , α , The -staircase algebra A ( λ ) of steps and of length is defined by A ( λ ) = K Q ( λ ) / ( β , α , − α , β , ) . .1 Link to graded nilpotent pairs Each graded nilpotent pair ( ϕ, ψ ) together with a bigraded vector space V as in 2.1 canbe considered as a representation M : = M ( ϕ, ψ, V ) of A ( λ ( V )) in a natural way. Wedenote dim V : = dim A ( λ ( V )) M which depends on the bigrading on V , but not on ( ϕ, ψ ). Example 3.3.
Consider the setup of Example 2.1. Then the A ((1 , , -representationM ( ϕ, ψ, V ) is given by KK K K K K ψ , ϕ , ψ , ϕ , ϕ , Representation-theoretically speaking, the graded nilpotent pairs of V are encoded inthe representation variety R dim V A ( λ ( V )). Therefore, certain criteria can be translatedfrom the representation theory of finite-dimensional algebras straight away. Theorem 3.4.
Let λ be a partition. Modulo Levi-base change, there are only finitelymany λ -graded nilpotent pairs if and only if A ( λ ) is of finite representation type. Other-wise, there are at most one-parameter families of non-decomposable graded nilpotentpairs if and only if A ( λ ) is tame. Lemma 3.5.
The number of graded nilpotent pairs (up to Levi-base change) of thefixed bigraded vector space V is finite if and only if R dim V A ( λ ( V )) admits only finitelymany GL dim V -orbits. Corollary 3.6.
The equivalent conditions of Lemma 3.5 hold true if and only if thenumber of isomorphism classes of representations in rep K A ( λ ( V ))(dim V ) is finite. These criteria motivate the classification of representation types of staircase algebrasin Section 4 and certain further considerations in Section 5. To start, we study generalproperties of staircase algebras.
For ( i , j ) ∈ Q ( λ ) , let S ( i , j ) be the standard simple representation at the vertex ( i , j )of A ( λ ). The (isomorphism classes of the) projective indecomposables of A ( λ ) areparametrized by P ( i , j ), ( i , j ) ∈ Q ( λ ) , which are given by P ( i , j ) k , l = (cid:40) K if k ≤ i and l ≤ j , . together with identity and zero maps, accordingly. Proposition 3.7.
The algebra A ( λ ) is triangular, fulfills the separation condition andis strongly simply connected. roof. Since Q ( λ ) does not contain oriented cycles, A ( λ ) is triangular.The radical of every projective indecomposable is indecomposable, such that each pro-jective indecomposable has separated radical and A ( λ ) fulfills the separation condition.Let λ be an increasing partition and consider a convex subcategory C of A ( λ ). We aimto show that C fulfills the separation condition.The radical of every projective P of the vertex ( i , j ) is either indecomposable or decom-poses into exactly two indecomposables. The latter is the case if and only if ( i − , j − Q of C . Let Q (cid:48) be the subquiver of Q which is obtainedby deleting all start vertices of all paths ending in ( i , j ). It is clear that Q (cid:48) decomposesinto two disjoint quivers, these correspond to the supports of the two indecomposabledirect summands of rad P . Thus, C fulfills the separation condition. The algebra A ( λ )is, thus, strongly simply connected. (cid:3) We approach the module categories by general techniques. Since Proposition 3.7 statesthat A ( λ ) fulfills the separation condition, the following lemma follows from Section2.2). Lemma 3.8.
The algebra A ( λ ) admits a preprojective component Π ( λ ) : = Π A ( λ ) . The knowledge of the orbit type, that is, of the type of the orbit quiver Υ ( λ ) : = Υ A ( λ ) gives first clues about the corresponding representation types, which we classify inTheorem 4.5. Lemma 3.9.
