Piecewise Hereditary algebras of Dynkin and extended Dynkin type
aa r X i v : . [ m a t h . R T ] M a r March 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin [email protected]@unifal-mg.edu.br arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin PIECEWISE HEREDITARY INCIDENCE ALGEBRAS OF DYNKINAND EXTENDED DYNKIN TYPE
EDUARDO DO N. MARCOSDto de Matem´atica, Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua doMat˜ao 1010, Cidade Universit´ariaS˜ao Paulo-SP, CEP 05508-090, BrasilMARCELO MOREIRADto de Matem´atica, Instituto de Ciˆencias Exatas, Universidade Federal de Alfenas, CampusSede, Rua Gabriel Monteiro da Silva 700, CentroAlfenas-MG, CEP 37130-001, BrasilReceived (Day Month 2019)Accepted (Day Month Year)Communicated by [editor]Let K ∆ be the incidence algebra associated with a finite poset ( ∆ , ⪯) over the alge-braically closed field K . We present a study of incidence algebras K ∆ that are piecewisehereditary, which we denominate PHI algebras.We describe the quiver with relations of the PHI algebras of Dynkin type and intro-duce a new family of PHI algebras of extended Dynkin type, which we call ANS family, inreference to Assem, Nehring, and Skowro´nski. In this description, the important methodwas the one of cutting sets on trivial extensions, inspired by this we made a computerprogram which shows exactly the cutting sets on the given trivial extension that resulton incidence algebras.Keywords: incidence algebra; piecewise hereditary algebra; trivial extension.Mathematics Subject Classification 2010: 16D10 16G10 16G20 Introduction
Throughout the paper, K denotes an algebraically closed field. All algebra will befinite dimensional basic associative K -algebra. Using Gabriel’s theorem we will as-sume all algebras to be of the form KQ / I , where Q is a finite quiver and I is anadmissible ideal. All modules will be finite dimensional right modules.Incidence algebras were introduced in the mid-1960s as a natural way of studyingsome combinatorial problems. In the representation theory of finite dimensionalalgebras, the incidence algebras have been the subject of many investigations (see,for instance, [27], [33], [6] and [2]). We will focus the study on incidence algebras K ∆ associated with a finite poset ∆ over K . We remark that K ∆ is isomorphic arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin to the algebra KQ / I , where the quiver Q is the Hasse diagram and I is the idealgenerated by all commuting relations, i.e. the difference of any pair of parallel pathsare in I .The classification of algebras is one of the most important problems in the rep-resentation of finite-dimensional algebras. In the 1980s, Assem and Happel began toclassify the piecewise hereditary algebras, see [3]. Many mathematicians contributedto the classification of piecewise hereditary algebras such as Keller [25], Fern´andez[14], among others.In the article [32], we consider the incidence algebras that are piecewise hered-itary, which we called PHI algebras, piecewise hereditary incidence algebras . Thepurpose of this paper is to give a description of some classes of the PHI algebras.We describe the PHI algebras through their quivers with relations.Let A and B be an abelian categories. In this paper the notation A ≅ ′ B meansthat A is derived equivalently to B , that is D b (A) and D b (B) are equivalent as tri-angulated categories, we also use the notation D b ( A ) for the category D b ( mod A ) ,where mod A denotes the category of finitely generated modules over a finite dimen-sional algebra A .The paper is organized as follows:Section 2 is devoted to fixing the notation and we briefly recall some definitionsand results. In Section 3, we characterize the PHI algebras of type A n and the PHIalgebras of type ̃ A n . The Sections 4 and 5 are dedicated to the description of PHIalgebras such that K ∆ ≅ ′ KQ where Q = D n and Q = E , respectively.In Sections 6 and 7, we give a presentation by quivers with relations of PHIalgebras such that K ∆ ≅ ′ KQ where Q = E and Q = E , respectively. In Section8, we introduce a family PHI algebras of extended Dynkin type called the ANSfamily, in reference to Assem, Nehring and Skowro´nski. In this description of thePHI algebras, the important method was that of cutting sets in trivial extensions,introduced by Fern´andez and Platzeck, [15], this method inspired the elaboration of acomputer program that produces exactly the cutting sets in the given trivial extensionthat result in incidence algebras.We also give the code of this computer program in the appendix,the code waswritten in java script.This is part of the PhD of the second named author, he thanks CAPES(Brazil),for financial help during this time of the PhD. Preliminaries
In this Section, for the sake of completeness, we will recall some definitions. Thereader should see the references for more detail.We begin with the definition of incidence algebras. There are several equivalentways of defining incidence algebras of finite posets, we give one of them below.
Definition 2.1 (incidence algebra).
Let ( ∆ , ⪯) be a poset with n elements. Theincidence algebra K ∆ is a quotient of the path algebra of the following quiver Q . arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin The set of vertices, Q , is in bijection with the elements of the poset ∆ and the setof arrows Q is defining by declaring that there is an arrow α from a vertex a to avertex b , whenever a ⪯ b and there is no a ⪯ c ⪯ b , with c ≠ a and c ≠ b . Let I be theideal generated by all commutativity relations γ − γ ′ , with γ and γ ′ parallel paths.The incidence algebra K ∆ is KQ / I .The quiver Q of the incidence algebra, in the former definition, is also called theHasse quiver of the poset.We are going to assume always that our incidence algebras are connected, thatis the Hasse quiver is connected.For more details in the subject of incidence algebras we refer to [31] and [11].We want to define next the notion of piecewise hereditary algebras. In order todo this we need to introduce, very briefly, some previous notions.Given an abelian category A we denote D b (A) its bounded derived category,as usual if A is a K -algebra then D b ( A ) denotes the bounded derived category of mod A .An abelian category H is called hereditary if the extension groups Ext n H ( X, Y ) are zero for all n ≥ for any pair of objects X and Y of H . Remark 2.2.
All hereditary categories considered in this paper have splittingidempotents, finite dimension Hom spaces, and tilting object. See below the defini-tion of tilting object.
Definition 2.3 (piecewise hereditary algebra).
We say that A is piecewisehereditary algebra of type H if there exists a hereditary abelian category H , withsplitting idempotents, finite dimension Hom spaces, such that D b (A) is triangle-equivalent to the bounded derived category D b (H) .For more details in the subject of piecewise hereditary algebra we refer to [19],[13], [20], [18], [26], [29], [30], [9], [10], [24], [22], [28] and [1].The definition of tilting modules inspired the definition of tilting object thatfollows: Definition 2.4 (tilting object).
Let H be a hereditary abelian K -category. Anobject T ∈ H is called tilting ifi) Ext H ( T, T ) = , andii) for every X ∈ H the condition Ext H ( T, X ) = = Hom H ( T, X ) implies that X = .Let A piecewise hereditary algebra of type H . It follows from Rickard’s theorem[34], the existence of a tilting object T in D b (H) such that A = End T .Given a sequence p , . . . , p n of positive integers, X ( p , . . . , p n ) , will denote theweighted projective line of type p , . . . , p n , in the sense of [16], and Coh X the cat-egory of coherent sheaves over X ( p , . . . , p n ) . Let Q be a finite, connected quiverwithout oriented cycles and let KQ denote the path algebra of Q . We state one of arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin the most important theorems about piecewise hereditary algebras. Theorem 2.5 (Happel [18]).
Let H be an abelian hereditary connected K -catego-ry with tilting object. Then H is derived equivalent to mod KQ or derived equivalentto Coh X for some weighted projective line X .An algebra A is called a piecewise hereditary algebras of quiver type (or of type Q ) or of sheaf type if A ≅ ′ KQ for some quiver Q or A ≅ ′ Coh X for some weightedprojective line X , respectively.Observe that an algebra can be, at the same time, of quiver and sheaf type.In order to study the PHI algebras of quiver type, it is enough to characterizethe iterated tilted incidence algebras of type Q , since, according to the followingtheorem: Theorem 2.6 (Happel-Rickard-Schofield [20]).
Let A be a finite dimensionalbasic associative K -algebra and Q be a finite quiver with no oriented cycles. Then A is piecewise hereditary of type Q if and only if A is iterated tilted of type Q .We recall that an algebra A is called iterated tilted of type Q if there exists asequence of algebras A = A , A , . . . , A n , where A n is the path algebra Q , and asequence of tilting modules T iA i , for ≤ i < n , such that A i + = End ( T iA i ) , and every A i -indecomposable module M satisfies Hom A i ( T i , M ) = or Ext A i ( T i , M ) = . Type A n and type ̃ A n We will devote this section to the study of PHI gentle algebras.
Definition 3.1 (gentle algebra [7]).
Let A be an algebra with acyclic quiver Q A . The algebra A ≅ KQ A / I is called gentle if the bound quiver ( Q A , I ) has thefollowing properties:i) each point of Q A is the source and the target of at most two arrows;ii) for each arrow α of Q A , there is at most one arrow β and one arrow γ suchthat αβ ∉ I and γα ∉ I ;iii) for each arrow α of Q A , there is at most one arrow δ and one arrow ζ suchthat αδ ∈ I and ζα ∈ I ;iv) the ideal I is generated by paths of length two.The following proposition, is probably well known, and we will use it to give acharacterization of the gentle incidence algebra. Note that to be gentle is preservedby derived equivalence, [40]. Definition 3.2 (bypass).
