An algorithm for the periodicity of deformed preprojective algebras of Dynkin types E 6 , E 7 and E 8
aa r X i v : . [ m a t h . R T ] J un AN ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OFDYNKIN TYPES E , E AND E JERZY BIA LKOWSKI
Dedicated to Andrzej Skowro´nski on the occasion of his seventieth birthday
Abstract.
We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojectivealgebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivialdeformed preprojective algebras of Dynkin types E and E exist only in characteristic 2. As a consequence, we show thatdeformed preprojective algebras of Dynkin types E , E and E are periodic and we obtain an algorithm for a classification ofsuch algebras, up to algebra isomorphism. We do it by a reduction of the conjecture to a solution of a system of equationsassociated with the problem of the existence of a suitable algebra isomorphism ϕ f : P f ( E n ) → P ( E n ) described in Theorem 2.1.One also shows that our algorithmic approach to the conjecture is also applicable to the classification of the mesh algebras ofgeneralized Dynkin type. Keywords:
Preprojective algebra, Deformed preprojective algebra, Self-injective algebra, Periodic algebra, System of equations Introduction and main results
In this paper, by K we denote a fixed algebraically closed field. Moreover, an by algebra mean an associative finite-dimensionalbasic connected K -algebra with identity, if not stated otherwise.We recall from [7] that the deformed preprojective algebras of generalized Dynkin type A n ( n ≥ D n ( n ≥ E , E , E , L n ( n ≥
1) (defined later in this section) were introduced as a new class of periodic K -algebras. Unfortunately, the original proofof their periodicity had a gap, see [3, 5]. However, the proof is correct for deformed preprojective K -algebras of generalizedDynkin type which are not isomorphic to preprojective K -algebras of generalized Dynkin, but only in the case of K being ofpositive characteristic. One of the main aims of this article is to complete the proof by showing that deformed preprojective K -algebras of generalized Dynkin type that are not isomorphic to preprojective K -algebras of generalized Dynkin type existonly for K of positive characteristic, see also Conjecture 1.7.We do it by applying the numeric algorithm presented in Section 3. On this way we obtain an affirmative solution ofConjecture 1.7 in a simple combinatorial data by showing that non-trivial deformed preprojective algebras of Dynkin types E and E exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types E , E and E are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it bya reduction of the conjecture to a solution of a system of equations associated with the problem of the existence of a suitable K -algebra isomorphism of the form ϕ f : P f ( E n ) → P ( E n ) described in Theorem 2.1, where P ( E n ) is the preprojective algebra(1.1) of Dynkin type E n and P f ( E n ) is the deformed preprojective algebra (1.4) of Dynkin type E n . On the other hand, ouralgorithmic approach to the conjecture is also applicable to the classification of the mesh algebras of generalized Dynkin type.The solution we obtain here shows an essential application of the computer algebra technique and computer computations insolving difficult and important theoretical problems of high complexity of modern representation theory.We recall from [7] and [1, 13, 14, 15] that the preprojective algebras are playing a crucial role in representation theoryof finite-dimensional algebras R , their derived categories D b (mod R ), their representation types, and in the Auslander-Reitentheory. Also they play an important role in a categorical study of isolated surface singularities of finite type and tame type.Our results of the paper can be viewed as a highly non-trivial application of the discrete mathematics technique, computeralgebra technique, and computer calculation in the modern representation theory and the Auslander-Reiten theory that weresuccessfully developed during last fifty years.On the other hand, we feel that our computational algorithmic approach leading to an affirmative solution of Conjecture 1.7can be also applied in: (i) the Coxeter-type classification of Φ-mesh root systems in the sense of [9, 8], (ii) the Grothendieck grouprecognition for derived categories D b (mod R ) of a module category mod R , (iii) the derived category study of the hypersurfacesingularities f = x a + x b + x a classified recently in [10, 11], in particular the tame singularities such that their orbifoldcharacteristic χ ( a,b,c ) = a + b + c is zero, and (iv) in the Coxeter spectral classification of connected symmetrizable integerCartan matrices of Dynkin types E , E and E and their generalized mesh root systems recently studied in [12].To introduce the reader to the problems we study and to outline the contents of this paper, we recall from [7] that, by theclassification of deformed preprojective algebras of generalized Dynkin type L n for an algebraically closed field K of characteristicdifferent from 2, each of the deformed preprojective K -algebras of type L n is isomorphic to the preprojective K -algebra P ( L n )of type L n . Moreover, each of the deformed preprojective algebras of type A n by the definition is isomorphic to the preprojectivealgebra P ( A n ) of type A n . Furthermore, by [3], all deformed preprojective algebras of Dynkin type E are isomorphic to thecanonical preprojective algebra P ( E ) of type E . This research was supported by the Research Grant DEC-2011/02/A/ST1/00216 of the Polish National Science Center.
On the other hand, it is known that for K with char( K ) = 2 there exist deformed preprojective K -algebras of the Dynkintypes D n , E and E (see [5, Theorem]) not isomorphic to the canonical preprojective K -algebras of these types. But thecomplete classification of these algebras seems to be a difficult problem.In our forthcoming article [2] we prove that, for K of characteristic different from 2, every deformed preprojective K -algebraof Dynkin type D n , with n ≥
4, is isomorphic to the preprojective K -algebra P ( D n ) of Dynkin type D n . The present paperis devoted to obtain the dual fact for the remaining types E and E . Unfortunately, calculations for these types are ratherlength and much more complicated than in the previous one and the main difficulty that appears is to find precise formulaeor patterns that are easy to formulate and to use in the proof handled “manually”. Theretofore we reduce the problem of theexistence of non-trivial deformed preprojective algebras P ( E n ) of Dynkin types E and E to a computational problem solvablein an algorithmic way, namely to the problem of solving an associated system of equations.We note that our main interest is to find a suitable form of coefficients of the solution to be able to determine in whichcharacteristics the expected solutions exist. Moreover, we would like to obtain possibly simple and “human readable” exemplaryformulae for a K -algebra isomorphism ϕ f : P f ( E n ) → P ( E n ) described in Theorem 2.1 between deformed preprojective algebraof type E n and the preprojective algebra of type E n for n ∈ { , } . Fortunately, we are able to show that such an isomorphism ϕ f exists in case when the characteristic of K is different from 2.In order to do that we present some methods of imposing additional restrictions on the associated systems of equations.However, the computations in the case of type E remain complicated. To get them more effective we need to modify theobtained sets of equations to be able to perform computations in a reasonable time, but a selection of suitable modificationsteps is quite non-trivial part of our work.We recall from [7] that the preprojective algebra P ( E n ) of Dynkin type E n , for n ∈ { , , } , is the bound quiver K -algebra P ( E n ) := KQ E n / I ( E n )(1.1)given by the quiver Q E n : 0 a (cid:15) (cid:15) a / / ¯ a o o a / / ¯ a o o a / / ¯ a O O ¯ a o o a · · · ¯ a o o a n − / / n − ¯ a n − and the ideal I ( E n ) of relations a ¯ a = 0 , a ¯ a = 0 , ¯ a a + a ¯ a = 0 , ¯ a a + ¯ a a + a ¯ a = 0 , ¯ a i a i + a i +1 ¯ a i +1 = 0 , for i ∈ { , . . . , n − } , ¯ a n − a n − = 0 . We also use in the paper the local commutative K -algebra R ( E n ) = K h x, y i / (cid:0) x , y , ( x + y ) n − (cid:1) (1.2)that is isomorphic to the K -algebra e P ( E n ) e , where e is the primitive idempotent in P ( E n ) associated to the vertex 3 of Q E n .An element f from the square rad R ( E n ) of the radical rad R ( E n ) of R ( E n ) is said to be admissible if f satisfies the followingcondition (cid:0) x + y + f ( x, y ) (cid:1) n − = 0 . (1.3)Following [7, Section 7], given an admissible element f ∈ rad R ( E n ), we define the deformed preprojective K -algebra of Dynkintype E n P f ( E n ) := KQ E n / I ( E n , f )(1.4)to be the bound quiver K -algebra given by the quiver Q E n and the ideal I ( E n , f ) of relations a ¯ a = 0 , a ¯ a = 0 , ¯ a a + a ¯ a = 0 , ¯ a a + ¯ a a + a ¯ a + f (¯ a a , ¯ a a ) = 0 , ¯ a i a i + a i +1 ¯ a i +1 = 0 , for i ∈ { , . . . , n − } , ¯ a n − a n − = 0 . For a general definition of a deformed preprojective algebra of generalized Dynkin type the reader is referred to [7].
Remarks . (a) Observe that P f ( E n ) is obtained from P ( E n ) by deforming the relation at the exceptional vertex 3 of Q E n ,and P f ( E n ) = P ( E n ) if f = 0.(b) In the original definition of the admissible element f for the types E , E , E the condition (1.3) was missing (see [3,Remark] for details).(c) The corrected definition of the admissible element has been presented in [7, Section 9] (on page 238) in the definition ofdeformed mesh algebras of generalized Dynkin type. As a consequence of this mistake, the originally constructed algebras (see[3, Corollary]) were not deformed preprojective algebras of generalized Dynkin and hence the proof of a corresponding theoremin the case of types E , E , and E was not correct, see [3, Corollary] for its correction and more details.To complete and correct the proof we construct in Section 3 an algorithm to test whether for a given type E n ∈ { E , E , E } every deformed preprojective algebra of type E n is isomorphic to the preprojective algebra P ( E n ) (1.1) of type E n . N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E By applying the algorithm we construct in Appendices A and B a K -algebra isomorphism ϕ : P f ( E n ) → P ( E n ) defined inTheorem 2.1 for the types E and E . As a consequence we obtain the following theorem that is one of the main results of thispaper. Theorem 1.6.
Assume that n ∈ { , } and K is a field of characteristic different from . Then every deformed preprojective K -algebra of Dynkin type E n is isomorphic to the preprojective K -algebra P ( E n ) (1.1) of Dynkin type E n . We note that the calculations for the types E and E (presented in A and B, respectively) are much more complicated thanthat one given in [3] and [6], for the types F and E , and cannot be handled manually, compare with their structure for types E n in Tables 2, 4–5.In the forthcoming article [2] we prove that every deformed preprojective algebra of Dynkin type D n , for n ≥
4, over a field ofcharacteristic different from 2, is isomorphic (as a K -algebra) to the preprojective algebra P ( D n ) of Dynkin type D n , and hencewe complete the proof of the following conjecture and as a consequence we prove that every deformed preprojective algebra ofgeneralized Dynkin type is periodic. Conjecture 1.7.