The orbit type of A ( λ ) is1. A n , if λ = (1 k , n − k ) for some k,2. D n , if λ ∈ { (1 n − , ) , (2 , n − } ,3. E , if λ ∈ { (1 , , , (2 ) , (3 ) } ,4. E , if λ ∈ { (1 , , , (1 , , , (1 , ) , (3 , , (1 , ) , (2 , } ,5. E , if λ ∈ { (1 , , , (1 , , , (1 , ) , (3 , , (1 , ) , (2 , , (2 ) , (4 ) } ,6. (cid:102) E , if λ ∈ { (1 , , , (1 , , , (2 , ) , (1 , , } ; and if λ ∈ { (3 ) , (1 , ) , (2 , } .7. (cid:102) E , if λ ∈ { (3 , , (4 , , (1 , , , (2 , , (1 , ) , (1 , ) , (1 , , , (1 , ) } ; and if λ = (5 ) .In every other case, the orbit type is wild. Note that for every listed case except for λ ∈{ (3 ) , (1 , ) , (2 , , (5 ) } , every indecomposable projective indecomposable comes upin the preprojective component. In the Euclidean cases, no injective indecomposableis contained in the preprojective component.Proof. The Auslander-Reiten quivers of every staircase algebra appearing in 1. to 5.,that is, of every staircase algebra of Dynkin orbit type, are depicted in the Appendix A.The orbit type can be read o ff directly.Let n ∈ { , } , then the orbit quivers listed in 6. and 7. can easily be calculated by10nitting the beginning of the preprojective components, since every projective inde-composable comes up. Every remaining case is seen to be of of wild orbit type by thesame method. There are two exceptions of this procedure:In case λ = (3 ) the claim follows from knitting - but one has to realize that the projec-tive indecomposable P (3 ,
3) is injective as well and does not appear in the preprojectivecomponent, since this case is representation-infinite, see Lemma 4.4.In case λ = (1 , ) = (2 , T knitting yields the claim, but it must be proved that P (2 , Γ , restricted to the vertex (3 , U , .., U ,such that every τ − -translation of these will be non-zero-dimensional at the vertex (3 , ffi ces to calculate the Auslander-Reiten quiver until U , .., U appear and re-alizes that the radical of P (2 ,
4) does not appear.For n ≥
10, every case except (5 ) is of wild orbit type. The case (5 ) is of type (cid:102) E ,which can be seen by knitting its preprojective component. Again, one of the projec-tive indecomposables is injective and does not appear in the preprojective component -since this case is representation-infinite by Lemma 4.4. (cid:3) In order to classify the representation types of all staircase algebra, it is of great valueto have certain reduction statements available.
Lemma 4.1.
Let A be a convex subcategory of A (cid:48) . Then1. ... if A is tame, then A (cid:48) is tame or wild.2. ... if A is wild, then A (cid:48) is wild.3. ... if A (cid:48) has finite representation type, A has finite representation type.4. ... if A (cid:48) is tame, then A is tame or of finite representation type.In particular, if λ ≤ λ (cid:48) is a subpartition, then these facts hold true for A = A ( λ ) and A (cid:48) = A ( λ (cid:48) ) .Proof. The claim follows from general representation theory of quivers with relations[1] by restricting in a natural way or expanding with zero. (cid:3)
Lemma 4.2.
For each partition λ , the representation type of A ( λ ) and A ( λ T ) is thesame. We provide a complete classification of the representation type of a staircase algebra.In order to assure a nice structure of the presentation, we begin by classifying therepresentation types of staircase algebras of length 2 in Lemma 4.3 and of length 3 in11emma 4.4, since most of the staircase algebras of finite and tame representation typecome up in these contexts. We then generalize the results to arbitrary staircase algebrasin Theorem 4.5 which completes the classification.
Lemma 4.3.