A bypass in a quiver Q is a pair of parallel paths ( α, γ ) where α is an arrow and γ is a path distinct from α and parallel to it. Remark 3.3.
The quiver of an incidence algebra has no bypass.
Proposition 3.4.
Assume that Q has no bypass and KQ / I ≅ KQ / I ′ , for admissibleideals I and I ′ . If a path γ ∈ I then γ ∈ I ′ . arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin Proof.
Since there is no bypass it follows that any isomorphism of KQ / I ≅ KQ / I ′ takes the class of arrow α + I to λ α α + I ′ , where λ α is in K ∗ . It follows that theisomorphism takes the class of a path γ + I to λ γ γ + I ′ where λ γ is in K ∗ . Thereforeif γ ∈ I then γ ∈ I ′ .This has the following corollary, which characterizes the gentle incidence alge-bras. Corollary 3.5.
If the incidence algebra K ∆ ≅ KQ / I is gentle, then K ∆ is hered-itary. Proof.
The quiver associated with an incidence algebra has no bypass and a gentlealgebra have a presentation with ideal generating by quadratic monomial relations.Then by the proposition 3.4 the generating ideal must be zero.We can state the following result:
Corollary 3.6.
If a PHI algebra K ∆ is gentle, then K ∆ ≅ KQ , where Q = A n or Q = ̃ A n . Proof.
From the previous proposition, we conclude that K ∆ is hereditary, that is, K ∆ is isomorphic to KQ . This implies that KQ is gentle, because K ∆ is a gentlealgebra, by hypothesis.We assume that there exists a subquiver of Q of the form: ●● ● ● with the edges oriented respecting the first condition of the definition of gentle alge-bra. We will analyze a case of orientation of these edges, the other cases are similar.Let ●● ● ● α θ β By the fourth condition of the definition of gentle algebra, the ideal I is generatedby paths of length two. In the subquiver above, we have the paths βα and βθ both oflength two. Thus, we have βα ∈ I or βθ ∈ I . Since the algebra KQ has the ideal I empty, this case does not happen.So we only have the two options below: arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin ● ● ⋯ ● ● ou ● ⋯ ●● ●● ⋯ ● We conclude that Q A = A n or Q A = ̃ A n .These two first results and the fact that the property of algebra is gentle is in-variant by derived equivalence, see [40], we get these corollaries. Corollary 3.7. If K ∆ is a PHI algebra of type A n , then K ∆ ≅ KQ , where Q = A n . Corollary 3.8. If K ∆ is a PHI algebra of type ̃ A n , then K ∆ ≅ KQ , where Q = ̃ A n . Type D n The characterization of the piecewise hereditary algebras of type D n was done byKeller in the article [25]. In her thesis [14], Elsa Fern´andez obtained the samecharacterization through an alternative tool, the concept of trivial extension. Wewill rewrite the Fern´andez’s theorem for the case of PHI algebras. Theorem 4.1.
Let K ∆ ≅ KQ / I be a PHI algebra of type D n such that Q is not oftree type. Then K ∆ or K ∆ op is isomorphic to one of the following algebras:(1) ∗ ●● ● ∗ ∗ ⋯ ∗ ∗ (2) ●● ● ∗ ∗ ⋯ ∗ ∗ (3) ● ∗ ● ● ∗ ● (4) ●● ● ∗ ● (5) ● ∗ ● ●● The notation ∗ , in the picture above, means that at the vertices of the graph wecan attach a diagram of the form ● ● ● ⋯ ● ● with no new relations. Type E The first effort to classify piecewise hereditary algebras of type E was made byHappel, in [17]. Years later, in her thesis [14], Elsa Fern´andez obtained some updates arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin in relation to this classification, determining a complete description for this classof algebras. This problem was also treated computationally by the Roggon’s project[39]. We rewrite the Fern´andez’s theorem restricted to the PHI algebras of type E . Theorem 5.1.
Let K ∆ ≅ KQ / I be a PHI algebra of type E such that Q is not oftree type. Then K ∆ or K ∆ op is isomorphic to one of the following algebras:(1) ● ●● ●● ● (2) ●● ●● ●● (3) ●●● ●●● (4) ●● ● ●●● (5) ● ● ●● ● ● (6) ● ● ●● ● ● (7) ● ●● ● ●● In the diagrams when there is a non oriented edge we can have two quivers, onefor each orientation. Type E The class of iterated tilted algebras of type E has an important relation with a classof representation-finite trivial extensions. Fern´andez, in her thesis [14], describesin detail these trivial extensions. We recall the definition of trivial extension of analgebra. Definition 6.1 (trivial extension).
Let A be an K -algebra. The trivial extensionof A is the algebra T ( A ) = A ⋉ D ( A ) , where D ( A ) = Hom K ( A, K ) , whose underlyingof K -vector space is A × D ( A ) with multiplication given by: ( a, f )( b, g ) = ( ab, ag + f b ) , for any a, b ∈ A and f, g ∈ D ( A ) . In this work, we will consider the trivial extensions which are finite representa-tion type. The trivial extension is a symmetric algebra. The self injective algebras offinite representation type are divided into classes, called Cartan classes. The
Cartanclass of an algebra B self injective of finite representation type was introduced byRiedtmann in [36]. More details, we refer the reader to [36], [35] and [37].The Cartan class is defined, briefly, as follows:Consider ( Γ , τ ) the stable Auslander-Reiten quiver of the algebra B , which is selfinjective of finite representation type. Riedtmann showed that ( Γ , τ ) is isomorphic arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin to Z Q / G for an automorphism G of the translation quiver Z Q , where the G -actionin Z Q is admissible that is each orbit of G finds { x } ∪ x − in at most one vertex andfinds { x } ∪ x + in at most one vertex for each x of ( Z Q ) , she also showed that Q is of Dynkin type. The Dynkin type of Q is uniquely determined and it is called theCartan class of B .In this next result, we will have an association of trivial extensions finite repre-sentation type with iterated tilted algebras. Theorem 6.2 (Assem-Happel-Rold´an [4]).
Let A be an algebra. The followingconditions are equivalent:a) T ( A ) is representation-finite of Cartan class Q ;b) A is iterated tilted algebra of Dynkin type Q .Recall that an algebra A is schurian if the following condition is satisfied: dim K ( Hom A ( P, P ′ )) ≤ for any indecomposable projectives P, P ′ .The focus of the study is the quiver with relations of the trivial extension T ( A ) of a schurian algebra A ≅ KQ A / I . Note that all incidence algebras are schurian.We define now the notion of maximal path in an algebra KQ / I with I admissible.We remark that when we say that a path is in an algebra of the form KQ / I we meanthat it is a path in Q whose class is not zero in KQ / I . Definition 6.3 (maximal path).
A path γ in KQ A / I will be called maximal if γ ≠ and αγ = = γα for every arrow α of ( Q A ) .The quiver of T ( A ) is described in the following theorem. Theorem 6.4 (Fern´andez [14]). If A ≅ KQ A / I is a schurian algebra, then thequiver of T ( A ) is given by:a) ( Q T ( A ) ) = ( Q A ) ,b) ( Q T ( A ) ) = ( Q A ) ∪ { β γ , . . . , β γ t } , where { γ , . . . , γ t } is a maximal set oflinearly independent maximal paths. Moreover, for each i , β γ i is an arrow such that s ( β γ i ) = t ( γ i ) and t ( β γ i ) = s ( γ i ) .It is necessary to describe the relations of the presentation associated with T ( A ) ,for this we will need some definitions. Definition 6.5 (elementary cycle).
Let M = { γ , . . . , γ t } be a maximal set oflinearly independent maximal paths. We say that a oriented cycle C of Q T ( A ) iselementary if C = β γ i α α . . . α n , where γ i ∈ M and α . . . α n = kγ i , for some k ∈ K ∗ .Note that the length of elementary cycle C is at least two.We will say that the supplement of γ in C is the trivial path e s ( γ ) , if s ( γ ) = t ( γ ) ,otherwise the supplement of γ is the path in C whose origin is the terminus of γ and it has terminus the origin of γ . Theorem 6.6 (Fern´andez [14]).
Let A ≅ KQ A / I A be a schurian triangularalgebra such that parallel paths in Q A are equal in A . Then T ( A ) = KQ T ( A ) / I T ( A ) where the admissible ideal I T ( A ) is generated by: arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin a) the paths consisting of n + arrows containing all arrows of an elementarycycle of length n ,b) the paths whose arrows does not belong to a unique elementary cycle,c) the difference γ − γ ′ of paths γ, γ ′ with the same origin and the same endpointand having a common supplement in elementary cycles of Q T ( A ) .Note that every path γ in T ( A ) is contained in an elementary cycle C .For future references in the text, the relations defined by items a), b) and c) arecalled relations of type 1, 2, and 3, respectively. Observe that the relations of type 1and 2 are monomial.We will use an important procedure on a trivial extension T ( A ) . The two defi-nitions below are required in order to give a description of this method. Definition 6.7 (cutting set [15]).