Every deformed preprojective K -algebra of generalized Dynkin type ∆ over a field K of characteristic differentfrom is isomorphic to the preprojective K -algebra of corresponding Dynkin type ∆ .Remarks . (a) It is easy to see that under a minor modifications our algorithm can be also used in testing whether ornot, for a given (arbitrary) generalized Dynkin type ∆, every deformed preprojective algebra of type ∆ is isomorphic to thepreprojective algebra P (∆)of type ∆, and to test whether or not, for a given (arbitrary) mesh type ∆ (in the sense of [7]) everydeformed mesh algebra of type ∆ is isomorphic to the mesh algebra of type ∆.(b) We recall that by the definition class of deformed mesh algebras of generalized Dynkin type introduced in [7] containsall deformed preprojective algebras of generalized Dynkin type.c) Our algorithm was already successfully used in [3] for the classification of the deformed preprojective K -algebras of Dynkintype E and deformed mesh K -algebras of type F in [6].The article is organized as follows. Section 2 contains mathematical background for the algorithmic approach of the problemand theorems proving the correctness of the algorithm presented in the next sections. Section 3 presents an outline of thealgorithm used in testing if P ( E n ) and P f ( E n ) are isomorphic as K -algebras. Section 4 contains a detailed description of thedata structures used in the calculations. In Section 5 we describe the calculations associated with the admissible conditionfor the algebras P f ( E n ). In Section 6 an algorithm constructing a suitable K -basis B of the preprojective P ( E n ) and for asuitable presentation of elements of P ( E n ) in the basis B . In Section 7 we construct a required set of equations and we discussa relationship between the complexity of its equations and the choice of the base B of the algebra P ( E n ). The final Section 8contains a discussion of solving the obtained system of equations.In A and B we present applications of our algorithms to the types E and E , as well as we discuss details of the calculations(performed according to our algorithmic procedures) for the types E and E , respectively.The reader is referred to the monographs [1, 13, 14, 15] for a general background on the representation theory and selfinjectivealgebras. We also refer to [9]–[11] for a discussion of potential areas of application mentioned earlier in this section of thealgorithmic procedures constructed in our paper.2. On algebra homomorphisms from P f ( E n ) to P ( E n )From now on we assume that n ∈ { , , } and f is a fixed admissible element from rad R ( E n ). We usually identifythe bound quiver Q E n with the quadruple Q E n = ( Q , Q , s, t ), where Q = { , . . . , n − } is the set of vertices of Q E n , Q = { a , . . . , a n − , ¯ a , . . . , ¯ a n − } is the set of arrows of Q E n , and s, t : Q → Q are the functions assigning to every arrow itssource and target, respectively.To simplify the notation we set in this section a n + i − = ¯ a i for i ∈ { , . . . , n − } . We also denote by m the length of themaximal non-zero path in P ( E n ). Hence we set m =
10 for n = 6 ,
16 for n = 7 ,
28 for n = 8 . An important role in our study is played by a K -algebra homomorphism ϕ : P f ( E n ) → P ( E n )(2.1)such that for every arrow a we have ϕ ( a ) = a + w a for some element w a ∈ e s ( a ) rad P ( E n ) e t ( a ) .Observe that in that case these w a , for a ∈ Q , in fact induce the homomorphism ϕ . Hence each of these homomorphismsis induced by the equalities ϕ ( a k ) = a k + m X l =2 i ( k,l ) − X j =0 α ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − , for k = 0 , . . . , n − , (2.2)for some coefficients α ( k, l, j ) ∈ K with k = 0 , . . . , n − l = 2 , . . . , m , j = 0 , . . . , i ( k, l ) −
1, where for k ∈ { , . . . , n − }{ a k } ∪ n a i ( k,l,j, . . . a i ( k,l,j,l − (cid:12)(cid:12) l ∈ { , . . . , m } , j ∈ { , . . . , i ( k, l ) − }} o J. BIAKOWSKI form a basis of e s ( a k ) P ( E n ) e t ( a k ) with some i ( k, l ) ∈ N (for k ∈ { , . . . , n − } , l ∈ { , . . . , m } ), and i ( k, l, j, s ) ∈ { , . . . , n − } ,(for k, s ∈ { , . . . , n − } , l ∈ { , . . . , m } , j ∈ { , . . . , i ( k, l ) − } ).We note that assumption of the existence of a homomorphism (2.1) of the above form is in fact very strong, and in thegeneral case, finding such a homomorphism is not easy. The conditions for the existence of such a homomorphism are describedby Corollary 2.2. In the subsequent sections we construct the algorithm that checks them.The following theorem explains the importance of these homomorphisms. Theorem 2.1. If ϕ : P f ( E n ) → P ( E n ) is a K -algebra homomorphism given by the equalities (2.2) then ϕ is invertible, andhence is a K -algebra isomorphism.Proof. We construct inductively elements β ( k, l, j ) ∈ K for k = 0 , . . . , n − l = 2 , . . . , m , j = 0 , . . . , i ( k, l ) −
1, such that ϕ (cid:18) a k + m X l =2 i ( k,l ) − X j =0 β ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − (cid:19) = a k , (2.3)for k ∈ { , . . . , n − } . We set R k,l = e s ( a k ) rad m − l − P ( E n ) e t ( a k ) and observe that ϕ ( a k ) + R k, = a k + R k, . Assume now that ϕ (cid:18) a k + t − X l =2 i ( k,l ) − X j =0 β ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − (cid:19) + R k,t − = a k + R k,t − , (2.4)for some t ∈ { , . . . , m − } . Now we construct β ( k, t, j ) ∈ K , for k ∈ { , . . . , n − } , j ∈ { , . . . , i ( k, l ) − } , such that ϕ (cid:18) a k + t X l =2 i ( k,l ) − X j =0 β ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − (cid:19) + R k,t = a k + R k,t . (2.5)Observe that (2.4) yields ϕ (cid:18) a k + t − X l =2 i ( k,l ) − X j =0 β ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − (cid:19) − a k + R k,t − = 0 + R k,t − , for k ∈ { , . . . , n − } , and hence there exist elements γ k,t,j ∈ K , with k ∈ { , . . . , n − } , j ∈ { , . . . , i ( k, l ) − } , such that ϕ (cid:18) a k + t − X l =2 i ( k,l ) − X j =0 β ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − (cid:19) − a k + R k,t = i ( k,t ) − X j =0 γ k,t,j a i ( k,t,j, . . . a i ( k,t,j,t − + R k,t , for k ∈ { , . . . , n − } . Therefore, if we set β ( k, t, j ) = − γ k,t,j , for k ∈ { , . . . , n − } , j ∈ { , . . . , i ( k, l ) − } , then (2.5) issatisfied. Hence, by induction we prove that (2.5) is satisfied also for t = m . But in this case (2.5) is equivalent to (2.3).Consequently, (2.3) is satisfied and we obtain a k = ϕ ( a k ) + m X l =2 i ( k,l ) − X j =0 β ( k, l, j ) ϕ ( a i ( k,l,j, ) . . . ϕ ( a i ( k,l,j,l − ) , (2.6)for k ∈ { , . . . , n − } , and hence the set { ϕ ( a i ) } i ∈{ ,..., n − } generates the K -algebra P ( E n ).Now we show that the K -linear map ψ : P ( E n ) → P f ( E n ) defined by ψ ( a k ) = a k + m X l =2 i ( k,l ) − X j =0 β ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − , (2.7)for k ∈ { , . . . , n − } , is a K -algebra homomorphism that is inverse to ϕ . Indeed, ψ is well defined, because ϕ is a homomorphismand (2.3) is satisfied. Moreover, we have ϕ ( ψ ( a k )) = a k for k ∈ { , . . . , n − } . Hence ϕ ◦ ψ = id P ( E n ) .To complete the proof that ϕ (and hence also ψ ) is an isomorphism it suffices to show that P f ( E n ) is generated by elements { ψ ( a i ) } i ∈{ ,..., n − } . Recall that P f ( E n ) is generated by elements { a k } k ∈{ ,..., n − } , so it suffices show that for each k ∈{ , . . . , n − } element a k ∈ P f ( E n ) is generated by the elements { ψ ( a i ) } i ∈{ ,..., n − } . In particular, we may dually constructinductively elements α ′ ( k, l, j ) ∈ K for k = 0 , . . . , n − l = 2 , . . . , m , j = 0 , . . . , i ( k, l ) −
1, such that ψ (cid:18) a k + m X l =2 i ( k,l ) − X j =0 α ′ ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − (cid:19) = a k , for k ∈ { , . . . , n − } . (cid:3) To apply above theorem to the computer algorithm we need a tool to look for a homomorphism given by the equalities ofthe form presented in (2.2). Hence we need to formulate the conditions for the existence of such a homomorphism in a formthat can be checked by such an algorithm. The following consequence of Theorem 2.1 is being useful in the construction of thatalgorithm.
N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E Corollary 2.2.
Let i ( k, l ) ∈ N for k ∈ { , . . . , n − } , l ∈ { , . . . , m } , and i ( k, l, j, s ) ∈ { , . . . , n − } , for k, s ∈ { , . . . , n − } , l ∈ { , . . . , m } , j ∈ { , . . . , i ( k, l ) − } such that { a k } ∪ n a i ( k,l,j, . . . a i ( k,l,j,l − (cid:12)(cid:12) l ∈ { , . . . , m } , j ∈ { , . . . , i ( k, l ) − }} o form a basis of e s ( a k ) P ( E n ) e t ( a k ) for each k ∈ { , . . . , n − } . Assume that there exist coefficients α ( k, l, j ) ∈ K for k =0 , . . . , n − , l = 2 , . . . , m , j = 0 , . . . , i ( k, l ) − satisfying the equalities: δ ( k , k ) = 0 for ( k , k ) ∈ { (0 , n − , (1 , n ) , (2 n − , n − } ; δ ( k + n − , k −
1) + δ ( k, k + n −
1) = 0 for k ∈ { , , . . . , n − } ; δ ( n − ,
0) + δ ( n + 1 ,
2) + δ (2 n − , n −
2) + f (cid:0) δ ( n − , , δ ( n + 1 , (cid:1) = 0 where δ : { , . . . , n − } → P ( E n ) are defined as follow δ ( k , k ) = m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a i ( k ,l,j, . . . a i ( k ,l,j,l − a k + m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a k a i ( k ,l,j, . . . a i ( k ,l,j,l − + (cid:18) m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a i ( k ,l,j, . . . a i ( k ,l,j,l − (cid:19) (cid:18) m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a i ( k ,l,j, . . . a i ( k ,l,j,l − (cid:19) . Then the map ϕ : P f ( E n ) → P ( E n ) defined by setting ϕ ( a k ) = a k + m X l =2 i ( k,l ) − X j =0 α ( k, l, j ) a i ( k,l,j, . . . a i ( k,l,j,l − for k = 0 , . . . , n − , is a K -algebra isomorphism.Proof. It follows from the assumption on δ that ϕ : P f ( E n ) → P ( E n ) defined on arrows as in the claim is a well definedhomomorphism. Hence the corollary is an immediate consequence of Theorem 2.1. (cid:3) We note that Corollary 2.2 says that for a given admissible element f ∈ rad R ( E n ) we can find an isomorphism ϕ : P f ( E n ) → P ( E n ) if we can solve a particular system of P n − i =0 dim rad e i P ( E n ) e i equations over K with P n − k =0 dim rad e s ( a k ) P ( E n ) e t ( a k ) variables. Details of the construction of these equations are presented in Sections 6 and 7.On the other hand, if we want to show that such a system of equations is solvable for all admissible elements f ∈ rad R ( E n ),then we also need some tool (preferable equations) to determine if a given element f ∈ rad R ( E n ) is admissible. We notethat not all elements f ∈ rad R ( E n ) are admissible (see [3, Remark] for details). Observe, that if we denote by B the basis ofrad R ( E n ) then we can identify element f ∈ rad R ( E n ) with coefficients θ b ∈ K , for b ∈ B , such that f = P b ∈ B θ b b . We showlater in Section 5 how to construct equations for θ b , b ∈ B , over K equivalent with P b ∈ B θ b b being an admissible element.We note also that in the above approach we may in fact reduce the scope of the calculations to P ( E n ). Obviously, it isconvenient, because calculations in P f ( E n ) are much more complicated.3. Alghoritm
We present here an outline of the constructed algorithm. Details of the particular steps of this algorithm are presented inthe subsequent sections.We split the calculations into following three parts:I. Computing the equations corresponding to the admissibility condition.II. Computing the equations equivalent to the existence of coefficients from Corollary 2.2.III. Solving the system composed of equations obtained in Parts I and II.In Part I we can extinguish the following steps:1. Choosing a base B of R ( E n ), and a base B ′ of rad R ( E n ) such that B ′ ⊂ B , and computing the presentation of (all) non-zeroelements of rad R ( E n ) in B .2. Constructing the presentation of an arbitrary element f ∈ rad R ( E n ) through the base elements. We are identifying f ∈ rad R ( E n ) with elements θ b ∈ K , for b ∈ B ′ , such that f = P b ∈ B ′ θ b b .3. Calculating ( x + y + f ( x, y )) n − . In other words, we are constructing terms ω b over K , for b ∈ B ′ , such that P b ∈ B ′ ω b b = (cid:0) P b ∈ B \ B ′ b + P b ∈ B ′ θ b b (cid:1) n − .4. From the previous step we obtain the set of equations { ω b = 0 } b ∈ B ′ . We reduce it to obtain the set Ω of independentequations.5. Further reduction of the obtained system of equations. For each equation of Ω we chose the variable to be substituted in theequations obtained in Part II.In Part II we have the following steps:1. Computation of • a base B of P ( E n ), • the set E of non-zero paths of P ( E n ), J. BIAKOWSKI • the presentation of elements from E in B .2. Constructing the presentation of arbitrary homomorphism ϕ : P f ( E n ) → P ( E n ) defined on arrows. We identify ϕ with theset of variables (cid:8) α a,b ∈ K | a ∈ Q , b ∈ B, b is a path from s ( a ) to t ( a ) (cid:9) .