A staircase algebra A ( λ ) of length , that is, λ = ( λ , λ ) (cid:96) n, is • of finite representation type if and only if n ≤ or λ ∈ { , } . • tame concealed if and only if λ = (3 , . • tame, but not tame concealed, if and only if λ ∈ { (4 , , (5 ) } .Otherwise, A ( λ ) is of wild representation type.Proof. The following table shows the structure of our proof; the marked ones showwhich cases need to be proved in order to classify every remaining non-marked casedue to reductions via Lemma 4.1 and Lemma 4.2 (W = Wild, T = Tame, TC = Tame con-cealed, F = Finite): ... ... ... ... ... ... ... ... ... ... · · · T W W W · · ·
F T W
W W · · ·
F TC W W · · · F F F · · · · · · λ /λ · · · Representation-finite cases: Let λ =
1, then Q ( λ ) is of type A n which is representation-finite. For the remaining maximal finite cases, the Auslander-Reiten quivers (or theirsymmetric versions), are attached in the Appendix A. In particular, if λ =
2, then theAuslander-Reiten quiver Γ ( λ ) for λ = (2 ,
6) is attached and it is easy to see that thequiver stays finite, if λ increases.Representation-infinite cases: The case (3 ,
6) is tame concealed by [15].The cases λ ∈ { (5 ) , (4 , } are tame, since the algebras A ( λ ) do not contain a hyper-critical convex subcategory (Unger’s list [20]) by Theorem 2.4 (see Lemma 5.1 for astraight-away proof of their tameness). They are not minimal tame and thus not tameconcealed [18], since they contain a tame concealed subquiver of the BHV-list [2]: • • • •• • • • The cases λ = (4 ,
6) and λ (cid:48) = (3 ,
7) are wild by Corollary 2.5, since q λ (cid:32) (cid:33) = q λ (cid:48) (cid:32) (cid:33) = − (cid:3) emma 4.4. A staircase algebra A ( λ ) of length , that is, λ = ( λ , λ , λ ) (cid:96) n, is • of finite representation type if and only if n ≤ or λ ∈ { (1 , , λ ) , (1 , , , (2 , } . • tame concealed if and only if λ ∈ { (1 , , , (1 , , , (2 , , (1 , ) } . • tame, but not tame concealed, if and only if λ ∈ { (1 , ) , (2 , ) , (2 , , (3 ) } .Otherwise, A ( λ ) is of wild representation type.Proof. Whenever λ ≥
3, either λ = (3 ), or A ( λ ) is of wild representation type: thecase λ = (2 , ,
4) is wild by Corollary 2.5 q λ = − . If λ = (3 ), then A ( λ ) is not wild, since it does not contain a hypercritical convexsubcategory (Unger’s list [20]). See Lemma 5.1 for a straight forward proof of itstameness. It is not tame concealed, since it contains the tame concealed algebra • •• • •• • By reduction via Lemma 4.1 we only consider λ ∈ { , } . The following tables visu-alize all cases and it su ffi ces to prove the marked ones by Lemma 4.1. λ = ... ... ... ... ... ... ... ... ... · · · · · · T W W W W · · ·
F TC W
W W W · · ·
F TC W W · · · F · · · λ = ... ... ... ... ... ... ... ... ... · · · T W
W W W W · · ·
F TC W
W W · · · λ /λ · · · Representation-finite cases: Either λ =
1, then we arrive at an A n -classification. Forthe remaining finite cases, the Auslander-Reiten quivers or their symmetric versions(by Lemma 4.2) are depicted in the Appendix A.Representation-infinite cases:Every marked tame concealed case is tame concealed by the BHV-list [15].13he case λ = (1 , ) is not wild by Theorem 2.4, since the algebra A ( λ ) does notcontain any hypercritical convex subcategory (Unger’s list [20]). It is representation-infinite and not tame concealed, that is, minimal tame, since it contains the case (1 , , λ = (2 , ) is tame, but not tame concealed since A (3 ) is tame and since itcontains the above depicted convex subcategory.For every minimal wild case, by Corollary 2.5 we provide dimension vectors v whichfulfill q λ ( v ) ≤ −
122 44 42 4 2 124686 104 8 6 12462 6 84 6 4 (1 , ,
5) (1 , ,
7) (2 , λ = (2 , ,
4) has been shown to be wild above. (cid:3)
We are able to deduce all remaining cases now which leads to a complete classificationof the representation types of all staircase algebras.
Theorem 4.5.