Let T ( A ) ≅ KQ T ( A ) / I T ( A ) be a trivial ex-tension and let Σ be a set of arrows of Q T ( A ) . We say that Σ is a cutting set if itconsists of exactly one arrow in each elementary cycle of T ( A ) . Definition 6.8 ([15]).
Let T ( A ) ≅ KQ T ( A ) / I T ( A ) be a trivial extension given bya quiver KQ T ( A ) and an admissible ideal I T ( A ) . We say that A ′ is defined by thecutting set Σ of T ( A ) , if A ′ is isomorphic to KQ T ( A ) / < I T ( A ) ∪ Σ > .Looking at the Theorem 6.6 and the definition of the cutting set, we see that therelations of type , are eliminate with the cutting and they have no influence onthe relations of the ideal which define A ′ . Therefore, in the study of these algebras,we will pay attention only relations of type and type . We can now state thefollowing theorem of Fern´andez. Proposition 6.9 (Fern´andez [14]).
Let A be a schurian triangular algebra. Thenan algebra A ′ is defined by a cutting set of T ( A ) if and only if T ( A ) ≅ T ( A ′ ) .Let Γ be a representation-finite trivial extension of Cartan class E . All algebras A defined by a cutting set of Γ have their trivial extensions isomorphic to Γ . There-fore, the trivial extension of A is representation-finite of Cartan class E , implyingthat the algebra A is iterated tilted of type E by Theorem 6.2.We want to classify all PHI algebras such that its trivial extension is a givenselfinjective finite representation type algebra with Cartan class E In particular wecan assume that A = KQ / I where Q has no bypass and I is generated by the allcommutative relations.We state now the following lemma. Lemma 6.10.
Let A ≅ KQ A / I A be a schurian triangular algebra such that parallelpaths in Q A are equal in A . Let A ′ ≅ KQ ′ A / I ′ be an algebra defined by a cutting setof T ( A ) . If an arrow of the cutting set belongs to two elementary cycles of Q T ( A ) then there is, at least, a commutativity relation in I ′ . Proof. arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin We give a proof only in the case of a trivial extension T ( A ) associated with aquiver which is the union of the vertices and arrows belonging to two elementarycycles, as in the picture. The general case reduces to this case.We use the following notation: C = θ θ . . . θ l α l + . . . α n and C = θ θ . . . θ l α ′ l + . . . α ′ m . ● ⋯ ●● ⋯ ●● ⋯ ● α n α n − α l + θ θ l α l + α ′ l + α ′ m α ′ m − α ′ l + with relations of type and : α ′ m θ . . . θ l α l + ; α n θ . . . θ l α ′ l + α l + . . . α n − α ′ l + . . . α ′ m Now, let Σ = { θ , . . . , θ l } be a cutting set such that θ i belonging to the twoelementary cycles C and C .Therefore the presentation of A ′ has only the relation α l + α l + . . . α n − α ′ l + α ′ l + . . . α ′ m , as desired.A question arises on the existence of a cutting set of trivial extension that definesan incidence algebra. We come across various forms of trivial extension diagrams,and we can find some particular patterns in the cutting sets that define incidencealgebras. We join these patterns in some lemmas that we will call cutting lemmas.Before stating these lemmas, we will introduce some necessary concepts.For each vertex h of quiver Q T ( A ) , let C h be the set of all elementary cycles C ,with origin and terminus in h . Example 6.11.
Let T ( A ) be a trivial extension whose quiver Q T ( A ) is the follow-ing:arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin ●● ... ● . .. ● ⋱● ⋯ ● h ● ⋯ ●⋱ ● ... ● ... ●● θ δ θ λ m δ θ n λ δ n µ µ m β ′ αβ α ′ γ m γ ρ n η σ n ρ η m σ ρ σ assume that C h = { C , C , C , C } . Then there are only two cutting sets definingtwo incidence algebras, which are { α, β } and { α ′ , β ‘ } . Proof.
We use the notation λ = λ . . . λ m , θ = θ . . . θ n , µ = µ . . . µ m , η = η . . . η m , ρ = ρ . . . ρ n , δ = δ . . . δ n , γ = γ . . . γ m and σ = σ . . . σ n .The relations of type and are: αλθ − βηρ ; θµβ ′ − δγα ′ ; αλδ − βησ ; ρµβ ′ − σγα ′ ; θ n µβ ′ β ; ρ n µβ ′ α ; δ n γα ′ β ; σ n γα ′ α ; β ′ αλδ ; α ′ αλθ ; β ′ βησ ; α ′ βηρ We consider the cutting set with the arrows α and β . This cutting set define thequiver with commutativity relations θµβ ′ − δγα ′ and ρµβ ′ − σγα ′ .The second incidence algebra with is defined by cutting set { α ′ , β ′ } , is isomorphicto quiver with commutativity relations αλθ − βηρ and αλδ − βησ .In the section 8, we present two more cutting lemmas 8.9 and 8.6.These lemmas inspired the search for a general algorithm for finding good cut-ting sets. We have been able to go one step further, in a computational way, weelaborate a program that shows exactly the cutting sets of given trivial extensionthat define incidence algebras. In addition, we implemented this algorithm in the arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin site . In the appendix, we present the sourcecode of the program. We briefly describe the ideas used in the program.The first step is to name the arrows of quiver associated with the given trivialextension. We start labeling by α , α , α , . . . the arrows which are in the supportof relations of type . The other arrows are named arbitrarily.In the next step we put the necessary information from the trivial extension intothe program. These initial data are: ● number of relations of type ; ● number of elementary cycles; ● number of different arrows in these relations of type .Then the program displays a table with checkboxes. In this table, the columns rep-resent the relations of type and the elementary cycles, respectively, and the linesrepresent the arrows involved in these elements. The user clicks on checkbox if thearrow is in the relation(s) or if it belongs to the cycle(s), and leaves unchecked oth-erwise. Press the “Ready!” button, and then the solution is shown. The solution isthe cutting sets that define incident algebras.Fern´andez’s thesis [14] showed all the representation-finite trivial extension ofCartan class E , totalling . Thus, for each trivial extension of the list E , we lookfor the cutting sets that define incidence algebras.Using the program we can see the following: Theorem 6.12.
Let K ∆ ≅ KQ / I be a PHI algebra of type E such that Q is notof tree type. Then K ∆ or K ∆ op is isomorphic to one of the following algebras:(1) ● ●● ●● ● ● (2) ●● ● ●● ● ● (3) ●● ● ●● ● ● (4) ●● ●● ● ● ● (5) ●● ● ●● ● ● (6) ●● ● ●● ● ● (7) ● ● ●● ●● ● (8) ● ● ●●● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (9) ● ● ●● ● ●● (10) ●● ● ●● ● ● (11) ●● ● ●●● ● (12) ● ● ●● ●● ● (13) ● ● ●●● ●● (14) ● ● ●● ● ● ● (15) ● ● ●●●● ● (16) ●● ● ●● ● ● (17) ● ● ●●● ● ● (18) ● ●● ● ●● ● (19) ● ● ●● ●● ● (20) ● ●● ● ●● ● (21) ● ●● ● ● ●● (22) ● ●● ●● ●● (23) ● ●● ● ● ●● (24) ● ●● ● ● ●● (25) ● ●● ● ●● ● (26) ● ●●●● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (27) ● ●● ● ● ●● (28) ● ●● ● ●● ● (29) ● ●● ● ●● ● (30) ●● ● ●● ● ● (31) ●● ● ●● ● ● (32) ●● ● ●● ●● (33) ●● ● ●●● ● (34) ●● ● ●● ● ● (35) ● ●● ● ●● ● (36) ● ●● ● ●● ● (37) ● ●● ● ●● ● (38) ● ●● ● ●● ● (39) ● ●● ● ●● ● (40) ●● ● ●● ● ● (41) ●● ● ●● ● ● (42) ●● ● ●● ● ● (43) ● ● ●● ●● ● (44) ● ●● ●● ●● (45) ● ● ●●● ● ● (46) ● ● ●●● ● ● (47) ● ● ●●● ● ● (48) ● ●● ● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (49) ● ●● ● ●● ● (50) ● ●● ● ●● ● (51) ● ● ●● ● ● ● (52) ●● ● ●● ● ● (53) ●● ● ●● ●● (54) ● ● ●● ● ●● (55) ●● ● ●● ● ● (56) ●● ● ●● ● ● (57) ● ● ●● ● ● ● (58) ● ● ●● ●● ● (59) ● ● ●● ●● ● (60) ● ●●● ●● ● (61) ●● ● ●● ● ● (62) ● ● ●●● ● ● (63) ● ●● ● ●● ● (64) ● ●● ● ●● ● (65) ●● ●● ● ●● (66) ● ●● ● ●● ● (67) ● ● ● ●● ●● (68) ● ● ●●● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (69) ● ● ● ●● ●● (70) ● ● ●● ● ●● (71) ● ● ● ●● ● ● (72) ● ● ● ●● ● ● (73) ● ● ●● ● ● ● Proof.