3. Calculating the actions of ϕ on the paths from the relations of P f ( E n ).4. Deriving from the previous step the equations in P ( E n ) corresponding to the relations of P f ( E n ).4. Data structures
In this section we consider the choice of data structures. Our priority is to reduce the computational and memory complexityof the necessary calculations.We note that we need to choose the data structure in such a way, so we can be able to • quickly perform operations such as sum, product, multiplication by scalar, on elements from P ( E n ) (respectively, onelements from R ( E n )); • quickly determine if the result of such an operation is non-zero; • effectively store the presentation of elements from P ( E n ) (respectively, elements from R ( E n )) in the limited amount ofmemory.We recall that in P ( E ) we have 14 different arrows and non-zero paths of length 28, so “brutal force” approach by computingand storing coefficients for all (not necessarily non-zero) paths of length less or equal 28 would require a big amount of memory.We note that the basis (of both R ( E n ) and P ( E n )) consist of relatively small amount of elements, so we can store them in(respectively) two and four-dimensional tables: • we store the basis of R ( E n ) in the 2-dimensional table, where the dimensions corresponds to – length n of the element, – number r of the element of length n ; • we store the basis of P ( E n ) in the 4-dimensional table, where the dimensions corresponds to – length n of the path, – source s of the path, – target t of the path, – number r of the element from s to t of length n .We note that the last dimension in the case of both bases has variable number of indices and we identify these tables withcomposition of one-dimensional tables.The presentation of elements of R ( E n ) we compute as a binary search tree, where • left branches corresponds to the element x , • right branches corresponds to the element y , • each of the nodes correspond to the element being a composition of elements corresponding to branches on the path(from root) to that node, • all leaves and only leaves correspond to zero elements, • with every node which is not a leaf we associate the vector of coefficients of the corresponding element in the base of R ( E n ).We identify the leaves with null pointers and denote them by symbol “ ⊠ ”. We note that if we wanted our trees to be compatiblewith the generally accepted definition of a tree, then we should remove all nodes “ ⊠ ”. In our case we keep them in order toemphasise their role to indicate all zero-paths. We note also that in the construction of such a tree (see Algorithm 1) thereappear some similar nodes to “ ⊠ ”, of a special role such as “under computation”.For R ( E ) such a tree is of the shape presented on Figure 1. On this figure we may observe the association of nodes of thistree with compositions of generators x , y , where ⊠ denote the leaves.The basis of R ( E ) over K is stored in the following table B = (cid:8) { R ( E ) } , { x, y } , { xy, yx, yy } , { xyx, xyy, yxy } , { xyxy, yxyy } , { xyxyy } (cid:9) . Presentation of elements in the above basis is stored in the tree illustrated on Figure 2.We note that both finding the basis B of R ( E n ) and the presentation of elements of R ( E n ) in B , even in the case of n = 8,has a relatively low computational and memory complexity. The basic properties of such trees has been presented in Table 1. Table 1.
Complexity of the trees for R ( E n ) Dynkin type E E E number of elements of basis B
12 24 60depth of the tree 6 9 15number of nodes 43 103 537number of non-leaf and non-root nodes 20 50 267
N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E id R ( E ) x ⊠ xyxyx ⊠ xyxy ⊠ xyxyy ⊠ ⊠ xyyxyyx ⊠ xyyxy ⊠ ⊠⊠ yyx ⊠ yxyyxyx ⊠ yxyxy ⊠ ⊠ yxyyyxyyx ⊠ ⊠ ⊠ yyyyx ⊠ yyxyyyxyx ⊠ ⊠ yyxyy ⊠ ⊠⊠ xx yxx yx yx y yxx yx yy y xx yxx yx y yxx y y y xx yxx y yx yy Figure 1.
Tree with non-zero paths of R ( E )[1][1 , ⊠ [1 , , , , ⊠ [1 , ⊠ [1] ⊠ ⊠ [0 , , − , ⊠ [ − ⊠ ⊠⊠ [0 , , , ⊠ [0 , , , ⊠ [1] ⊠ ⊠ [0 , − ⊠ ⊠ ⊠ [0 , , − , − , − ⊠ [ − , − ⊠ ⊠ [ − ⊠ ⊠⊠ xx yxx yx yx y yxx yx yy yxx yxx yx y yxx y y y xx yxx y yx yy Figure 2.
Presentation of elements of R ( E )Much more computational complexity has the step in which we compute ( x + y + f ( x, y )) n − . We recall that f ( x, y ) = P b ∈ B ′ θ b b for some coefficients θ b ∈ K . Hence the powers of x + y + f ( x, y ) are sums of the elements ω of the form ω = k ω Y b ′ ∈ B θ ω b ′ b ′ b for some integer k ω ∈ Z , natural numbers ω b ′ ∈ N , and b ∈ B .Hence each of these summands we may associate with the triple ( k ω , d ω , b ) ∈ Z × N | B | × B , where d ω = [ ω b ′ ] b ′ ∈ B . We notealso that in fact in our calculation vectors d can be treated as sparse matrix, and detailed analysis show that even in the caseof E it can be stored as 128-bit vector. On the other hand b can be associated with natural number less then | B | and storedas a 6-bit value. Observe that we can easily multiply such elements. Indeed, for ( k , d , b ) , ( k , d , b ) ∈ Z × N | B | × B , with b b = P b ∈ B c b b for some c b ∈ Z , b ∈ B . We obtain then( k , d , b )( k , d , b ) = X b ∈ B ( c b k k , d + d , b ) . We note also that if b b = 0, then we may simply set ( k , d , b )( k , d , b ) = 0. Moreover, in the above computations itis highly recommended to perform normalizations, that means, to replace pairs of elements of the form ( k , d, b ), ( k , d, b ) by( k + k , d, b ), and remove elements of the form (0 , d, b ). Otherwise computational complexity of computing ( x + y + f ( x, y )) n − in the case of type E would be very high. We want to emphasize, that although keeping elements of the form (0 , d, b ) mayreduce amount of operations of allocating and freeing memory and in the case of low number of such elements (i.e. in the caseof type E ) can be useful, in the case of type E it can results in allocating much more elements than necessary (even thousands J. BIAKOWSKI times more), and in the consequence cause a big impact on memory and time complexity of the algorithm. We recall that theproduct of a two basis elements is a combination of (often many) basis elements.Dually (to the computation of the presentation R ( E n )), we compute the presentation of elements of P ( E n ) as the forest ofsearch threes, where • roots corresponds to the (source) vertices, • branches corresponds to arrows, • every node correspond to path consisting of arrows corresponding to branches on path to that node (from root), • all leaves and only leaves correspond to zero elements, • with every node which is not a leaf we associate the vector of coefficients of the corresponding element in the base of P ( E n ).We note that in general, trees in such a forest are not binary trees. In particular the tree with root associated with theexceptional vertex 3 have three branches coming from root and is of the shape presented on Figure 3. e ¯ a ¯ a ¯ a ... ¯ a a ... ... ... ¯ a ... a ... ... ¯ a ¯ a a a ¯ a ¯ a a ¯ a a a ¯ a a Figure 3.
Shape of a tree associated with vertex 3 for P ( E n )On Figure 4 we present also more detailed shape of the tree for P ( E ) associated with the vertex 0. e a a ¯ a a ¯ a ¯ a a ¯ a ¯ a a ⊠ a ¯ a ¯ a a a ... ⊠ ... a ¯ a a a ¯ a a ¯ a ⊠ a ¯ a a ¯ a a ... ⊠ ... ⊠ a ¯ a a a a ¯ a a a ¯ a ... ⊠ ... a ¯ a a a a ... ⊠ a a ... ... a ¯ a ¯ a a ¯ a a ¯ a ¯ a ¯ a a ¯ a ¯ a a ¯ a ¯ a ¯ a ¯ a a ¯ a ¯ a ¯ a ¯ a a ¯ a ¯ a a ¯ a a Figure 4.
Shape of the tree associated with vertex 0 for P ( E )For such a forest and the chosen basis B of P ( E n ) we construct the corresponding forest describing the presentation ofelements of P ( E n ) in B (consisting of the trees of structure dual to the structure of trees from Figure 2).We note that the structure of such a forest is much more complicated than the structure of the tree for R ( E n ) (see Table 2for details and compare with Table 1). Table 2.