A staircase algebra A ( λ ) is • representation-finite if and only if one of the following holds true:1. λ ∈ { ( n ) , (1 k , n − k ) , (2 , n − , (1 n − , ) } for some k ≤ n,2. n ≤ and λ (cid:60) { (1 , , , (2 , ) , (1 , , , (1 , , } . • tame concealed if and only if λ comes up in the following list: (3 , , (1 , , , (1 , , , (2 , , (1 , , , (1 , , , (1 , ) , (1 , ) , (1 , , . • tame, but not tame concealed if and only if λ comes up in the following list: (4 , , (5 ) , (1 , ) , (2 , ) , (3 ) , (2 , , (1 , ) , (2 ) .Otherwise, A ( λ ) is of wild representation type.Proof. All listed representation-finite cases are in fact of finite representation type:Let λ = (1 k , n − k ) for some k , then we obtain an A n -classification problem, thus,representation-finiteness [13]. Given the two symmetric cases λ = (2 , n − = (1 n − , ) T ,finiteness follows from Lemma 4.3. For every remaining finite case, the Auslander-Reiten quivers can be found in the Appendix A.Assume, l ( λ ) ≤
3, then the claimed classification follows from Lemma 4.3 and Lemma4.4. Thus, let without loss of generality λ = ( λ , ..., λ k ), such that k ≥
4; and λ k − > λ k − ≥
4, then A ( λ ) is of wild representation type by Lemma 4.1: The case A (1 , , , λ
22 44 41 2 4 2 = − . If λ k − =
2, then λ is of the form (1 , ..., , , ..., , x ) for some x . • If x ∈ { , } , then the transpose of every case has been considered in Lemma 4.3or Lemma 4.4. • If x ≥
4, then – for λ k − = A ((1 , , , T ) = A (1 , , ,
4) is wild via symmetry of Lemma 4.2. – if λ k − = x =
4, the algebra A ( λ ) is tame concealed [15]. – for λ k − = x ≥ q λ = − . If λ k − =
3, then (since A (1 , , ,
4) is wild) A ( λ ) is wild whenever λ k ≥
4. If λ k = λ is of length 3 and has, thus, been considered in Lemma 4.4. (cid:3) Corollary 4.6.
The tensor algebra KA m ⊗ K KA l is of finite representation type ifand only if ( m , l ) ∈ { (1 , l ) , ( m , , (2 , , (2 , , (3 , , (4 , , (2 , } . It is tame exactlyif ( m , l ) ∈ { (2 , , (5 , , (3 , } and wild, otherwise.Proof. The tensor algebra KA m ⊗ K KA l equals the 1-step staircase algebra A ( k l ), thus,the proof follows from Theorem 4.5. (cid:3) The hierarchy of cases is depicted in the Appendix B. Note that the representation typeof the staircase algebra A ((2 k )) has been examined - it is already known by Asashiba;and by Escolar and Hiraoka [12].By Lemma 3.9, Theorem 4.5 yields the following classification via orbit types. Lemma 4.7. A ( λ ) is of finite / tame / wild representation type if and only if its orbitquiver is Dynkin / Euclidean / wild. These results were obtained in the GL n -setup. We think that it should be possible togeneralize them to arbitrary classical Lie types by methods similar to [16]. We have a closer look at the representation theory of staircase algebras now and divideour considerations by representation types, which are known from Theorem 4.5.15 .1 Finite cases