We put the necessary information for each representation-finite trivial ex-tension of Cartan class E [14] in the program and we show the non-hereditarysolutions. As an example we will describe here this procedure to a particular trivialextension the other cases follows the same pattern. ● ●● ●● ● ● α α α α α α α α The relations of type are: r = α α α and r = α α α . The ele-mentary cycles are: C = α α α α and C = α α α α α . Therefore, theprogram shows us the cutting set thatdefines the solution of the theorem. Type E In the previous section, we explained the results and methods applied in the classi-fication of all PHI algebras of type E . We will repeat the same procedure for thecase E .From the thesis of Fern´andez [14], we have the list of all such trivial extensions,counting . From this, we will present all non-hereditary incidence algebras definedby the cutting sets of each trivial extension, as we did for the case E . We did themanual work of finding the incidence algebras defined by the cutting sets for eachof the trivial extensions, except that in order to write the proof we prefer to use theprogram. Theorem 7.1.
Let K ∆ ≅ KQ / I be a PHI algebra of type E such that Q is not oftree type. Then K ∆ or K ∆ op is isomorphic to one of the following algebras:(1) ● ●● ●● ● ● ● (2) ● ● ●●● ● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (3) ● ●●● ●●●● (4) ●●●●● ●●● (5) ●●●● ●●●● (6) ●●● ●●●●● (7) ●● ●●●●●● (8) ● ● ●● ●● ● ● (9) ● ● ●● ●● ● ● (10) ● ●● ● ●● ● ● (11) ● ●● ● ●● ● ● (12) ● ●● ● ●● ●● (13) ● ●● ● ●● ●● (14) ● ● ● ●●● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (15) ●● ●● ● ●●● (16) ● ● ●●● ● ● ● (17) ●●● ● ●● ●● (18) ● ● ●● ●● ● ● (19) ●●● ● ●●● ● (20) ●● ● ● ●● ● ● (21) ●●● ●● ●● ● (22) ●● ●● ●● ●● (23) ● ● ●●● ●● ● (24) ●● ● ● ● ●● ● (25) ●●●● ●● ●● (26) ●●●●● ● ●● (27) ● ●● ● ●● ● ● (28) ● ● ● ●● ● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (29) ●● ● ●● ● ●● (30) ● ●● ● ●● ● ● (31) ●● ● ●●● ● ● (32) ●● ● ●●● ● ● (33) ● ● ●● ●● ● ● (34) ● ● ●● ●● ● ● (35) ●●● ●● ● ●● (36) ●● ● ●● ●● ● (37) ●● ●● ● ●●● (38) ●● ●● ●● ●● (39) ●● ● ●● ●●● (40) ●● ●●●● ●● (41) ●●●● ● ●●● (42) ●● ●● ●● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (43) ●● ●●● ●●● (44) ●●● ●●● ●● (45) ●●● ● ●●●● (46) ●●●● ●● ●● (47) ●● ●● ●●●● (48) ●●● ●● ●●● (49) ●●●● ●● ●● (50) ●●●●● ● ●● (51) ●● ● ●● ● ● ● (52) ●● ● ●● ● ● ● (53) ●● ● ●● ●● ● (54) ●● ● ●● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (55) ● ●● ● ●●● ● (56) ● ●● ● ●●● ● (57) ●● ● ●●● ● ● (58) ●● ● ●● ● ● ● (59) ● ●● ● ●● ● ● (60) ● ●● ● ●●● ● (61) ●● ● ● ●● ● ● (62) ●● ● ● ●● ● ● (63) ●●● ● ●● ●● (64) ● ● ●●● ● ● ● (65) ● ●● ● ●● ● ● (66) ● ●● ● ●● ● ● (67) ●● ● ●● ● ●● (68) ● ● ● ●● ● ● ● (69) ●● ● ● ●● ● ● (70) ●● ● ●● ● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (71) ● ●● ● ●● ●● (72) ●● ●● ●● ●● (73) ●● ●● ● ●● ● (74) ●● ● ●●● ●● (75) ●● ●● ●● ●● (76) ● ●● ●● ● ●● (77) ●●● ● ●●● ● (78) ● ●● ● ●● ●● (79) ●● ●● ●● ●● (80) ● ●● ● ●● ●● (81) ●● ●● ● ●● ● (82) ● ● ●● ● ●● ● (83) ●● ●● ●● ●● (84) ●● ● ●● ● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (85) ● ●● ●● ● ●● (86) ● ●● ● ●● ● ● (87) ● ●● ● ●● ● ● (88) ● ●● ● ●● ● ● (89) ● ●● ● ●● ● ● (90) ● ●● ● ● ●● ● (91) ● ●● ● ● ●● ● (92) ● ● ● ●● ● ● ● (93) ●● ● ●● ●● ● (94) ●●● ●● ●● ● (95) ● ●● ● ● ●● ● (96) ● ● ● ●● ● ● ● (97) ● ● ●● ● ● ● ● (98) ● ● ● ● ●● ● ● (99) ● ●● ● ● ●● ● (100) ● ●● ● ●● ● ● (101) ● ●● ● ●● ● ● (102) ● ●● ● ● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (103) ● ●● ● ●● ● ● (104) ●● ● ● ●● ● ● (105) ●● ● ●● ●● ● (106) ● ● ●● ●● ● ● (107) ● ● ●● ●● ● ● (108) ● ● ●● ●● ● ● (109) ● ●●● ● ●● ● (110) ● ●●● ● ●● ● (111) ● ● ●● ● ●● ● (112) ● ● ●● ● ●● ● (113) ●● ●● ● ●● ● (114) ●● ●● ● ●● ● (115) ● ●● ● ●● ● ● (116) ● ●● ● ●● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (117) ● ●● ● ●● ● ● (118) ● ●● ● ●● ●● (119) ● ●● ● ●● ●● (120) ● ●● ● ●● ●● (121) ● ●●● ● ●● ● (122) ● ● ●● ● ●● ● (123) ● ●●● ● ●● ● (124) ●● ● ●● ●● ● (125) ●● ● ●● ●● ● (126) ●● ● ●● ●● ● (127) ● ●● ● ●● ● ● (128) ● ●● ● ●● ● ● (129) ● ●● ● ●● ● ● (130) ●● ● ●● ● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (131) ●● ● ●● ● ●● (132) ● ●● ● ●● ● ● (133) ● ●● ● ●● ● ● (134) ●● ● ●● ● ●● (135) ● ●●● ● ●● ● (136) ● ●●● ● ●● ● (137) ● ●● ● ●● ●● (138) ● ● ●● ● ●● ● (139) ● ● ●● ●● ● ● (140) ● ●● ● ●● ● ● (141) ● ●● ●● ●● ● (142) ● ●● ●● ●● ● (143) ● ●● ●● ●● ● (144) ● ●●● ● ●● ● (145) ● ●●● ● ●● ● (146) ● ●●● ● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (147) ● ●●● ● ●● ● (148) ●●● ● ● ●● ● (149) ●● ● ●● ●● ● (150) ●● ● ●● ●● ● (151) ● ●● ● ●● ●● (152) ● ● ● ●● ● ● ● (153) ● ● ●●● ● ● ● (154) ● ● ● ●● ● ● ● (155) ● ●●● ● ● ● ● (156) ● ● ● ●● ● ● ● (157) ● ● ●●● ● ● ● (158) ● ● ●●● ● ● ● (159) ● ● ●● ● ● ● ● (160) ● ● ●●● ● ● ● (161) ● ● ●●● ● ● ● (162) ● ● ●●● ● ● ● (163) ● ● ●●● ● ● ● (164) ● ● ●●● ● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (165) ● ● ● ●● ● ● ● (166) ● ● ●●● ● ● ● (167) ●● ●● ●● ● ● (168) ●● ●● ●● ● ● (169) ●● ●● ●● ● ● (170) ● ● ●● ●● ● ● (171) ● ● ●● ●● ● ● (172) ● ● ●● ●● ● ● (173) ●● ● ●●● ● ● (174) ●● ● ●● ● ● ● (175) ●● ● ●●● ● ● (176) ● ●● ● ●● ● ● (177) ● ●● ● ●●● ● (178) ● ●● ●● ● ● ● (179) ● ●● ●● ●● ● (180) ● ●● ● ●● ● ● (181) ● ●● ● ●●● ● (182) ●● ● ● ●● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (183) ●● ● ●●● ●● (184) ● ●● ● ●● ● ● (185) ● ●● ● ●● ● ● (186) ● ●● ● ●● ● ● (187) ● ● ●● ● ●● ● (188) ● ● ●● ● ●● ● (189) ● ● ●● ● ●● ● (190) ● ●● ● ●● ● ● (191) ● ●● ● ●● ● ● (192) ● ●● ● ●● ● ● (193) ● ● ●●● ●● ● (194) ● ●● ●● ●● ● (195) ● ● ●●● ●● ● (196) ● ●● ● ●● ● ● (197) ●● ● ● ●● ● ● (198) ●●● ● ●● ● ● (199) ● ● ●● ●● ● ● (200) ●● ● ●●● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (201) ●● ● ●● ● ●● (202) ●● ● ●●● ● ● (203) ●● ● ●● ● ●● (204) ●● ● ●●● ● ● (205) ●● ● ●● ● ● ● (206) ●● ● ● ●● ● ● (207) ● ●● ● ●● ● ● (208) ● ● ●●● ● ● ● (209) ●● ● ●●●● ● (210) ● ● ●●● ● ● ● (211) ●● ● ●● ● ●● (212) ● ● ●● ● ●● ● (213) ●● ● ●●● ● ● (214) ● ●● ● ● ●● ● (215) ● ● ●● ● ●● ● (216) ● ● ●●● ● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (217) ●● ● ●● ● ● ● (218) ● ● ●● ● ● ● ● (219) ● ●●● ● ● ● ● (220) ● ● ●● ● ● ●● (221) ● ●● ●● ● ● ● (222) ● ●● ● ●● ● ● (223) ● ● ● ●● ● ● ● (224) ● ● ●●● ● ● ● (225) ● ● ●●● ● ● ● (226) ● ● ● ●● ● ● ● (227) ● ● ●● ● ●● ● (228) ● ●● ● ●● ●● (229) ●●● ● ●● ● ● (230) ●● ● ●● ● ●● (231) ●● ● ● ●● ● ● (232) ●● ● ●●● ● ● (233) ● ● ●● ● ●● ● (234) ●● ● ● ●● ● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (235) ●● ● ●● ● ●● (236) ● ● ●●● ● ● ● (237) ● ●● ●● ● ● ● (238) ● ● ● ●●● ● ● (239) ●●● ● ●● ● ● (240) ●● ● ● ●● ● ● (241) ● ● ● ● ●● ● ● (242) ● ● ●● ● ● ● ● Proof.