Complexity of the forests for P ( E n ) Dynkin type E E E number elements of the basis B of P ( E n ) 156 399 1240depth of the trees 11 17 29number of non-leaf and non-root nodes 1551 48887 47137364number of tree nodes in total 3395 108322 104753728 N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E Computation of admissibility condition
We recall that we have the following computations related to admissibility condition:1. Computation of a base of R ( E n ) and presentation of elements of R ( E n ) in that base.2. Presentation of the function f through the base elements.3. Calculating the corresponding power of the function f .4. Derivation of equations from the calculated power of f .5. Reduction of the obtained system of equations.6. Extraction of the substitutions from the reduced system of equation.We start with computation of a base of R ( E n ) and the presentation of elements of R ( E n ) in that base. Outline of thesecomputation is presented as Algorithm 1.In Algorithm 1 we denote by B the computed set of basis elements of R ( E n ) and by P res the tree of structure presented inSection 4. We recall that every node of
P res is associated to the element of R ( E n ) corresponding to the path from root to thatnode. To every node of P res we assign either a mapping from B to K (the presentation of the associated element in the basis B ) or ⊠ indicating zero-path or (cid:3) indicating “calculation in progress”. If in P res there is the node induced by a path a , thenwe set P res [ a ] to the value assigned to that node and we set P res [ a ] = ∅ otherwise. Algorithm 1
Finding the basis of R ( E n ) Require: S ← { x, y } , R ← relations , i ← E ← S , B ← S ∪ { R ( E n ) } , P res [] ← { ( s, δ s ) | s ∈ S } while E i = ∅ do for all a ∈ E i , s ∈ S do P res ( as ) ← (cid:3) end for i ← i + 1 while U = ∅ do while ∃ u ∈ U ∃ ∼ r ∈ R ∃ P ⊆ S i \ U ∀ c ∈ P ∃ α c ∈{− , +1 } u ∼ r P c ∈ P α c c do if P c ∈ P ∩ E i α c P res [ c ] = 0 then P res [ u ] ← P c ∈ P ∩ E i α c P res [ c ] else P res [ u ] ← ⊠ end if end while if U = ∅ then b ← choose ( U ) B ← B ∪ { b } P res [ b ] ← δ b end if end while end while return B , B \ ( S ∪ { R ( E n ) } ), P res , i − U the set of paths corresponding to the nodes with value ⊠ ( U = { a | P res [ a ] = (cid:3) } )and by E i ⊆ { x, y } i the set of paths of length i associated with nodes of P res of values different from ⊠ and (cid:3) . Moreover foreach element b of B we denote by by δ b the mapping δ b : B → K such that δ b ( b ) = 1 and δ b ( b ′ ) = 0 for b ′ ∈ B \ { b } .The function choose () chooses some path from the given set of paths. Although this choice does not affect the correctnessof the algorithm, it may have an impact on the later calculations (and their computational complexity) and we deal with theappropriate selection of this function in the next section.We note that the statement “ U = ∅ ” from Lines 7 and 15 is equivalent to the statement “ ∃ a P res [ a ] = (cid:3) .” The statementfrom Line 8 means that there exist a path u for which presentation still were not computed, but which is in the relation ∼ r with some paths (forming the set P ) with known presentation in the base B . Moreover, the statement from Line 9 means, thatthe combination of these paths in the relation ∼ r is non-zero. We note, that we use the fact that all relations are of a specialform. In particular in each of the relations all coefficients belong to the set {− , +1 } and all paths are of the same length.Observe also that the operation P res [ u ] ← ⊠ from Line 12 acts as U ← U \ { u } , and the operation from Line 10 acts as U ← U \ { u } and E i ← E i ∪ { u } . Similar side-effects on the sets U and E i are also performed in Lines 4 and 18.We also note that above algorithm contains some simplifications. For example in the real implementation P res [ a ] shouldcontain only mappings on the elements of B of the same length as a (see Section 4). We may also reduce a little the timecomplexity of the calculations when we replace Lines 4–6 by setting P res [ u ] ← (cid:3) for u ∈ { as | a ∈ E i − , s ∈ S } ∩ { sa | a ∈ E i − , s ∈ S } , and P res [ u ] ← ⊠ for u ∈ { as | a ∈ E i − , s ∈ S } \ { sa | a ∈ E i − , s ∈ S } .In the next step we compute ( x + y + f ( x, y )) n − . We start with some preparation. Let B be the basis of R ( E n ), B ′ ⊆ B the basis of rad R ( E n ), and f the fixed admissible element of rad R ( E n ). We recallthat f ( x, y ) = P b ∈ B ′ θ b b for some fixed elements θ b ∈ K , and hence ( x + y + f ( x, y )) n − is the sum of elements ω of the form ω = k ω Y b ′ ∈ B θ ω b ′ b ′ b for some integer k ∈ Z , natural numbers ω b ′ ∈ N , and b ∈ B ′ . We may divide these summands depending on b . We denote byΩ b the set of such summands indicated by the basis element b . So we have (cid:0) x + y + f ( x, y ) (cid:1) n − = X b ∈ B X ω ∈ Ω b ω = X b ∈ B (cid:18) X ω ∈ Ω b k ω Y b ′ ∈ B θ ω b ′ b ′ (cid:19) b. Hence the admissible condition (cid:0) x + y + f ( x, y ) (cid:1) n − = 0is equivalent to the set of equations (cid:26) X ω ∈ Ω b k ω Y b ′ ∈ B θ ω b ′ b ′ = 0 (cid:27) b ∈ B . (5.1)Clearly, the coefficients k ω , ω b ′ , for b ′ ∈ B , ω ∈ P b ∈ B Ω B , can be easily computed with the methods introduced in Section 4.Finally, we need to apply some changes to the admissibility condition to reduce set of equations (5.1) and modify it in such away (see (5.2)), so obtained equations could be easily used in the final computation. We start with removing trivial equations.Further, we sort these equations in the ascending order of “lengths” of the base elements b . We denote by E start the set of theseequations. We want to obtain a minimal set Subst of equivalent “substitutions” of the form (cid:26) θ b = X ω ∈ Ω ′ b k ′ ω Y b ′ ∈ B \ B ′′ θ ω b ′ b ′ (cid:27) b ∈ B ′′ (5.2)with B ′′ ⊆ B , k ′ ω quotient numbers with denominator being a power of 2, and natural numbers ω b ′ ∈ N . Algorithm 2
Finding “substitutions” equivalent to the admissibility condition
Require: Eq ← E start , Subst ← ∅ , while Eq = ∅ do eq ← min ( Eq ) Eq ← Eq \ { eq } if app ( eq, Subst ) is non-trivial then eqsubst ← app ( eq, Subst ) if ∃ u ∈ max ( eqsubst ) u base ( u ) ∈ {− , } then Subst ← Subst ∪ { substitution of θ base ( u ) derived from eqsubst } else if ∃ u ∈ max ( eqsubst ) u base ( u ) ∈ {− , } then Subst ← Subst ∪ { substitution of θ base ( u ) derived from eqsubst } else throw error end if end if end while return Subst
We denote by min ( E ) a function which choose the equation associated with the element b ∈ B of minimal length fromsome set of equations E ⊆ E start . (We note that this function is not necessary unique.) Moreover by appl ( eq, S ) we meanthe equation obtained from the equation eq ∈ E start by applying all substitutions from the set S ⊆ Subst . We also denote by max ( eq ) the set of coefficients u = Q a ∈ B θ u a a of the equation eq X u ∈ Ω k u Y a ∈ B θ u a a = 0(with k u ∈ Q , u a ∈ N ), such that there exists a ∈ B satisfying u a = 0, u a ′ = 0 for all a ′ ∈ B \ { a } , and a is of maximal lengthof that property. By base ( u ) we denote then such an element a ∈ B . Then Algorithm 2 describes the procedure of constructionof a sequence of substitutions of selected coefficients θ b of f which are equivalent to the admissibility condition.We note that in general the result of the above steps may depend on the characteristic of the field K . In particular, in thecase of type F we obtain the equation which is trivial if and only if the field K is of characteristic 2 (see the the proof of [6,Lemma]). We recall that in our computations (for the types E and E ) we assume that K is not of characteristic 2. We notethat in the case of the types E and E choosing element u satisfying the condition from line 7 or the condition from line 9 ofthe above algorithm it always is possible (we refer for the details to A.1 and B.1). N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E We note that above computations in the case of Dynkin types E and F are trivial, because in these cases we have very fewequations with very few coefficients, and all of them being of simple form (see [3, Lemma] and [6, Lemma] for details). But onthe other hand, it is much more complicated in the case of type E . In the Table 3 we comparing the complexity of computationsfor types E , E and E . The last row of this table contain numbers corresponding to the choices of the calculations described Table 3.
Complexity of the computation of the admissibility condition
Dynkin type E E E number of elements of the basis B
12 24 60number of equations in E start Subst in [3, Lemma], A.1 and B.1, respectively. We note that the coefficients of equations for the type E are usually also of muchmore complicated form. 6. Computation of a base of P ( E n )In this section we construct Algorithm 3 computing a suitable K -basis B of the K -algebra P ( E n ) and a K -linear presentationof elements of P ( E n ) in the basis B . We note that it is similar to the dual Algorithm 1 presented in Section 5.Let Q E n be the Gabriel quiver of P ( E n ) and I be the set of relations of KQ E n defined in Section 1. Then P ( E n ) ∼ = KQ E n /I for the ideal I of KQ E n generated by R . Following Section 2 we denote Q E n = ( Q , Q , s, t ), where Q is the set of vertices of Q , where Q is the set of arrows of Q , and s, t : Q → Q assigns to every arrow its source and target, respectively.In our algorithm B denote the set of basis elements of P ( E n ) and P res denote the forest of trees of the structure presentedin Section 4. We recall that each node of
P res is associated to some uniquely determined path of KQ E n (corresponding to thepath from root of the tree to that node). To each node of P res we assign either a mapping from B to K (the presentation of theassociated element in the basis B ) or ⊠ indicating zero-path or (cid:3) indicating “calculation in progress”. If in P res there is thenode induced by a path ω of KQ E n , then we set P res [ ω ] to the value assigned to that node and otherwise we set P res [ ω ] = ∅ .To simplify the notation we denote as before by U the set of paths corresponding to the nodes with value ⊠ ( U = { ω | P res [ ω ] = (cid:3) } ) and by E i ⊆ Q i the set of paths of length i associated with nodes of P res of values different from ⊠ and (cid:3) .Moreover, for each element b of B , we denote by by δ b the mapping δ b : B → K such that δ b ( b ) = 1 and δ b ( b ′ ) = 0 for b ′ ∈ B \ { b } . Algorithm 3
Finding the basis of P ( E n ) Require: Q E n = ( Q , Q , s, t ), R such that KQ E n / h R i ∼ = P ( E n ) i ← E ← Q , B ← Q ∪ { KQ/ h R i } , P res [] ← { ( α, δ α ) | α ∈ Q } while E i = ∅ do for all ω ∈ E i , α ∈ Q with t ( ω ) = s ( α ) do P res ( ωα ) ← (cid:3) end for i ← i + 1 while U = ∅ do while ∃ ω ∈ U ∃ ∼∈ R ∃ P ⊆ Q i \ U ∀ c ∈ P ∃ α c ∈{− , +1 } u ∼ P c ∈ P α c c do if P c ∈ P ∩ E i α c P res [ c ] = 0 then P res [ ω ] ← P c ∈ P ∩ E i α c P res [ c ] else P res [ ω ] ← ⊠ end if end while if U = ∅ then b ← choose ( U ) B ← B ∪ { b } P res [ b ] ← δ b end if end while end while return B , P res , i − We again made some simplifications. We omitted details on the structure of B and simplified the initialization of P res , asbefore. It also should be mention that in a practical implementation the calculations of the loop while from Lines 7–20 should bedivided in such a way, so the calculations on the paths with different source (respectively, target) could be performed separately.We note that the choice of the basis generators of R ( E n ) and P ( E n ) (performed by the function chose () in Algorithms 1and 3) should be synchronized. Moreover, this choice have a significant impact on the complexity of generated equations andin the consequence on the computational complexity of the final calculations. In Tables 4 and 5 we comparing number of thecoefficients in the obtained equations for different implementations of the function chose ().In our implementation of the function choice () we threat the paths as words over the alphabet consisting of the arrows withthe order defined by the following sequence a , ¯ a , a , ¯ a , ¯ a , a , a , ¯ a , ¯ a , a , . . . , ¯ a n − , a n − , and we chose the first path in thelexicographical order. In other words we use the rule “closest to the exceptional vertex first” and then the rule “from theshortest branch first”. We omit the detailed explanation of why this choice is expected to be “good”. But in the next section wecompare the complexity of the formulas obtained in this way with the complexity of formulas obtained with the implementationof the “first found” approach (see Tables 4 and 5).7. Construction of the system of equations
We keep the notation from Corollary 2.2. We recall that by Corollary 2.2 to prove that for a given admissible f ∈ rad R ( E n ) K -algebras P f ( E n ) and P ( E n ) are isomorphic it suffices to find coefficients α ( k, l, j ) ∈ K such that there are satisfied thefollowing n equalities in P ( E n ) δ ( k , k ) = 0 for ( k , k ) ∈ { (0 , n − , (1 , n ) , (2 n − , n − } ; δ ( k + n − , k −
1) + δ ( k, k + n −
1) = 0 for k ∈ { , , . . . , n − } ; δ ( n − ,
0) + δ ( n + 1 ,
2) + δ (2 n − , n −
2) + f (cid:0) δ ( n − , , δ ( n + 1 , (cid:1) = 0(one equality for each of the vertices) where δ : { , . . . , n − } → P ( E n ) are defined as follows δ ( k , k ) = m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a i ( k ,l,j, . . . a i ( k ,l,j,l − a k + m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a k a i ( k ,l,j, . . . a i ( k ,l,j,l − + (cid:18) m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a i ( k ,l,j, . . . a i ( k ,l,j,l − (cid:19) (cid:18) m X l =2 i ( k ,l ) − X j =0 α ( k , l, j ) a i ( k ,l,j, . . . a i ( k ,l,j,l − (cid:19) . We recall also that the basis elements P ( E n ) can be determined by Algorithm 3 and the operations in P ( E n ) were describedin Section 4 (with the use of the presentation also provided by Algorithm 3). Moreover, f can be presented as a polynomialwith coefficients bound by the relations determined by the admissibility condition, and hence satisfying the equations Subst obtained by Algorithm 2. So in the consequence we should find the coefficients α ( k, l, j ) ∈ K for each set of coefficients of f satisfying Subst .For every non-exceptional vertex k ∈ { , . . . , n − } \ { } the corresponding equality in P ( E n ) can be presented in the formof dim rad e k P ( E n ) e k equations in K (one equation for each of the basis elements of rad e k P ( E n ) e k ). These equations can beeasily obtained by the symbolic calculations.In the case of exceptional vertex 3 we need to calculate f ( δ ( n − , , δ ( n +1 , Subst . Then we may observe that the equation δ ( n − ,
0) + δ ( n + 1 ,
2) + δ (2 n − , n −
2) + f (cid:0) δ ( n − , , δ ( n + 1 , (cid:1) = 0can be presented as the set of dim rad e P ( E n ) e equations over K with dim rad R ( E n ) − Subst invariables. We also notethat in some cases (in particular in the case of E ) it is convenient to refrain from applying equations from Subst directly todim rad e P ( E n ) e (as it can significantly enlarge its notation) and instead to apply them in the final stage (see B.4 and [4, commands.txt ] for details).In Tables 4 and 5 we compare the numbers of coefficients in the obtained equations for different implementations of thefunction chose () (see the notes from the previous section) before applying the equations implied by admissibility condition (seeAlgorithm 2). Table 4.