All finite cases are listed in the table, together with their orbit quiver Υ ( λ ) (see Lemma3.9) and the link to their Auslander-Reiten quiver Γ ( λ ) in the Appendix A). In thelatter, the corresponding maximal dimension vectors are marked. Since Γ ( λ ) alwayscontains a sincere representation, each such algebra A ( λ ) is tilted Dynkin and the orbitquiver indicates the frame. By Theorem 2.3 up to isomorphism, every indecomposableappears in Γ ( λ ); such that they can be constructed directly or by means of methodsfrom Auslander-Reiten Theory [1]. This way, a complete representative system ofindecomposable modules is obtained for each finite case. Clearly, all orbits of a fixeddimension vector can be deduced from these by Krull-Remak-Schmidt. By results ofBongartz [6], the explicit calculation of orbit closures can nicely be done. n λ Υ ( λ ) Γ ( λ ) n λ Υ ( λ ) Γ ( λ ) n (1 n − k , k ) A n - n (1 n − , )(2 , n − D n A.2 7 (1 , , , , E A.78 (1 , , , , E A.8 7 (3 , , ) E A.18 (3 , , ) E A.6 7 (1 , )(2 , E A.58 (1 , )(2 , E A.9 6 (1 , , E A.38 (2 ) , (4 ) E A.10 6 (2 ) , (3 ) E A.4
Given a tame concealed algebra A ( λ ), we know that there is an algebra B and a pre-projective B -tilting module T , such that End B ( T ) = A ( λ ). n λ Min. nullroot Tubular type Υ ( λ )8 (1 , , , (1 , ,
11 2 21 2 2 1 (4 , , f E , ,
122 31 2 3 2 (4 , , f E , , (1 , ) (5 , , f E , , , (1 , ,
23 41 2 3 4 5 3 (5 , , f E , , (1 , ) (5 , , f E The table shows every such algebra together with some information. Any tame con-cealed algebra has a unique one-parameter family of indecomposable modules X with16nd( X ) = K , Ext ( X , X ) = K , and the dimension vector of these modules is the minimalpositive nullroot (see [15]). It generates the radical of q λ . The orbit quiver Υ ( λ ) equalsthe frame of the tilted algebra, that is, the Gabriel quiver of B and the tubular type isthe same as the tubular type of B [18]. Each tilting module is given by a direct sumof preprojective indecomposables and there is a stable separating tubular P K -familyof the tubular type of the algebra which separates the preprojective and preinjectivecomponent. In these components, all indecomposables come up (up to isomorphism).The regular component equals the Hom B ( T , _)-translation of the regular component ofthe Euclidean quiver [18]. The simple regular modules are those at the mouths of thetubes; and they have explicitly been calculated for (cid:102) E and (cid:102) E in [10]. Thus, all simpleregular modules and all tubes are obtained. Note that by results of Bongartz [5], orbitclosures can be calculated. The non-maximal non-concealed staircase algebras are corresponding to the partitions λ = (2 , ), λ (cid:48) = (4 ,
5) and λ (cid:48) T = (1 , ). Both are Euclidean tilted by Theorem 2.8 andCorollary 2.7 and their orbit type ( (cid:102) E for λ , (cid:102) E for λ (cid:48) ) equals their frame. The tiltingmodule is given by a direct sum of preprojective indecomposables plus at least onenon-preprojective module, since the algebras are not tame concealed. Each contains atame concealed subcategory and the radical of the quadratic form is always generatedby the induced minimal nullroot, since these algebras are Euclidean tilted and, thus,their quadratic form is isometric to a quadratic form of a quiver. By Lemma 2.8 thequadratic form is non-negative and there is no quiver of non-negative quadratic formwith 2-dimensional radical.Concerning the maximal tame staircase algebras, we prove tameness without using thelist of hypercritical algebras [20], now. We know from Theorem 2.4 that A ( λ ) is tameif and only if the quadratic form q λ is weakly non-negative. Lemma 5.1.