The scheme of the demonstration consists in analyze each representation-finite trivial extension of Cartan class E [14], displaying its relations of type andits elementary cycles. With this information, we apply in our computer programand we have the cutting sets that define the incidence algebras. Thus, we show inthe list of our theorem only the non-hereditary incidence algebras. We will show thisprocedure to a trivial extension and the remaining work is analogous.The following is a particular example: ● ●● ● ● ●● ● α α α α α α α α α α α The relations of type are: r = α α α α α , r = α α α α α , r = α α α α , r = α α α α , r = α α α , r = α α α , r = α α α and r = α α .The elementary cycles are: C = α α α α α , C = α α α α α , C = α α α α and C = α α α .Therefore, the program shows us thecutting set that defines the solutions and of the theorem. Type ̃ D n , type ̃ E , type ̃ E , type ̃ E The PHI algebras of extended Dynkin type are derived equivalent to the category ofcoherent sheaves of certain weighted projective line. This category is derived equiv-alent to the category of modules over canonical algebras, see [23], [16], [38]. arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin We do not have a complete description of the PHI algebras of extended Dynkintype as in the Dynkin type, but we have been able to identify some families. Oneof them is the PHI concealed algebras of extended Dynkin type. We can identifythrough the works of Happel-Vossieck [21] and Bongartz [12].The other family of PHI algebras of extended Dynkin type are the membersof a new set which we call the
ANS family , in reference to Assem, Nehring andSkowro´nski.The algebras which belong to this family are the algebras which are obtained bya cutting of the trivial extensions of concealed algebras of type Q where Q = ̃ D n forsome n , or Q = ̃ E p for some p .In the Tsukuba Journal of Mathematics [5], Assem, Nehring and Skowro´nskipublished the work entitled “Domestic trivial extensions of simply connected alge-bras”. The main result of this article has an important role in the characterizationof the ANS family. Theorem 8.1 (Assem-Nehring-Skowro´nski).
Let A be a finite-dimensional,basic and connected algebra over an algebraically closed field K . If A is simplyconnected, then the following conditions are equivalent:(1) A is an iterated tilted algebra of Euclidean type Q , where Q = ̃ D n for some n , or Q = ̃ E p for some p .(2) There exists a representation-infinite tilted algebra B of the type Q suchthat T ( A ) ≅ T ( B ) . Proposition 8.2.
The theorem above is valid without the hypothesis of A simplyconnected. Proof.
To show that item 1 implies item 2 we observe that since A is iterated tiltedof type Q it is automatically simply connected [32] and we get the validity of item2. The implication of item 2 to item 1 is done as follows:The algebra B is tilted of type Q so it is simply connected, change in the item 1the role of A by B . Since B is simply connected the first two items are equivalent,with the simply connected hypothesis, changing A for B so the validity of item 2implies that item 1 is valid for A and also that A is simply connected.We will describe the algebras B which are the members of the ANS family.We will work with the concealed algebras A of Euclidean type ̃ D m , ̃ E , ̃ E ou ̃ E .Therefore, the difficulty is to obtain the PHI algebras K ∆ such that T ( K ∆ ) ≅ T ( A ) .We will use the list of Happel-Vossieck [21] to work with all the concealed algebras A of Euclidean type, so the algebras B come from the list of Happel-Vossieck.The algebras which we consider are the trivial extension of schurian algebras.Thus, from the list of Happel-Vossieck [21] we will explore only the schurian algebras.In turn, we observed that the schurian concealed algebras A of Euclidean type fit arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin perfectly in the strategy used in the full description of PHI algebras of Dynkin type.The main difference is that we do not have a list of the trivial extensions of A .Therefore, the machinery to find PHI algebra K ∆ will have one more step thanthat which was done in the previous sections. We will describe the quivers of thetrivial extensions of A . An important observation is that we will not study the trivialextensions of A op since the trivial extension of A op is isomorphic to T ( A ) op . Lemma 8.3.
Let A be a finite dimensional algebra. We denote the trivial extensionof A by T ( A ) . We have T ( A ) op ≅ T ( A op ) . Proof.
We consider the morphism Φ ∶ T ( A ) op Ð → T ( A op )( a, f ) z→ ( a, f ) The reader can check that this is an isomorphism of algebras.Let A schurian concealed algebra Euclidean type ̃ D m , ̃ E , ̃ E or ̃ E , we will dothe following: ● compute the quiver of trivial extension A ; ● describe the elementary cycles of Q T ( A ) ; ● describe the relations of type in presentation of T ( A ) ; ● identifies the arrows in these relations of type ; ● use our computer program; ● if there is a non-hereditary solution, then we need to verify if the cutting setdefines a PHI algebra K ∆ that is not a concealed algebra of the Euclideantype.This will be the script of the proof of the characterization theorems of the PHIalgebras of ANS family.The chapter XIV of the book “Elements of the Representation of AssociativeAlgebras” [41] gives a list of all frames of the list of Happel-Vossieck [21]. We willuse this list to write the series of results for the PHI algebras of type ̃ D m , ̃ E , ̃ E and ̃ E .We will start the work of describing the PHI algebras of the ANS family of type ̃ D m . First, we’ll start with members of type ̃ D . Theorem 8.4.
The algebras associated with the quivers with relations below:(1) ●● ● ●● (2) ● ●● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (3) ● ●●● ● are PHI algebras of the ANS family of type ̃ D . Proof.
Given an algebra coming from an admissible operation on a frame, see [41],we will use the trivial extension of that algebra and extract the necessary informationto use our computer program. We will do this for each admissible operation in eachframe. Since we are only interested in type ̃ D , the only frame that satisfies thiscondition is F r .Observe that we are not interested in the algebras whose trivial extensions onlyhave cutting sets that define hereditary PHI algebra. This will be examined morecarefully in frame F r . The adjacent quiver ̃ D has possibilities of admissibleoperations , that is, we have Euclidean graphs ̃ D . Now, our goal is to studythe trivial extension of each graph and see if there is any cutting set that defines anon-hereditary incidence algebra. For this, the trivial extension must have at leasttwo elementary cycles that have at least one common arrow in accordance with thelemma 6.10. As an example we will describe here this procedure to a particulartrivial extension the other cases follows the same pattern. ● ●●● ● ● ●●● ● α α α α α α α By symmetry, there are four quivers in this set. The relations of type are: r = α α α , r = α α α , r = α α α , r = α α α , r = α α α and r = α α α .The elementary cycles are: C = α α α , C = α α α and C = α α α . With thisinformation, the program shows us the solution .In general, it is difficult to describe the PHI algebras of type ̃ D m . Thus, we beginwith the type ̃ D and the next type is ̃ D . Theorem 8.5.