Number of coefficients for “first found” approach
Dynkin number of the coefficients for the vertextype 0 1 2 3 4 5 6 7 E
11 2 25 351 25 2 – – E
42 11 116 12892 176 43 6 – E
243 48 590 7155925 1173 429 96 11
We note that the chosen approach let us not only to reduce the total number of coefficients for the equations, but also signif-icantly reduce the maximal number of coefficients peer equation. Indeed, observe that dim rad e P ( E ) e = dim rad R ( E ) = N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E Table 5.
Number of coefficients in equations for the choosen approach
Dynkin number of the coefficients for the vertextype 0 1 2 3 4 5 6 7 E
11 2 21 347 27 2 – – E
39 11 100 7173 168 43 6 – E
238 53 539 3790572 1190 455 92 11
57, so in the case of E we obtain 57 equations associated with vertex 3. For the “first found” approach they have respectively6 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Moreover, in the latter approach we obtain more equations with “small” number of coefficients. We expect that this shouldsimplify the substitutions and in result reduce the complexity of solving the set of equations.8.
On solutions of the obtained system of equations
We note that obtained systems of equations in the case of types E and E are to complicated to be solved manually. Sonatural approach is to use some dedicated software package in order to try to solve them.Simple comparing number of equations to number of variables make us expect that if such a system of equations has asolution, then the space of solutions is rather big. On the other hand, we are not interested in finding all solutions but only • to determine if the obtained system of equations has a solution, • if it is the case, then to find one, possibly “simple” exemplary solution.If problem of solving the obtained system of equation is still of acceptable computational complexity, then we may use thefollowing approach: • obtain the general form of the solution, • replace some of the “not-bounded” variables by “0” until we obtain a single solution (a 0-dimensional space of solutions).It works fine in the case of type E (see A.3).Unfortunately, in the case of type E computational complexity of the problem of solving the original obtained system ofequations seems to be a little too high. In that case we may try to act differently:(1) Try to replace some of the variables by “0” in order to reduce the computational complexity of the problem of solvingsuch a system and check if the obtained system of equations is solvable and if we can find its solution.(2) If we obtain a system for which we can find some non-empty space of solutions, then replace some of the (remaining)“not-bounded” variables by “0” and “1” until we obtain a 0-dimensional space of solutions.This approach seems to work in the case of E (see B.3 and B.4), but it has some disadvantages: • The process of finding the “proper” variables to be replaced by “0” in the step (1) is unintuitive and verifying its effectstakes a lot of time (but still less then trying to solve the initial system of equations). • The process of finding the “proper” variables to be replaced by “0” and “1” in the step (2) requires prior analysis ofbig amount of data and verifying its correctness takes a lot of computation (and time). • The structure of the obtained solution seems to be a little artificial and more complicated then necessary. We note thatit can be the result of not optimal choices made in the step (1).
Appendix A. Calculations for type E In this appendix we present construction of a K -algebra isomorphism ϕ : P f ( E ) → P ( E ) for a given admissible deformingelement f according to the algorithm described in the main part of the article. This construction is organized as follows. In A.1we derive the equations for the coefficients of f indicated by the admissibility condition of f . In A.2 we present the general formof the homomorphisms from P f ( E ) to P ( E ) (in particular we give the chosen basis elements of P ( E )). In A.3 we describethe chosen method (see Section 8 for details) of simplifying the system of equations for the coefficients of an isomorphism from P f ( E ) to P ( E ). Finally, in A.4 we present obtained coefficients of a (simplified) isomorphism from P f ( E ) to P ( E ). We estimate, that in the case of E , where the difference in the complexity of equations is smaller than in the case of E , solving the set ofequations obtained with the chosen approach (without applying the equations obtained from the admissibility condition) is about 7–8 times fasterthan solving the set of equations obtained with the “first found” approach. Unfortunately, the complexity of this problem in the case of E is tohigh to be able to make the similar estimations. We refer to the main part of the article for description of applied algorithms as well as for theoretical background. We recallthat considered case is much more complicated than the case of type E (described in [3]) and it is not possible to performmanually calculations described in the main part of the article. But on the other hand the case of type is type E is even morecomplicated and need a little different approach (see Section 8 and B for details).A.1. Equations derived from the admissibility condition.
We carry out the steps described in Section 5. First we computea base of R ( E ) (see Algorithm 2 for details). We take as a basis the set { , x, y, xy, yx, yy, xyx, xyy, yxy, yyx, xyxy, xyyx, yxyx,yxyy, xyxyx, xyxyy, yxyxy, yxyyx, xyxyxy, xyxyyx, yxyxyx, xyxyxyx, yxyxyxy, xyxyxyxy } . Hence we know that each element f ∈ rad R ( E ) is of the form f ( x, y ) = θ xy + θ yx + θ yy + θ xyx + θ xyy + θ yxy + θ yyx + θ xyxy + θ xyyx + θ yxyx + θ yxyy + θ xyxyx + θ xyxyy + θ yxyxy + θ yxyyx + θ xyxyxy + θ xyxyyx + θ yxyxyx + θ xyxyxyx + θ yxyxyxy + θ xyxyxyxy for some θ , . . . , θ ∈ K . We fix these coefficients. Further, we compute (cid:0) x + y + f ( x, y ) (cid:1) = ( θ − θ + θ ) yxyxy + (3 θ − θ + θ − θ + 2 θ − θ θ + θ − θ ) xyxyxy + (3 θ − θ + θ − θ + θ + 2 θ − θ θ − θ ) yxyxyx + (3 θ − θ + 3 θ − θ + 3 θ θ − θ θ + 3 θ θ − θ θ − θ θ + 2 θ θ − θ θ + θ θ + θ θ − θ θ θ + θ + θ ) xyxyxyx + (3 θ − θ + 3 θ − θ + 2 θ θ − θ θ + θ θ + 2 θ θ + θ θ − θ θ + 2 θ θ − θ θ − θ θ + θ θ + θ θ − θ θ θ + θ + θ ) yxyxyxy + (4 θ θ − θ θ + 2 θ θ − θ θ − θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ θ + 2 θ θ θ + 2 θ θ − θ θ − θ θ + 2 θ θ − θ θ θ + 2 θ θ θ + 2 θ θ θ + 2 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 2 θ θ θ + θ − θ θ θ + 4 θ θ θ − θ ) xyxyxyxy (see Sections 5 and 6 for details). Hence f satisfy the admissibility condition if and only if the following equalities are satisfied: θ − θ + θ = 0 , θ − θ + θ − θ + 2 θ − θ θ + θ − θ = 0 , θ − θ + θ − θ + θ + 2 θ − θ θ − θ = 0 , θ − θ + 3 θ − θ + 3 θ θ − θ θ + 3 θ θ − θ θ − θ θ + 2 θ θ − θ θ + θ θ + θ θ − θ θ θ + θ + θ = 0 , θ − θ + 3 θ − θ + 2 θ θ − θ θ + θ θ + 2 θ θ + θ θ − θ θ + 2 θ θ − θ θ − θ θ + θ θ + θ θ − θ θ θ + θ + θ = 0 , θ θ − θ θ + 2 θ θ − θ θ − θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ θ + 2 θ θ θ + 2 θ θ − θ θ − θ θ + 2 θ θ − θ θ θ + 2 θ θ θ + 2 θ θ θ +2 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 2 θ θ θ + θ − θ θ θ + 4 θ θ θ − θ = 0 . We recall that following [5, Theorem] there is a non-isomorphic deformation of P ( E ) in characteristic 2. Hence we may assumethat K is of characteristic different from 2. Then, applying Algorithm 2 to these equations we obtain the following equivalentset of equations: θ = 2 θ − θ ,θ = − θ + 6 θ θ − θ − θ + 2 θ + 2 θ ,θ = θ θ − θ θ + θ − θ θ + θ θ + θ θ − θ θ − θ + ( θ + θ ) . A.2.
General form of homomorphism.
Let f be a given admissible element of the structure described in A.1. In order toconstruct an isomorphism ϕ : P f ( E ) → P ( E ) we want to find the coefficients satisfying the assumptions of Corollary 2.2. Westart with computing the base of P ( E ), according to Algorithm 3. In particular, for each arrow α ∈ Q we compute a basis of N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E e s ( α ) P ( E ) e t ( α ) . Then we conclude that our constructed isomorphism should be given by the following formulas ϕ ( a ) = a + α (0)1 a ¯ a a + α (0)2 a ¯ a a ¯ a a + α (0)3 a ¯ a a ¯ a a + α (0)4 a ¯ a a ¯ a a ¯ a a + α (0)5 a ¯ a a ¯ a a ¯ a a + α (0)6 a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)7 a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)8 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)9 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)10 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)11 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ,ϕ (¯ a ) = ¯ a + ¯ α (0)1 ¯ a a ¯ a + ¯ α (0)2 ¯ a a ¯ a a ¯ a + ¯ α (0)3 ¯ a a ¯ a a ¯ a + ¯ α (0)4 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)5 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)6 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)7 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)8 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)9 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)10 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)11 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ,ϕ ( a ) = a + α (1)1 a a ¯ a a ¯ a + α (1)2 a a ¯ a a ¯ a a ¯ a + α (1)3 a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)4 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)5 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ,ϕ (¯ a ) = ¯ a + ¯ α (1)1 a ¯ a a ¯ a ¯ a + ¯ α (1)2 a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)3 a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)4 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)5 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a ,ϕ ( a ) = a + α (2)1 a ¯ a a + α (2)2 a ¯ a a + α (2)3 a ¯ a a ¯ a a + α (2)4 a ¯ a a ¯ a a + α (2)5 a ¯ a a ¯ a a ¯ a a + α (2)6 a ¯ a a ¯ a a ¯ a a + α (2)7 a ¯ a a ¯ a a ¯ a a + α (2)8 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)9 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)10 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)11 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)12 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)13 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)14 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)15 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ,ϕ (¯ a ) = ¯ a + ¯ α (2)1 ¯ a a ¯ a + ¯ α (2)2 ¯ a a ¯ a + ¯ α (2)3 ¯ a a ¯ a a ¯ a + ¯ α (2)4 ¯ a a ¯ a a ¯ a + ¯ α (2)5 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)6 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)7 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)8 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)9 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)10 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)11 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)12 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)13 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)14 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)15 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ,ϕ ( a ) = a + α (3)1 ¯ a a a + α (3)2 ¯ a a a + α (3)3 ¯ a a ¯ a a a + α (3)4 ¯ a a ¯ a a a + α (3)5 ¯ a a ¯ a a a + α (3)6 ¯ a a ¯ a a ¯ a a a + α (3)7 ¯ a a ¯ a a ¯ a a a + α (3)8 ¯ a a ¯ a a ¯ a a a + α (3)9 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)10 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)11 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)12 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)13 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)14 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)15 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)16 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)17 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a ,ϕ (¯ a ) = ¯ a + ¯ α (3)1 ¯ a ¯ a a + ¯ α (3)2 ¯ a ¯ a a + ¯ α (3)3 ¯ a ¯ a a ¯ a a + ¯ α (3)4 ¯ a ¯ a a ¯ a a + ¯ α (3)5 ¯ a ¯ a a ¯ a a + ¯ α (3)6 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)7 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)8 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)9 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)10 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)11 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)12 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)13 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)14 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)15 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)16 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)17 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ,ϕ ( a ) = a + α (4)1 ¯ a a a + α (4)2 ¯ a ¯ a a a a + α (4)3 ¯ a ¯ a a ¯ a a a a + α (4)4 ¯ a ¯ a a ¯ a a a a + α (4)5 ¯ a ¯ a a ¯ a a ¯ a a a a + α (4)6 ¯ a ¯ a a ¯ a a ¯ a a a a + α (4)7 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)8 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)9 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a ,ϕ (¯ a ) = ¯ a + ¯ α (4)1 ¯ a ¯ a a + ¯ α (4)2 ¯ a ¯ a ¯ a a a + ¯ α (4)3 ¯ a ¯ a ¯ a a ¯ a a a + ¯ α (4)4 ¯ a ¯ a ¯ a a ¯ a a a + ¯ α (4)5 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a + ¯ α (4)6 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a + ¯ α (4)7 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)8 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)9 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a ,ϕ ( a ) = a + α (5)1 ¯ a ¯ a ¯ a a a a a + α (5)2 ¯ a ¯ a ¯ a a ¯ a a a a a + α (5)3 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a ,ϕ (¯ a ) = ¯ a + ¯ α (5)1 ¯ a ¯ a ¯ a ¯ a a a a + ¯ α (5)2 ¯ a ¯ a ¯ a ¯ a a ¯ a a a a + ¯ α (5)3 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a , with coefficients α ( i ) j i , ¯ α ( i ) j i ∈ K , for i = 0 , . . . , j = 1 , . . . , j i , j = 11 , j = 5 , j = 15 , j = 17 , j = 9 , j = 3.A.3. Simplified system of equations.