Let λ = (3 ) and λ (cid:48) ∈ { (5 ) , (2 ) } , then q λ and q λ (cid:48) are non-negative and rad q λ = (cid:42) u : = − − − , v : = (cid:43) . rad q λ (cid:48) = (cid:42)(cid:32) −
11 0 − − − (cid:33) , (cid:32) (cid:33)(cid:43) . In particular, A ( λ ) and A ( λ (cid:48) ) are tame and non-tilted.Proof. Let λ (cid:48)(cid:48) : = (2 , ), then A ( λ (cid:48)(cid:48) ) is tilted by Lemma 2.7. It is Euclidean tilted,since its preprojective component is of type (cid:102) E ("contains a (cid:102) E -slice") and contains allprojective indecomposables. By Lemma 2.8, q λ (cid:48)(cid:48) is non-negative.Note that u is orthogonal to every other vector concerning the symmetric form q λ (_ , _);in particular, q λ ( u ) = x ∈ R = R Q ( λ ) , then ˜ x : = x − x u ∈ R (cid:27) R Q ( λ (cid:48)(cid:48) ) and q λ ( x ) = q λ (cid:48)(cid:48) ( ˜ x ) ≥
0. If17 ∈ rad q λ , then (cid:101) x ∈ rad q λ (cid:48)(cid:48) , and, thus, (cid:101) x = ξ · v for some ξ as has been shown before.Since q λ is non-negative, A ( λ ) is tame by Lemma 2.4. If A ( λ ) was tilted, its quadraticform would be isometric to a quadratic form of a quiver. But there is no such non-negative form with 2-dimensional radical.The proof for λ (cid:48) can be done analogously. (cid:3) Tameness of (1 , ) = (2 , T follows from tameness of (5 ,
5) by Lemma 4.1, since q (1 , ) = q (4 , . Note that the radical of the quadratic form q λ for λ = (1 , ,
4) is 1-dimensional, since the algebra A (4 ,
5) is tilted. The orbit type of the algebras A (1 , ), A (2 ,
3) and A (3 ) is (cid:102) E and the orbit type of A (5 ) and A (2 ) is (cid:102) E .By Lemma 2.8 and Lemma 5.1, we have proved that the Tits form of every tame algebra A ( λ ) is non-negative. Lemma 5.2.
Let A ( λ ) be tame, then it has finite growth.Proof. Let A ( λ ) be a tame staircase algebras for λ (cid:60) { (3 ) , (2 ) , (5 ) } . Then rad q λ is1-dimensional whence Proposition 2.6 yields the claim. Let λ ∈ { (3 ) , (2 ) , (5 ) } , thenwe know explicit generators of rad q λ from Lemma 5.1. Clearly, corank q λ ≤
1, suchthat the claim follows from Proposition 2.6, as well. (cid:3)
The minimal wild cases can be found in the graphic in Appendix B. Some of themare hypercritical (see [20]), namely (1 , , , , , , , ,
7) and theirtransposes. Furthermore, all orbit quivers of the minimal wild cases are of extendedEuclidean type (cid:102)(cid:102) E or (cid:102)(cid:102) E .Let n ≥
2, then concrete n -parameter families can be produced nicely. Let A ( λ ) be aminimal wild staircase algebra and let A ( λ (cid:48) ) be a convex subcategory which is tameconcealed and is obtained from A ( λ ) by cancelling a source vertex x in which onearrow α : x → y starts.Let M (cid:48) be a preprojective indecomposable A ( λ (cid:48) )-representation, such that dim K M (cid:48) y = n +
1. Then the A ( λ )-representations defined by M i = M (cid:48) i if i (cid:44) x and M x = K togetherwith the induced maps of M (cid:48) and the embedding M α : K a ... a n + −−−−−−−−→ M x gives a P n -family of pairwise non-isomorphic indecomposables. Remark 5.3.
Explicit -parameter families are induced by these dimension vectors:
65 9 the marked entries show, where an embedding of K leads to a -parameter family).This way, for each wild staircase algebra, a -parameter family can be constructed. We obtain certain results for graded nilpotent pairs, which immediately follow fromour examinations of staircase algebras. Let λ be a partition and let V be a bigradedfinite-dimensional vector space of shape λ . Theorem 6.1.
The number of λ -graded nilpotent pairs (up to Levi-base change) isfinite if and only if1. λ ∈ { ( n ) , (1 k , n − k ) , (2 , n − , (1 n − , ) } for some k ≤ n,2. n ≤ and λ (cid:60) { (1 , , , (2 , ) , (1 , , , (1 , , } .Proof. Follows from Theorem 3.4 and Theorem 4.5: The staircase algebra A ( λ ) hasfinite representation type if and only if the number of isomorphism classes of repre-sentations is finite. The latter correspond to λ -graded nilpotent pairs (up to Levi-basechange) bijectively. (cid:3) If A ( λ ) is tame, then there are only one-parameter families of λ -graded nilpotent pairs(up to Levi-base change). Lemma 6.2.