The PHI algebras of the ANS family of type ̃ D are described, below, as quiverand relations.(4) ●● ●● ●● (5) ● ●● ●●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (6) ●● ● ●●● (7) ● ●●● ●● (8) ● ●●● ●● Proof.
We proceed as in the former theorem, now we are interested in the type ̃ D .The frame that satisfies this condition is F r , see [41]. The adjacent quiver ̃ D has of possibilities of admissible operations , that is, we have Euclidean graphs ̃ D . As before, to exemplify the work that has to be done. we will show this procedurein a particular case of trivial extension. ● ●● ●● ● ● ●● ●● ● ● ●● ●● ● α α α α α α α α α The relations of type are: r = α α α α , r = α α α α , r = α α α α , r = α α α α , r = α α α α , r = α α α α , r = α α α α and r = α α α α .The elementary cycles are: C = α α α α , C = α α α α , C = α α α α e C = α α α α . With this information, the program shows us the solutions , andthe opposit algebra of the algebra in item , above.After describing above the PHI algebras of ANS family of type ̃ D and ̃ D , wewere able to generalize the procedure and we give a description of the PHI algebrasof ANS family of type ̃ D n for n ≥ .The following lemma is used several times in proof of the theorem describing thePHI algebras of ANS family of type ̃ D n . Lemma 8.6.
Given the trivial extension Γ of the hereditary algebra of type A n ,with at least two elementary cycles, we consider its diagram below, where β i and β i + have opposit order for every i . Γ ∶ ● ● ● ∗ ∗ ∗ ● ● ● α α β β n α n α n + We will use the dashed arrow to represent a path which may occur in the quiver. arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin Moreover, the symbol ∗ ∗ ∗ in the quiver represents the possibility of attaching sub-quivers of the form ● ● ● or of the opposit form.i) There are cutting sets Σ of Γ such that KQ Γ / < I Γ ∪ Σ > is an hereditaryalgebra whose quiver has underlying graph an A n .ii) We also assume that there is an elementary cycle with three arrows or morein Γ .If α ∈ Σ or α n ∈ Σ , then KQ Γ / < I Γ ∪ Σ > is not an incidence algebra. If α ∈ Σ or α n + ∈ Σ , then KQ Γ / < I Γ ∪ Σ > is an incidence algebra with a quiver whoseunderlying graph is a A n . Proof.
The aim is to find a cutting set Σ of Γ where KQ Γ / < I Γ ∪ Σ > is anincidence algebra. First, let us assume that there is an elementary cycle with threearrows or more. We will show the existence of Σ by induction in the number ofelementary cycles. For clarity, we will begin the first step of the induction processwith two elementary cycles: ● ● ● ● ● α α β β α α The relations of type are: r = α β ′ , where β ′ is the first arrow of the path β ,and r = α β ′ , where β ′ is the first arrow of the path β α .The elementary cycles are: C = α β α and C = α β α .Without loss of generality, we assume that exists the path β . Also, we startconstructing the cutting set Σ choosing the arrow of the elementary cycle C .We consider α ∈ Σ . Then, eliminating the relation r = α β , we get α ∈ Σ . Sousing Σ = { α , α } we have the incidence algebra: KQ Γ / < I Γ ∪ Σ > ∶ ● ● ● ● ● Let Γ be a trivial extension of the lemma with m + elementary cycles. Wecall the set Σ = { α , α , . . . , α m } for m elementary cycles, where it satisfies thehypothesis of induction. We need to define which arrow of the elementary cycle C m + would complete the cutting set Σ of Γ : ● ● ● ∗ ∗ ∗ ● ● ● ● ● α α β α m β m − β m + α m − α m + α m + The relations of type are: r = α β , r = α β , . . . , rk = α m β m + and r ( k + ) = α m + α m − , for some natural number k . The elementary cycles are: C = β α α , . . . , C m + = β m + α m + α m + .Therefore, by the relation r ( k + ) = α m + α m − , we determine the arrow α m + ∈ Σ . We conclude this part, and the proof is analogous to α n + ∈ Σ .Now, if the trivial extension Γ have only elementary cycles with two arrows: arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin ● ● ∗ ∗ ∗ ● ● ● α α α m α m + α m − α m + The relations of type are: r = α α , r = α α , . . . , rk = α m α m + and r ( k + ) = α m + α m − , for some natural number k . The elementary cycles are: C = α α , . . . , C m + = α m + α m + .The process is analogous to have the cutting set Σ = { α , α , . . . , α m + } that define the incidence algebra. And we can verify that the cutting set Σ = { α , α , . . . , α m + } defines the opposit incidence algebra.The last part is missing. We assume that there is an elementary cycle with threearrows or more in Γ , with α ∈ Σ or α n ∈ Σ . Without loss of generality, we assumethat exists the path β . We consider the subquiver: ● ● ● ● ● α α β β α α .The relations of type are: r = α β and r = α β . The elementary cycles are: C = α β α and C = β α α .By hypothesis, we have α ∈ Σ . Then we have to choose an arrow from theelementary cycle C = β α α . Independently of the choice, we cannot rule out therelation r = α β or r = α β . Therefore, KQ Γ / < I Γ ∪ Σ > is not a incidencealgebra. Similarly, we get the same the result if α n ∈ Σ .Next, we will display the PHI algebras of the ANS family of type ̃ D n . The morecomplicated case in the proof is in the case of the hereditary algebra A of type ̃ D n ,that is when there is no relation. The possible quiver algebras, with a given frameis very large. So we can get a large amount of possible trivial extensions associatedwith a given frame. Theorem 8.7.
The PHI algebras of the ANS family of type ̃ D n , for n ≥ , aredescribed below, via presentations of quivers and relations:(1) ● ... ●● ●● ●● (2) ● ... ●● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (3) ●● ● ●● ... ● (4) ● ●● ⋯ ● ●● ● (5) ● ●● ● ⋯ ●● ● (6) ● ●● ⋯ ● ● ⋯ ●● ● (7) ● ●● ⋯ ● ● ●● ● (8) ● ●● ⋯ ● ● ● ⋯ ●● ● (9) ● ●● ⋯ ● ● ● ⋯ ● ● ● ⋯ ●● ● odd number (10) ● ●● ⋯ ● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (11) ● ●● ● ● ⋯ ● ●● ● odd number (12) ● ... ● ⋯ ●●● ●● There are two cases in the substitution of subquiver ●● ⋯ ● ● .If we have an even number of edges then we replace by ●● ● ⋯ ● ● ● otherwise, we replace by ●● ● ⋯ ● ● ●
Proof.
We describe the script of the proof for each PHI algebra of the ANS familyof type ̃ D n for n ≥ : ● we consider each schurian concealed algebra A of type ̃ D n ; ● we describe the trivial extensions of A ; ● we analyze the cutting sets that define an incidence algebra K ∆ ; ● we use the theorem 8.1, in order to proof that K ∆ is a PHI algebra of type ̃ D n .We begin with the hereditary algebra A of type ̃ D n , with n ≥ . First, we collectthe configurations of the elementary cycles of the trivial extensions of A of type ̃ D and ̃ D . arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin Next we use these configurations in the trivial extensions of the hereditary algebra A , of type ̃ D n .We list one trivial extension for each par of algebras ( A, A op ) . The label F r isthe same as the one on the list giving in chapter XIV of the book “Elements of therepresentation theory of associative algebras” [41].The dashed arrow symbolizes an oriented path of whose length depends on n .We will show in detail two cases, the other cases can be obtained in an analogousway. ● ●● ● ● ●● ● α α α α β α α α α α α The relations of type are: r = α α βα α α , r = α α βα α α , r = α α α βα α , r = α α α βα α , r = α α α βα α , r = α α α βα α , r = α α βα α α and r = α α βα α α .The elementary cycles are: C = α α α α βα , C = α α α α βα , C = α α α α βα and C = α α α α βα .As in the previous theorems referring to the type ̃ D and ̃ D , there are fourcutting sets defining the PHI algebras and , and their opposit algebras. The PHIalgebras and are defined from the cutting sets { α } e { α , α } , respectively.The dashed arrows represent a possible oriented path to be added to the quiver.Moreover, the symbol ∗ ∗ ∗ in the quiver represents the possibility of placingsubquivers of the form ● ● ● or opposit form. For example, we wouldreplace ∗ ∗ ∗ by ● ● ● ● In the search for non-hereditary PHI algebras, in the trivial extensions, we areusing the lemma 6.10. This result requires that there is an arrow which belongs twoelementary cycles, at least. We will show now the trivial extensions satisfying thefollowing two hypothesis of lemma 6.10: ● there is, at least, one pair of elementary cycles which share an arrow; ● there is, at most, two pairs of elementary cycles, sharing one arrow.We will show the possible configurations, assuming the restriction above: arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin ● ● ● ●● ● ● ● ●● The trivial extension has at least five elementary cycles. In this part of the proof,we will strongly use lemma 8.6. ● ●● ● ● ● ● ∗ ∗ ∗ ●● ● α β ′ α ρ α α ρ β α The relations of type are: r = α β ′ α α , r = α β ′ α α , r = α ρ , r = ρ α , r = ρ α all other relations of type 2 have support on the right part of the quiver,(which we give bellow, in order to make it clear). ●● ● ● ∗ ∗ ∗ ● ● ρ ρ β The elementary cycles are: C = β ′ α α α , C = β ′ α α α and the others arepart of the previous subquiver.Next, we will examine the possible configuration, that is the possible elementarycycles and relations of type 2 in this trivial extension Γ . First we assume that thepath β has length greater than or equal to or that the symbol ∗ ∗ ∗ is replaced bysome elementary cycle with three arrows or more.We will look for a cutting set Σ such that KQ Γ / < I Γ ∪ Σ > is a non-hereditaryincidence algebra.In order to get that it is necessary to have α ∈ Σ . Otherwise, if the path β ′ , (inthe picture), has length at least one, then for any arrow in the path β ′ which is in Σ , we get that α ρ or ρ α belong to the ideal < I Γ ∪ Σ > , so the quotient wouldnot be an incidence algebra. If α ∈ Σ , then ρ ∈ Σ , another time, in this case usinglemma 8.6, we obtain that KQ Γ / < I Γ ∪ Σ > is not an incidence algebra.Next we look at configurations where the path β does not exist, and the symbol ∗∗∗ is replaced only by elementary cycle(s) with two arrows. As seen in the previousparagraph, we need to have α ∈ Σ , implying ρ ∈ Σ . Again we use the lemma 8.6,implying a single cutting set. Consequently, we get a solution.The solution has the quiver which depends on the length of the path β ′ and thenumber of elementary cycles in the middle of the trivial extension Γ . For example, arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin If Γ has the path β ′ with length one and only one elementary cycle in the middle of Γ , we get the following PHI algebra: ● ●● ● ● ●● ● We get two solutions. If the path β ′ , has length bigger or equal 1 then we havethe solution , otherwise we get the solution .Next we will proof two theorems related to the description of the PHI algebrasof the ANS family of type ̃ E and ̃ E . Theorem 8.8.