In order to obtain the simplified system of equations (see Section 7 for details) wewill assume that in the above definition of ϕ the following coefficients are equal zero: α (0)9 , α (0)10 , α (0)11 , ¯ α (0)2 , ¯ α (0)3 , ¯ α (0)4 , ¯ α (0)5 , ¯ α (0)6 , ¯ α (0)7 , ¯ α (0)8 , ¯ α (0)9 , ¯ α (0)10 , ¯ α (0)11 , α (1)1 , α (1)3 , α (1)5 , ¯ α (1)1 , ¯ α (1)2 , ¯ α (1)3 , ¯ α (1)4 , ¯ α (1)5 , α (2)1 , α (2)2 ,α (2)5 , α (2)6 , α (2)8 , α (2)9 , α (2)11 , α (2)12 , ¯ α (2)1 , ¯ α (2)2 , ¯ α (2)5 , ¯ α (2)7 , ¯ α (2)9 , ¯ α (2)10 , ¯ α (2)12 , ¯ α (2)14 , α (3)2 , α (3)3 , α (3)4 , α (3)5 , α (3)6 , α (3)7 , α (3)8 , α (3)9 , α (3)10 ,α (3)11 , α (3)12 , α (3)13 , α (3)14 , α (3)15 , α (3)16 , α (3)17 , ¯ α (3)1 , ¯ α (3)6 , ¯ α (3)15 , α (4)2 , α (4)3 , α (4)4 , α (4)5 , α (4)6 , α (4)7 , α (4)8 , α (4)9 , ¯ α (4)1 , ¯ α (4)2 , α (5)2 , α (5)3 . We note that this choice is crucial to obtain the system of equation with the solution of “acceptable” size. Then, applying therelations of P ( E ) we obtain the following equations − ¯ α (3)2 − α (3)1 + α (0)1 + θ = 0 , ¯ α (0)1 + θ = 0 , − ¯ α (3)2 + θ = 0 , − ¯ α (3)4 + α (0)2 + θ = 0 , − ¯ α (3)5 − α (3)1 ¯ α (3)2 + ¯ α (2)3 + α (0)3 + α (0)1 θ + θ = 0 , − ¯ α (3)3 + α (2)3 + ¯ α (2)4 + ¯ α (0)1 α (0)1 + ¯ α (0)1 θ + α (0)1 θ + θ = 0 , − ¯ α (3)4 + α (2)4 + ¯ α (0)1 θ + θ = 0 , − α (3)1 ¯ α (3)3 − α (2)7 + α (0)4 + α (2)3 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , ¯ α (3)8 − α (3)1 ¯ α (3)4 − α (2)7 + α (0)5 + α (2)4 θ − ¯ α (0)1 α (0)1 θ + ¯ α (2)3 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , ¯ α (3)8 − α (2)7 + ¯ α (0)1 α (0)2 + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ + ¯ α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , − ¯ α (3)7 + ¯ α (3)8 − α (2)7 + ¯ α (2)6 + ¯ α (0)1 α (0)3 + ¯ α (0)1 α (0)1 θ + α (0)3 θ − ¯ α (0)1 α (0)1 θ − α (2)3 θ + ¯ α (2)3 θ − ¯ α (2)4 θ + α (2)3 θ − α (2)4 θ +¯ α (2)4 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , − ¯ α (3)11 + α (3)1 ¯ α (3)8 − α (2)10 − ¯ α (2)3 α (2)3 + α (0)6 − α (2)7 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)5 θ − ¯ α (0)1 α (0)2 θ + ¯ α (0)1 α (0)3 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ − ¯ α (2)3 α (0)1 θ + ¯ α (2)6 θ + α (2)7 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)2 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ + ¯ α (0)1 α (0)1 θ − ¯ α (2)3 θ − α (0)2 θ + α (0)3 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ − ¯ α (0)1 θ − α (0)1 θ − α (0)1 θ − ¯ α (0)1 θ + ¯ α (0)1 θ + θ = 0 , − ¯ α (3)11 − α (3)1 ¯ α (3)7 + α (3)1 ¯ α (3)8 − ¯ α (2)3 α (2)3 + ¯ α (2)8 + α (0)7 − α (2)7 θ + ¯ α (2)6 θ − α (0)1 α (2)3 θ + α (0)1 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 θ + α (0)4 θ − α (0)5 θ − ¯ α (2)3 α (0)1 θ + α (0)3 θ − ¯ α (0)1 α (0)1 θ − α (0)1 α (0)1 θ − α (2)3 θ + ¯ α (2)3 θ − ¯ α (2)4 θ + α (2)3 θ − α (2)4 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (2)3 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ − α (0)1 θ + θ = 0 , − ¯ α (3)9 + ¯ α (3)11 + ¯ α (2)4 α (2)3 + ¯ α (0)1 α (0)4 + ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 θ + α (0)4 θ − ¯ α (0)1 α (0)3 θ + α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − ¯ α (2)6 θ − α (2)7 θ + ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)3 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ + θ = 0 , − ¯ α (3)10 − α (2)10 + ¯ α (2)4 α (2)4 + ¯ α (0)1 α (0)5 + ¯ α (0)1 α (2)4 θ + α (0)5 θ − ¯ α (0)1 α (0)2 θ + α (2)3 ¯ α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)6 θ +¯ α (0)1 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 θ + α (2)4 θ − α (0)2 θ − α (2)3 θ + ¯ α (2)3 θ − ¯ α (2)4 θ + α (2)3 θ − α (2)4 θ + ¯ α (2)4 θ + ¯ α (0)1 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ + θ = 0 , − ¯ α (3)13 − α (3)1 ¯ α (3)9 + α (3)1 ¯ α (3)11 − ¯ α (2)4 α (2)7 − ¯ α (2)6 α (2)3 + ¯ α (2)11 + α (0)8 + ¯ α (0)1 α (0)7 + ¯ α (2)4 α (2)3 θ − α (0)1 ¯ α (2)6 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ − α (0)3 ¯ α (2)3 θ + α (0)6 θ − ¯ α (0)1 α (2)7 θ + ¯ α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ + α (0)7 θ + α (2)3 α (0)3 θ − α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (2)3 α (0)3 θ + ¯ α (2)4 α (0)3 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)6 α (0)1 θ − ¯ α (2)3 α (2)3 θ + ¯ α (2)8 θ − α (2)3 α (2)3 θ + α (2)3 ¯ α (2)3 θ − α (2)3 ¯ α (2)4 θ − ¯ α (2)3 ¯ α (2)3 θ − ¯ α (2)3 α (2)3 θ − ¯ α (2)4 α (2)3 θ + ¯ α (2)4 ¯ α (2)3 θ − ¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)4 α (2)4 θ + α (0)4 θ − ¯ α (0)1 α (0)3 θ + α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − α (0)1 α (0)3 θ + α (0)2 α (0)1 θ + ¯ α (0)1 α (0)3 θ − ¯ α (0)1 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (0)1 α (0)1 θ − ¯ α (2)6 θ − α (2)7 θ − α (0)1 ¯ α (2)3 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)5 θ − ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 α (2)4 θ + ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 θ − α (2)7 θ +¯ α (2)6 θ − α (0)1 α (2)3 θ + α (0)1 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 θ + α (0)4 θ − α (0)5 θ + α (2)3 α (0)1 θ − α (2)3 ¯ α (0)1 θ − ¯ α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ − ¯ α (2)3 α (0)1 θ − α (2)3 α (0)1 θ − ¯ α (2)4 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)2 θ +¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ − α (0)3 θ − ¯ α (0)1 α (0)1 θ + α (0)3 θ − ¯ α (0)1 α (0)1 θ − α (0)1 α (0)1 θ − α (2)3 θ + ¯ α (2)3 θ − ¯ α (2)4 θ + α (2)3 θ − α (2)4 θ + ¯ α (2)4 θ + α (0)2 θ + α (2)3 θ − ¯ α (2)3 θ + ¯ α (2)4 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ − α (0)1 θ + θ = 0 , − ¯ α (3)14 − α (3)1 ¯ α (3)10 − α (2)10 θ + ¯ α (2)4 α (2)4 θ + α (0)2 α (2)4 θ − ¯ α (2)3 α (0)2 θ − ¯ α (2)3 α (2)3 θ + ¯ α (2)8 θ + α (0)5 θ − ¯ α (0)1 α (0)2 θ + α (2)3 ¯ α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)6 θ − α (0)1 α (0)2 θ − α (0)1 α (2)3 θ + α (0)1 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 θ + α (0)2 ¯ α (0)1 θ + α (0)4 θ − α (0)5 θ + α (2)4 θ − α (0)2 θ − α (2)3 θ + ¯ α (2)3 θ − ¯ α (2)4 θ + α (2)3 θ − α (2)4 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (2)3 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , − ¯ α (3)12 + ¯ α (3)14 − ¯ α (2)4 α (2)7 − ¯ α (2)6 α (2)3 + ¯ α (0)1 α (0)6 − ¯ α (0)1 α (2)7 θ − ¯ α (0)1 α (0)1 α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)5 θ + α (0)6 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ + α (2)3 α (0)2 θ − α (2)3 ¯ α (0)1 α (0)1 θ + α (2)4 ¯ α (0)1 α (0)1 θ + α (2)7 α (0)1 θ − α (2)7 ¯ α (0)1 θ + ¯ α (2)4 α (0)2 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)6 α (0)1 θ − ¯ α (2)3 α (2)3 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)4 α (2)4 θ − α (2)3 α (2)3 θ − α (2)3 ¯ α (2)4 θ + α (2)4 α (2)3 θ − α (2)4 α (2)4 θ + α (2)4 ¯ α (2)4 θ + α (2)10 θ − ¯ α (2)4 α (2)3 θ − ¯ α (2)4 ¯ α (2)4 θ − ¯ α (2)4 α (2)4 θ + ¯ α (0)1 α (0)2 θ − ¯ α (0)1 ¯ α (0)1 α (0)1 θ + ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)1 α (0)1 θ + ¯ α (0)1 α (0)2 θ − ¯ α (0)1 α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (2)4 θ − α (2)7 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)5 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 