Assume that λ ∈ { (1 , , , (1 , , , (1 , , , (3 , , (1 , ) , (1 , , , (1 , , , (2 , , (1 , ) } . Then there are only finitely many graded nilpotent pairs onV (modulo base change in the homogeneous components) if and only if dim V does notcontain a minimal nullroot as in 5.2.Proof. If A ( λ ) is tame concealed, then the number of isomorphism classes of a fixeddimension vector d is infinite if and only if d does not contain a minimal nullroot. Thus,the claim follows from Theorem 4.5. (cid:3) Lemma 6.3. If dim V contains a minimal nullroot (see Section 5 for the explicit list),then the number of graded nilpotent pairs on V is (up to isomorphism) infinite.
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A Auslander-Reiten quivers
A.1 The case λ = (3 , &% &% % &% &%→ & %& → % %&→ %& & %& → %& %& %→ &% & %& → %& %& %→ &% & %& → %& %& %→ &% & %& → %& & %→ &% %& → %& & %→ &% %& → %& & % &% %& % A.2 The case λ = (2 , Generalizes easily to λ = (1 , ..., , ,
2) and λ = (2 , n − &% &% % &% &%→ %& %& → % %& %& %&→ & %& %& → % &% %& %→& & %& %& → % &% %& %→& & %& %& → % &% %& %→& & %& %& → % &% %& %& & %& %& % &% % & % % .3 The case λ = (1 , , &% &% %& & &%→ % %& → %& &→ %& % %& → %& &→ %& % %& → %& &→ %& % %& → %& & %& % %& %& A.4 The case λ = (2 , , &% & %& &%→ % & → %& %&→ % %& → %& &→ %& % & → %& %&→ % & → %& %& % & % A.5 The case λ = (1 , , &% &% %& & &%→ % %& → %& &→ %& % %& → %& &→ %& %& %& → %& % &→ %& %& %& → %& % &→ %& %& %& → %& % &→ %& %& & → %& % %&→ %& & → %& % %& %& & % A.6 The case λ = (3 , &% &% % &% &%→ %& %& → %%& & %&→ %& %& %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% &% %& → %& & %& %→ &% & %& → %& %& %→ &% & %& → %& %& %→ &% & %& → %& %& % &% & %& % & % % % .7 The case λ = (1 , , &% &% %& &% &%→ % & %& → %& &→% %& % & %& → %& %&→ %& % & %& → %& %&→ %& % & %& → %& %&→ %& % & %& → %& %&→ %& % & %& → %& %&→ %& % & %& → %& %& %& % & %& %& % A.8 The case λ = (1 , , &% &% %& &% &%→ % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %& %& % %& %& %& & % % A.9 The case λ = (2 , &% &% %& &% &%→ % & %& → %& &→% %& % & %& → %& %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → %& & %&→ %& % %& %& → % & %&→ %& %& %& → % & %& %& % & % A.10 λ = (2 ) or λ = (4 ) &% &% % &% &%→ & %& → % &→% %& & %& → %& %&→ %& % & %& → %& %&→ %& % & %& → %& %&→ %& % %& %& → %& & %&→ %& % & %& → %& %&→ %& % & %& → %& %&→ %& % & %& → %& &% %&→ % & %& → % %& %&→ & %& → % & %& & % Hierarchy of representation types M i n i m a l w il d WILD T a m e n o n - c o n ce a l e d T a m ec o n ce a l e d TAME M ax i m a l fin i t e FINITE N o n - m ax i m a l fin i t e (3 , ,
23) (4 , ,
24) (1 , , , ,
3) (12 , , , ,
4) (2 , , , ,
32) (1 , , , ,
3) (22 , ,
32) (12 , , , , , ,
24) (52)(25)(1 , ,
3) (2 ,
32) (33)(3 , ,
23) (1 , , , ,
3) (1 , , , ,
3) (22 , ,
32) (12 , , , ,
23) (42)(24) (1 , , , ,
3) (22 , , , n − n − ,
22) (1 ,..., , k )(3 , ,
23) (2 , ,
22) (1 , , , ,
3) (22 , , , ,
22) (1 , , , , , , ,
3) (4)(14)(1 , ,
2) (3)(13)(1 ,
2) (2)(12)(1)2) (2)(12)(1)