The PHI algebras of the ANS family of type ̃ E , are described as quiver andrelations, as follows.(1) ●● ●● ●●● (2) ●●● ●● ●● (3) ●●● ●● ●● (4) ●● ●● ●●● (5) ●● ● ●● ● ● (6) ●● ● ●● ●● (7) ● ● ●● ● ●● (8) ● ● ●● ● ●● (9) ● ●● ● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (10) ●● ● ●● ●● (11) ●● ●● ●● ● Proof.
The list has five frames, see [41]: F r , F r , F r , F r and F r . We willanalyze only the schurian frames, satisfying the main hypothesis of the theory ofthe section 6. We will describe the trivial extension of each algebra originated fromeach one of the frames. That is, given a frame F r , we will apply the admissibleoperation and then describe the trivial extension of that algebra. Next we will inputthe necessary information for each trivial extension in the computer program andshow the non-hereditary solutions.We note that the concealed algebra of the type ̃ E is always one of the solutionsand we will omit it. We will show the argument just for F r , the argument for theother algebras which we obtain, is analogous.The adjacent Euclidean quiver ̃ E has of possibilities of admissible operation , that is, we can have Euclidean graphs ̃ E . Now, our goal is to analyze thetrivial extension of each graph and see if there is any cutting set that defines a non-hereditary PHI algebra. For this, in the trivial extension, we must have at least twoelementary cycles that have at least one common arrow, according to lemma 6.10.So we need to have two maximal paths that have at least length two and onearrow in common. Up to duality, there are three cases:(1) ●●● ● ● ● ● (2) ●●● ● ● ● ● (3) ●●● ● ● ● ● Up to symmetry, we will make all combinations of admissible operations fromthe three previous cases.We start with the frame F r . , above. With one possible orientation of theunoriented edge, on the frame, we get the following algebra and its trivial extension. ●●● ● ● ● ● ●●● ● ● ● ● α α α α α α α α The relations of type are: r = α α α α and r = α α α α . arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin The elementary cycles are: C = α α α α α and C = α α α α α .Then, the computer program shows us the cutting sets that define the algebras and of the theorem.Before stating the theorem describing the PHI algebras of the ANS family of type ̃ E , we need the following lemma. Lemma 8.9.
Let Γ be a trivial extension of a schurian algebra with at least threeelementary cycles with the following conditions:a) two elementary cycles have at least one arrow in common, we shall call thepath in the intersection by λ ;b) the third elementary cycle has a vertex in common with one of the two pre-vious elementary cycles and has no intersection with the other;c) the common vertex of the previous condition does not belong to the path λ ;These conditions are ilustrated in the following picture: ●● ● ⋯ ●● ●● ⋯ ● β ′ θ β γ γ n − γ n α γ λα α m − α m Let Σ be a cutting set of Γ . If any arrow of λ belongs to Σ then KQ Γ / < I Γ ∪ Σ > is not an incidence algebra. Proof.
Without loss of generality, we will consider Γ as a trivial extension withthree elementary cycles satisfying the conditions of the statement of the lemma.Thus, the relations of type are: r = γ β , r = β ′ γ , r = α m λγ e r = γ n λα .The elementary cycles are: C = α α . . . α m − α m λ , C = λγ γ . . . γ n − γ n e C = θβ ′ β .Assume that some arrow in the path λ belongs to the cutting set Σ . We will seethat, in this case, independently of the choice of the arrow of the elementary cycle C , the KQ Γ / < I Γ ∪ Σ > is not an incidence algebra.We assume that β ′ ∈ Σ , then the relation r belongs to < I Γ ∪ Σ > . Consequently, KQ Γ / < I Γ ∪ Σ > is not an incidence algebra.Analogously, we arrive at the same result if β ∈ Σ or any arrow of θ belongs tothe cutting set. Theorem 8.10.
The PHI algebras of the ANS family of type ̃ E are described below. arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (1) ●● ●● ●● ●● (2) ●●● ●● ● ● ● (3) ● ●●● ●● ● ● (4) ●●● ●● ●● ● (5) ●●● ●● ●●● (6) ●●●● ●● ●● (7) ●●●●● ●● ● (8) ●●●● ●● ●● (9) ●●●●● ●● ● (10) ●●●●● ●● ● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (11) ● ●● ●●● ●● (12) ●● ● ●● ●●● (13) ●●● ● ●● ●● (14) ●● ● ●● ●●● (15) ●● ● ●● ●●● (16) ●● ●● ●● ● ● (17) ●● ● ●● ●●● (18) ●● ● ●● ●●● (19) ●●● ● ●● ●● (20) ●● ● ●● ●●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (21) ●● ●● ●● ●● (22) ●●● ●● ● ●● (23) ●●● ●● ● ●● (24) ●● ●● ●● ●● (25) ●● ● ●● ●●● (26) ●● ● ●● ●●● (27) ● ● ● ●● ●●● (28) ●● ●●● ● ●● (29) ● ● ● ●● ●●● (30) ●● ●●● ●● ●● (31) ●●● ● ●● ●● (32) ●●● ●● ● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (33) ●●● ●● ● ●● (34) ●●● ● ●● ●● (35) ●● ● ●●● ●● (36) ● ●● ●● ● ●● (37) ● ●● ●● ● ●● (38) ●● ● ●● ● ● ● (39) ●● ● ●● ● ● ● (40) ●● ● ●● ●● ● (41) ● ●●● ●● ●● (42) ●● ● ● ● ●● ● (43) ●● ● ● ● ●● ● (44) ●● ●●● ● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (45) ●●● ● ●● ●● (46) ● ●● ●● ● ●● (47) ●● ●● ● ●● ● (48) ●● ● ●● ● ●● (49) ●● ●●● ● ●● (50) ●● ●● ● ●●● (51) ●● ● ●●● ● ● (52) ●● ●● ● ●●● (53) ●● ● ●● ● ●● (54) ●● ● ●● ●●● (55) ● ● ●● ● ● ●● (56) ● ●●● ● ● ●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (57) ● ●●● ●● ●● (58) ●●●●● ● ●● (59) ●●●●● ● ●● (60) ● ●● ● ● ● ●● (61) ●● ●● ● ●● ● (62) ● ●● ● ● ● ●● (63) ●● ●● ● ●● ● (64) ● ●● ● ●● ●● (65) ●●● ●● ● ●● (66) ●● ● ●● ●●● arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin (67) ●●● ●● ● ●● (68) ●●● ●● ● ●● (69) ● ●●● ● ● ●● (70) ●● ● ● ● ●● ● (71) ●● ● ● ●● ● ● (72) ●● ●● ● ● ●● (73) ●● ● ● ● ●● ● (74) ●● ● ● ● ●● ● (75) ●● ● ● ●● ● ● (76) ●● ● ● ●● ● ● (77) ●● ● ● ●● ● ● (78) ●● ● ● ●● ● ● Proof.