α (2)4 θ + ¯ α (0)1 ¯ α (2)4 θ +¯ α (0)1 α (0)2 θ − α (2)3 ¯ α (0)1 θ + α (2)4 ¯ α (0)1 θ + α (2)7 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ + α (0)4 θ − α (0)5 θ + α (2)3 α (0)1 θ − α (2)3 ¯ α (0)1 θ − ¯ α (2)3 α (0)1 θ +¯ α (2)4 α (0)1 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ − α (2)3 α (0)1 θ + α (2)4 α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ − ¯ α (2)4 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (0)1 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)2 θ + ¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ + ¯ α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + ¯ α (0)1 θ − α (0)1 θ − ¯ α (0)1 θ + θ = 0 , − ¯ α (3)16 − α (3)1 ¯ α (3)12 + α (3)1 ¯ α (3)14 + α (2)14 − ¯ α (2)4 α (2)10 − ¯ α (2)8 α (2)3 − ¯ α (2)13 − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 α (2)3 θ − α (0)1 ¯ α (2)3 α (2)3 θ + α (0)1 ¯ α (2)4 α (2)3 θ − α (0)1 ¯ α (2)4 α (2)4 θ − α (0)2 α (2)7 θ − α (0)4 α (2)3 θ − α (0)4 ¯ α (2)4 θ + α (0)5 α (2)3 θ − α (0)5 α (2)4 θ + α (0)5 ¯ α (2)4 θ − ¯ α (0)1 α (2)10 θ + ¯ α (0)1 ¯ α (2)4 α (2)4 θ +¯ α (0)1 α (0)2 α (2)4 θ + α (2)3 α (0)5 θ − α (2)3 ¯ α (0)1 α (0)2 θ + ¯ α (2)3 α (0)4 θ − ¯ α (2)3 α (0)5 θ + ¯ α (2)3 α (2)3 α (0)1 θ − ¯ α (2)3 α (2)3 ¯ α (0)1 θ + ¯ α (2)4 α (0)5 θ − ¯ α (2)4 ¯ α (0)1 α (0)2 θ +¯ α (2)4 α (2)3 ¯ α (0)1 θ − ¯ α (2)4 α (2)4 ¯ α (0)1 θ − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 α (0)2 θ − ¯ α (2)6 α (2)3 θ − ¯ α (2)8 α (0)1 θ + ¯ α (2)11 θ + α (0)6 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ + α (2)3 α (0)2 θ − α (2)3 ¯ α (0)1 α (0)1 θ + α (2)4 ¯ α (0)1 α (0)1 θ + α (2)7 α (0)1 θ − α (2)7 ¯ α (0)1 θ + ¯ α (2)4 α (0)2 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)6 α (0)1 θ + α (0)1 α (0)4 θ − α (0)1 α (0)5 θ + α (0)1 α (2)3 α (0)1 θ − α (0)1 α (2)3 ¯ α (0)1 θ − α (0)1 ¯ α (2)3 α (0)1 θ + α (0)1 ¯ α (2)4 α (0)1 θ − α (0)1 ¯ α (2)4 ¯ α (0)1 θ − α (0)1 ¯ α (2)6 θ + α (0)2 α (0)2 θ − α (0)2 ¯ α (0)1 α (0)1 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ − α (0)3 ¯ α (2)3 θ − α (0)4 α (0)1 θ + α (0)5 α (0)1 θ − α (0)5 ¯ α (0)1 θ + α (0)6 θ + ¯ α (0)1 α (0)5 θ − ¯ α (0)1 ¯ α (0)1 α (0)2 θ +¯ α (0)1 α (2)3 ¯ α (0)1 θ − ¯ α (0)1 α (2)4 ¯ α (0)1 θ − ¯ α (0)1 α (2)7 θ + ¯ α (0)1 ¯ α (2)4 ¯ α (0)1 θ + ¯ α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (0)1 α (0)2 θ − ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 ¯ α (0)1 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ − ¯ α (2)3 α (2)3 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)4 α (2)4 θ − α (2)3 α (2)3 θ − α (2)3 ¯ α (2)4 θ + α (2)4 α (2)3 θ − α (2)4 α (2)4 θ + α (2)4 ¯ α (2)4 θ + α (2)10 θ − ¯ α (2)4 α (2)3 θ − ¯ α (2)4 ¯ α (2)4 θ − ¯ α (2)4 α (2)4 θ − α (0)2 α (2)3 θ − α (0)2 ¯ α (2)4 θ − α (0)2 α (2)4 θ − α (2)10 θ +¯ α (2)4 α (2)4 θ + α (0)2 α (2)4 θ + α (2)3 α (2)4 θ + ¯ α (2)3 α (2)3 θ − ¯ α (2)3 α (2)4 θ + ¯ α (2)3 ¯ α (2)4 θ + ¯ α (2)3 α (0)2 θ + ¯ α (2)3 α (2)3 θ + ¯ α (2)4 α (2)4 θ − ¯ α (2)8 θ − ¯ α (2)3 α (0)2 θ − ¯ α (2)3 α (2)3 θ + ¯ α (2)8 θ − α (2)3 α (0)2 θ − α (2)3 α (2)3 θ + α (2)3 ¯ α (2)3 θ − α (2)3 ¯ α (2)4 θ − ¯ α (2)3 ¯ α (2)3 θ − ¯ α (2)3 α (2)3 θ − ¯ α (2)4 α (0)2 θ − ¯ α (2)4 α (2)3 θ + ¯ α (2)4 ¯ α (2)3 θ − ¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)4 α (2)4 θ − α (2)7 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)5 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 α (2)4 θ +¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 θ − α (2)3 ¯ α (0)1 θ + α (2)4 ¯ α (0)1 θ + α (2)7 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ + α (0)1 α (2)3 θ − α (0)1 α (2)4 θ + α (0)1 ¯ α (2)4 θ + α (0)1 α (0)2 θ + α (0)1 α (2)3 θ − α (0)1 ¯ α (2)3 θ + α (0)1 ¯ α (2)4 θ − α (0)2 ¯ α (0)1 θ − α (0)4 θ + α (0)5 θ + ¯ α (0)1 α (2)4 θ + α (0)4 θ − α (0)5 θ + α (2)3 α (0)1 θ − α (2)3 ¯ α (0)1 θ − ¯ α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ − α (2)3 α (0)1 θ + α (2)4 α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ − ¯ α (2)4 α (0)1 θ − α (0)1 ¯ α (2)3 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)2 α (0)1 θ − α (0)5 θ − ¯ α (0)1 α (0)2 θ − ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 α (2)4 θ + ¯ α (0)1 ¯ α (2)4 θ +¯ α (0)1 α (0)2 θ + α (0)5 θ − ¯ α (0)1 α (0)2 θ + α (2)3 ¯ α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)6 θ − α (0)1 α (0)2 θ − α (0)1 α (2)3 θ + α (0)1 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 θ + α (0)2 ¯ α (0)1 θ + α (0)4 θ − α (0)5 θ + α (2)3 ¯ α (0)1 θ + α (2)3 α (0)1 θ − α (2)3 ¯ α (0)1 θ + ¯ α (2)3 α (0)1 θ − ¯ α (2)3 ¯ α (0)1 θ − ¯ α (2)3 α (0)1 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)4 α (0)1 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)2 θ +¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ + α (0)1 α (0)1 θ − α (0)1 ¯ α (0)1 θ − α (0)1 α (0)1 θ + α (0)2 θ + ¯ α (0)1 ¯ α (0)1 θ +¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ − α (2)3 θ − ¯ α (2)4 θ − α (2)4 θ + α (2)4 θ + ¯ α (2)3 θ − α (0)2 θ − α (2)3 θ + ¯ α (2)3 θ − ¯ α (2)4 θ + α (2)3 θ − α (2)4 θ + ¯ α (2)4 θ + α (0)2 θ + α (2)3 θ − ¯ α (2)3 θ + ¯ α (2)4 θ − ¯ α (0)1 θ − α (0)1 θ + ¯ α (0)1 θ + α (0)1 θ − α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + ¯ α (0)1 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E α (2)13 + ¯ α (0)1 α (0)8 + ¯ α (0)1 ¯ α (2)4 α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)6 θ + ¯ α (0)1 α (0)2 α (2)3 θ + ¯ α (0)1 α (0)2 ¯ α (2)4 θ − ¯ α (0)1 α (0)3 ¯ α (2)3 θ + ¯ α (0)1 α (0)6 θ + α (0)8 θ + ¯ α (0)1 α (0)7 θ + α (2)3 α (0)4 θ − α (2)3 ¯ α (0)1 α (0)3 θ + α (2)4 ¯ α (0)1 α (0)3 θ + α (2)7 α (0)3 θ − α (2)7 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 α (0)4 θ − ¯ α (2)4 ¯ α (0)1 α (0)3 θ + ¯ α (2)4 α (2)3 α (0)1 θ − ¯ α (2)6 α (0)3 θ − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 α (2)3 θ + ¯ α (2)11 θ − α (2)3 ¯ α (2)6 θ − α (2)4 α (2)7 θ + α (2)4 ¯ α (2)6 θ − α (2)7 α (2)3 θ + α (2)7 ¯ α (2)3 θ − α (2)7 ¯ α (2)4 θ − ¯ α (2)4 ¯ α (2)6 θ − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 ¯ α (2)3 θ − ¯ α (2)6 α (2)3 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 ¯ α (0)1 α (0)3 θ + ¯ α (0)1 α (2)3 α (0)1 θ + ¯ α (0)1 ¯ α (2)4 α (0)1 θ − ¯ α (0)1 α (0)1 α (0)3 θ + ¯ α (0)1 α (0)2 α (0)1 θ − ¯ α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (2)7 θ − ¯ α (0)1 α (0)1 ¯ α (2)3 θ − ¯ α (0)1 α (0)1 α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)5 θ + ¯ α (2)4 α (2)3 θ − α (0)1 ¯ α (2)6 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ − α (0)3 ¯ α (2)3 θ + α (0)6 θ − ¯ α (0)1 α (2)7 θ + ¯ α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ + α (2)3 α (2)3 θ + α (2)3 ¯ α (2)4 θ + α (2)3 α (0)2 θ − α (2)3 ¯ α (0)1 α (0)1 θ + α (2)4 ¯ α (0)1 α (0)1 θ + α (2)7 α (0)1 θ − α (2)7 ¯ α (0)1 θ + ¯ α (2)4 α (2)3 θ +¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (0)2 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)6 α (0)1 θ + α (0)7 θ + α (2)3 α (0)3 θ − α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (2)3 α (0)3 θ + ¯ α (2)4 α (0)3 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)6 α (0)1 θ − α (2)3 α (0)3 θ + α (2)4 α (0)3 θ − α (2)4 ¯ α (0)1 α (0)1 θ − α (2)7 α (0)1 θ − ¯ α (2)4 α (0)3 θ + ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 