We consider the list of frames of concealed algebras, see [8]. In the framepart ̃ E has frames: F r , . . . , F r . As we explained at the beginning of thissection, we will use only schurian frames. For each frame, we will make the ad-missible operation resulting in a concealed schurian algebra A . Next, we will putthe necessary information on the trivial extension T ( A ) in the computer program.Finally, the members of this ANS family of type ̃ E will be the algebras K ∆ definedby a cutting sets of T ( A ) . We will show the work on the case of F r , the necessarywork on the other cases follows an analogous path.We apply the admissible operation in the Euclidean frame ̃ E , resulting inthe hereditary algebra of type ̃ E , this algebra is a concealed algebra. Thanks to the arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin theorem 8.1, we will analyze the trivial extension of the concealed algebra by notingthe existence of some cutting set which defines a non-hereditary PHI algebra. Forthis, in the trivial extension, we must have at least two elementary cycles that haveat least one common arrow, according to lemma 6.10.Therefore, we must have two maximal paths in hereditary algebras that have atleast length and one arrow in common. Up to dual graphs, the possibilities are: ●∗ ∗ ∗ ● ● ● ● ●● ● ● ● ∗ ∗ ∗●∗ ∗ ∗ ● ∗ ∗ ∗ The symbol ∗ ∗ ∗ can be replaced by ● ● , or ● ● ● , or ● ● ● , or ● ● ● .We will investigate all the trivial extensions of the case 1. We will put aside thetrivial extensions that satisfy the hypothesis of the lemma 8.9:One last remark, the path ● ● has length one or two.Therefore, as a consequence of this filter, we obtain only the trivial extension,on the right of the picture below: ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● α A direct application of the lemma 6.10, we have the cutting set { α } that definesthe solution 1.A work for the future is to apply the theorem 8.1 in frames F r , . . . , F r from the list of Happel and Vossieck [21] with an analogous demonstration of theprevious theorem. This would solve the description of PHI algebras of the ANSfamily of type ̃ E . Appendix A.
The source code of the main page of the site
The cutting sets of given trivial extension thatdefine incidence algebras
Complete the form below to calculate the cutting sets
Appendix B.The source code for app.js function Enviar() {// input variablesflechas = document.getElementById("flechasid");relacoes = document.getElementById("relsid");ciclos = document.getElementById("ciclosid");// previous variables only readingdocument.getElementById(’flechasid’).readOnly = true;document.getElementById(’relsid’).readOnly = true;document.getElementById(’ciclosid’).readOnly = true;// transform these variables into integersvar fls = parseInt(flechas.value, 10);var rels = parseInt(relacoes.value, 10); arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin var cics = parseInt(ciclos.value,10);// verification of incoming informationif (fls > 0 && fls < 30 && cics > 0 && cics < 10 &&rels > 0 && rels < 50) {// preparation for table creationvar cab1 = [];var cab2 = [];var lin = [];for (var i=1; i<=rels; i++){cab1.push(’r’+i);};for (var i=1; i<=cics; i++){cab2.push(’c’+i);};for (var i=1; i<=fls; i++){lin.push(’alfa’+i);};var cab = cab1;for (var i=0; i < cab2.length; i++){cab.push(cab2[i]);};var data = {celulas: cab,idLinhas: lin};var table = document.createElement(’table’);// table headervar tr = document.createElement(’tr’);var th = document.createElement(’th’);tr.appendChild(th);data.celulas.forEach(function (celula) {var th = document.createElement(’th’);th.innerHTML = celula;tr.appendChild(th);});table.appendChild(tr);// bodydata.idLinhas.forEach(function (id) {// create new linevar tr = document.createElement(’tr’);tr.dataset.id = id;// first cell with line namevar td = document.createElement(’td’); arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin td.innerHTML = id;tr.appendChild(td);// scroll array of TDsdata.celulas.forEach(function (celula) {var td = document.createElement(’td’);var input = document.createElement(’input’);input.type = ’checkbox’;input.name = celula + [];td.appendChild(input);tr.appendChild(td);}); table.appendChild(tr);});// add table to documentdocument.body.appendChild(table);// disable submit buttondocument.getElementById("enviarid").disabled = true;// create process buttonvar botao = document.createElement(’input’);botao.type = ’button’;botao.value = ’Ready!’;// processbotao.onclick = function() {// function to add the vectorsArray.prototype.add = function( b ) {var a = this,c = [];if( Object.prototype.toString.call( b ) ===’[object Array]’ ) {if( a.length !== b.length ) {throw "Array lengths do not match.";} else {for( var i = 0; i < a.length; i++ ) {c[ i ] = a[ i ] + b[ i ];}}} else if( typeof b === ’number’ ) {for( var i = 0; i < a.length; i++ ) {c[ i ] = a[ i ] + b;}}return c;}; arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin // function to make the combinationfunction combinations( inpSet, qty ) {var inpLen = inpSet.length;var combi = [];var result = [];var recurse = function( left, right ) {if ( right === 0 ) {result.push( combi.slice(0) );} else {for ( var i = left; i <= inpLen - right; i++ ) {combi.push( inpSet[i] );recurse( i + 1, right - 1 );combi.pop();}}};recurse( 0, qty );return result;} // create data objectvar et = { };for (var i=0; i < lin.length; i++){et[’alfa’+(i+1)]=[];};// transform checkbox in 0 or 1for (var i in cab){var cols = document.getElementsByName(cab[i]);var temp = [];for (var j = 0, l = cols.length; j < l; j++) {if (cols[j].checked) {et[’alfa’+(j+1)].push(1);} else {et[’alfa’+(j+1)].push(0);};};};// column with the total of each arrow in the cyclesfor (var i in et){var temp = [];for (var j=rels; j < rels + cics; j++){temp.push(et[i][j]);}et[i].push( arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin temp.reduce(function(prev, cur) {return prev + cur;}));}// create the variable matvar mat = [];for (var i in et){mat.push(et[i]);}// reference array with total column TotRe2var TotRe2 = [];for (var i in et){var temp = [];temp.push(i);var temp2 = [];for (var j=0; j < rels; j++){temp2.push(et[i][j]);}temp.push(temp2.reduce(function(prev, cur) {return prev + cur;}));TotRe2.push(temp);}// descending organization of TotRe2TotRe2.sort(function (a, b) {if (a[1] < b[1]) {return 1;}if (a[1] > b[1]) {return -1;}// a must be equal to breturn 0;});// selection of the arrow with more presence in therelations of type 2 and variables arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin var ref = TotRe2[0][0];var corte = cics - et[ref][rels+cics];var soma = et[ref];var sol = [ref];// cutting set of trivial extensialwhile (cics * TotRe2[0][1] >= rels){// separate the arrows that are not in the cyclesof the ref arrowfor (var j= rels; j < rels + cics; j++){if (et[ref][j] !== 0){for (var i in mat){if (mat[i][j] !== 0 && mat[i] !== ’fora’ && i !== ref){mat[i]=’fora’;}}}}// function that returns false if sum is not solutionfunction solucao(){for (var i=0; i < rels; i++){if (soma[i] === 0){return false;}}for (var i=rels; i < rels + cics; i++){if (soma[i] !== 1){return false;}}return true;}// see if the initial sum is a solutionif (corte === 0){if (solucao()){alert(’Answer ’ + sol);} else {alert("Some typing error");}// otherwise go to the cutting process} else {// variables to make the combinationvar x = [];for (i in mat){ arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin if (mat[i] !== ’fora’ && i !== ref){var aux = parseInt(i,10) + 1;x.push(’alfa’ + aux);}}// cutting setwhile (corte > 0){// combinationvar d = combinations(x,corte);// sumfor (var i=0; i < d.length; i++){for (var j=0; j < corte; j++){var a = et[d[i][j]];var c = soma.add( a );soma = c;} // see if the sum is a solutionif (solucao()){// organize the solutionfor (var j=0; j < corte; j++){sol.push(d[i][j]);}// show the solutionalert(’Answer ’ + sol);}soma = et[ref];sol = [ref];}corte = corte - 1;}}// preparation for a new referenceet[ref] = ’fora’;var mat = [];for (var i in et){mat.push(et[i]);}// remove the ref from the array to put the new refTotRe2.shift();ref = TotRe2[0][0];corte = cics - et[ref][rels+cics];soma = et[ref];sol = [ref]; arch 12, 2019 1:3 WSPC/INSTRUCTION FILE PHI˙algebra˙Dynkin if (TotRe2.length === 1){break;}} };// Add that buttondocument.body.appendChild(botao);} else {alert(’Invalid information!’);// free previous variablesdocument.getElementById(’flechasid’).readOnly = false;document.getElementById(’relsid’).readOnly = false;document.getElementById(’ciclosid’).readOnly = false;};}function Limpar(){history.go(0);document.getElementById("enviarid").disabled = false;} Acknowledgements
The first named author has been supported by the tematic project of Fapesp2014/09310-5.The second named author acknowledges support from CAPES, in the form of aPhD Scholarship, PhD made at programa de Matem´atica, IME-USP, (Brazil).
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