θ +¯ α (0)1 α (0)2 θ − ¯ α (0)1 ¯ α (0)1 α (0)1 θ + ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)1 α (0)1 θ + ¯ α (0)1 α (0)2 θ − ¯ α (0)1 α (0)3 θ + α (0)4 θ − ¯ α (0)1 α (0)3 θ + α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − α (0)1 α (0)3 θ + α (0)2 α (0)1 θ + ¯ α (0)1 α (0)3 θ − ¯ α (0)1 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (0)1 α (0)1 θ + α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − ¯ α (2)6 θ − α (2)7 θ − α (0)1 ¯ α (2)3 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)5 θ − ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 α (2)4 θ + ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 θ − α (2)3 ¯ α (0)1 θ + α (2)4 ¯ α (0)1 θ + α (2)7 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ + ¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)2 θ + ¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)3 θ − ¯ α (0)1 α (0)1 θ + ¯ α (0)1 θ + α (0)1 θ + θ = 0 , ¯ α (3)17 − α (2)15 + ¯ α (2)11 α (2)3 + ¯ α (2)15 + α (2)13 θ − α (0)1 ¯ α (2)4 α (2)7 θ − α (0)1 ¯ α (2)6 α (2)3 θ + α (0)1 ¯ α (2)11 θ + α (0)2 ¯ α (2)4 α (2)3 θ − α (0)3 ¯ α (2)3 α (2)3 θ + α (0)3 ¯ α (2)8 θ − α (0)4 ¯ α (2)6 θ − α (0)5 α (2)7 θ + α (0)5 ¯ α (2)6 θ + α (0)6 α (2)3 θ + α (0)6 ¯ α (2)4 θ − α (0)7 ¯ α (2)3 θ + ¯ α (0)1 ¯ α (2)4 α (2)7 θ + ¯ α (0)1 ¯ α (2)6 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)4 α (2)4 θ + ¯ α (0)1 α (0)2 α (2)7 θ + ¯ α (0)1 α (0)4 α (2)3 θ + ¯ α (0)1 α (0)4 ¯ α (2)4 θ − ¯ α (0)1 α (0)5 α (2)3 θ + ¯ α (0)1 α (0)5 α (2)4 θ − ¯ α (0)1 α (0)5 ¯ α (2)4 θ − α (2)3 α (0)6 θ − α (2)3 ¯ α (0)1 α (0)4 θ + α (2)3 ¯ α (0)1 α (0)5 θ − α (2)4 ¯ α (0)1 α (0)5 θ − α (2)7 α (0)5 θ + α (2)7 ¯ α (0)1 α (0)2 θ − α (2)13 θ + ¯ α (2)3 α (0)7 θ +¯ α (2)3 α (2)3 α (0)3 θ − ¯ α (2)3 α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (2)4 α (0)6 θ − ¯ α (2)4 ¯ α (0)1 α (0)4 θ + ¯ α (2)4 ¯ α (0)1 α (0)5 θ − ¯ α (2)4 α (2)3 α (0)2 θ + ¯ α (2)4 α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (2)4 α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)4 α (2)7 α (0)1 θ + ¯ α (2)4 α (2)7 ¯ α (0)1 θ − ¯ α (2)6 α (0)4 θ + ¯ α (2)6 α (0)5 θ − ¯ α (2)6 α (2)3 α (0)1 θ + ¯ α (2)6 α (2)3 ¯ α (0)1 θ − ¯ α (2)8 α (0)3 θ +¯ α (2)11 α (0)1 θ − α (2)14 θ + ¯ α (2)4 α (2)10 θ + ¯ α (2)8 α (2)3 θ + ¯ α (2)13 θ + α (2)3 ¯ α (2)3 α (2)3 θ − α (2)3 ¯ α (2)4 α (2)3 θ + α (2)3 ¯ α (2)4 α (2)4 θ + α (2)4 α (2)10 θ − α (2)4 ¯ α (2)4 α (2)4 θ − α (2)10 α (2)4 θ + α (2)14 θ − ¯ α (2)3 ¯ α (2)3 α (2)3 θ + ¯ α (2)3 ¯ α (2)8 θ − ¯ α (2)3 α (2)3 α (2)3 θ − ¯ α (2)3 ¯ α (2)3 α (2)3 θ + ¯ α (2)3 ¯ α (2)8 θ − ¯ α (2)3 α (2)3 α (2)3 θ +¯ α (2)3 α (2)3 ¯ α (2)3 θ − ¯ α (2)3 α (2)3 ¯ α (2)4 θ + ¯ α (2)4 ¯ α (2)3 α (2)3 θ − ¯ α (2)4 ¯ α (2)4 α (2)3 θ + ¯ α (2)4 ¯ α (2)4 α (2)4 θ + ¯ α (2)4 α (2)3 α (2)3 θ + ¯ α (2)4 α (2)3 ¯ α (2)4 θ − ¯ α (2)4 α (2)4 α (2)3 θ +¯ α (2)4 α (2)4 α (2)4 θ − ¯ α (2)4 α (2)4 ¯ α (2)4 θ − ¯ α (2)4 α (2)10 θ − ¯ α (2)8 ¯ α (2)3 θ − ¯ α (2)8 α (2)3 θ − ¯ α (2)13 θ + α (0)8 θ + ¯ α (0)1 α (0)7 θ + α (2)3 α (0)4 θ − α (2)3 ¯ α (0)1 α (0)3 θ + α (2)4 ¯ α (0)1 α (0)3 θ + α (2)7 α (0)3 θ − α (2)7 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 α (0)4 θ − ¯ α (2)4 ¯ α (0)1 α (0)3 θ + ¯ α (2)4 α (2)3 α (0)1 θ − ¯ α (2)6 α (0)3 θ + α (0)1 α (0)7 θ + α (0)1 α (2)3 α (0)3 θ − α (0)1 α (2)3 ¯ α (0)1 α (0)1 θ − α (0)1 ¯ α (2)3 α (0)3 θ + α (0)1 ¯ α (2)4 α (0)3 θ − α (0)1 ¯ α (2)4 ¯ α (0)1 α (0)1 θ − α (0)1 ¯ α (2)6 α (0)1 θ + α (0)2 α (0)4 θ − α (0)2 ¯ α (0)1 α (0)3 θ + α (0)2 α (2)3 α (0)1 θ + α (0)2 ¯ α (2)4 α (0)1 θ − α (0)3 ¯ α (2)3 α (0)1 θ − α (0)4 α (0)3 θ + α (0)5 α (0)3 θ − α (0)5 ¯ α (0)1 α (0)1 θ + α (0)6 α (0)1 θ − ¯ α (0)1 α (0)6 θ − ¯ α (0)1 ¯ α (0)1 α (0)4 θ + ¯ α (0)1 ¯ α (0)1 α (0)5 θ − ¯ α (0)1 α (2)3 α (0)2 θ + ¯ α (0)1 α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (2)7 α (0)1 θ +¯ α (0)1 α (2)7 ¯ α (0)1 θ − ¯ α (0)1 ¯ α (2)4 α (0)2 θ + ¯ α (0)1 ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (2)4 α (2)3 θ + ¯ α (0)1 ¯ α (2)6 α (0)1 θ − ¯ α (0)1 α (0)1 α (0)4 θ + ¯ α (0)1 α (0)1 α (0)5 θ − ¯ α (0)1 α (0)1 α (2)3 α (0)1 θ + ¯ α (0)1 α (0)1 α (2)3 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 α (0)1 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 α (0)1 θ + ¯ α (0)1 α (0)1 ¯ α (2)4 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (0)2 α (0)2 θ + ¯ α (0)1 α (0)2 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (0)2 α (2)3 θ − ¯ α (0)1 α (0)2 ¯ α (2)4 θ + ¯ α (0)1 α (0)3 ¯ α (2)3 θ + ¯ α (0)1 α (0)4 α (0)1 θ − ¯ α (0)1 α (0)5 α (0)1 θ +¯ α (0)1 α (0)5 ¯ α (0)1 θ − ¯ α (0)1 α (0)6 θ − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 α (2)3 θ + ¯ α (2)11 θ − α (2)3 ¯ α (2)6 θ − α (2)4 α (2)7 θ + α (2)4 ¯ α (2)6 θ − α (2)7 α (2)3 θ + α (2)7 ¯ α (2)3 θ − α (2)7 ¯ α (2)4 θ − ¯ α (2)4 ¯ α (2)6 θ − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 ¯ α (2)3 θ − ¯ α (2)6 α (2)3 θ − α (0)1 ¯ α (2)3 α (2)3 θ + α (0)1 ¯ α (2)8 θ − α (0)1 α (2)3 α (2)3 θ + α (0)1 α (2)3 ¯ α (2)3 θ − α (0)1 α (2)3 ¯ α (2)4 θ − α (0)1 ¯ α (2)3 ¯ α (2)3 θ − α (0)1 ¯ α (2)3 α (2)3 θ − α (0)1 ¯ α (2)4 α (2)3 θ + α (0)1 ¯ α (2)4 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 ¯ α (2)4 θ + α (0)1 ¯ α (2)4 α (2)3 θ − α (0)1 ¯ α (2)4 α (2)4 θ − α (0)2 ¯ α (2)6 θ − α (0)2 α (2)7 θ − α (0)4 ¯ α (2)3 θ − α (0)4 α (2)3 θ − α (0)4 ¯ α (2)4 θ − α (0)5 α (2)3 θ + α (0)5 ¯ α (2)3 θ − α (0)5 ¯ α (2)4 θ + α (0)5 α (2)3 θ − α (0)5 α (2)4 θ + α (0)5 ¯ α (2)4 θ + ¯ α (0)1 ¯ α (2)3 α (2)3 θ − ¯ α (0)1 ¯ α (2)4 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 α (2)4 θ + ¯ α (0)1 α (2)3 α (2)3 θ + ¯ α (0)1 α (2)3 ¯ α (2)4 θ − ¯ α (0)1 α (2)4 α (2)3 θ + ¯ α (0)1 α (2)4 α (2)4 θ − ¯ α (0)1 α (2)4 ¯ α (2)4 θ − ¯ α (0)1 α (2)10 θ + ¯ α (0)1 ¯ α (2)4 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 ¯ α (2)4 θ + ¯ α (0)1 ¯ α (2)4 α (2)4 θ + ¯ α (0)1 α (0)2 α (2)3 θ + ¯ α (0)1 α (0)2 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 α (2)4 θ +¯ α (2)4 α (2)7 θ + ¯ α (2)6 α (2)3 θ + α (0)1 ¯ α (2)3 α (2)3 θ − α (0)1 ¯ α (2)4 α (2)3 θ + α (0)1 ¯ α (2)4 α (2)4 θ + α (0)2 α (2)7 θ + α (0)4 α (2)3 θ + α (0)4 ¯ α (2)4 θ − α (0)5 α (2)3 θ + α (0)5 α (2)4 θ − α (0)5 ¯ α (2)4 θ + ¯ α (0)1 α (2)10 θ − ¯ α (0)1 ¯ α (2)4 α (2)4 θ − ¯ α (0)1 α (0)2 α (2)4 θ + α (2)3 α (2)7 θ + α (2)3 α (0)1 α (2)3 θ + α (2)3 α (0)1 ¯ α (2)4 θ + α (2)3 α (0)5 θ − α (2)3 ¯ α (0)1 α (2)3 θ + α (2)3 ¯ α (0)1 α (2)4 θ − α (2)3 ¯ α (0)1 ¯ α (2)4 θ − α (2)3 ¯ α (0)1 α (0)2 θ − α (2)4 ¯ α (0)1 α (2)4 θ − α (2)7 α (2)4 θ − ¯ α (2)3 α (2)7 θ + ¯ α (2)3 ¯ α (2)6 θ − ¯ α (2)3 α (0)1 α (2)3 θ + ¯ α (2)3 α (0)1 ¯ α (2)3 θ − ¯ α (2)3 α (0)1 ¯ α (2)4 θ + ¯ α (2)3 α (0)4 θ − ¯ α (2)3 α (0)5 θ + ¯ α (2)3 α (2)3 α (0)1 θ − ¯ α (2)3 α (2)3 ¯ α (0)1 θ + ¯ α (2)4 α (2)7 θ +¯ α (2)4 α (0)1 α (2)3 θ + ¯ α (2)4 α (0)1 ¯ α (2)4 θ + ¯ α (2)4 α (0)5 θ − ¯ α (2)4 ¯ α (0)1 α (2)3 θ + ¯ α (2)4 ¯ α (0)1 α (2)4 θ − ¯ α (2)4 ¯ α (0)1 ¯ α (2)4 θ − ¯ α (2)4 ¯ α (0)1 α (0)2 θ + ¯ α (2)4 α (2)3 ¯ α (0)1 θ − ¯ α (2)4 α (2)4 ¯ α (0)1 θ − ¯ α (2)4 α (2)7 θ − ¯ α (2)6 α (2)3 θ + ¯ α (2)6 α (2)4 θ − ¯ α (2)6 ¯ α (2)4 θ − ¯ α (2)6 α (0)2 θ − ¯ α (2)6 α (2)3 θ − ¯ α (2)8 α (0)1 θ + ¯ α (2)11 θ − α (2)3 α (0)5 θ + α (2)3 ¯ α (0)1 α (0)2 θ − ¯ α (2)3 α (0)4 θ + ¯ α (2)3 α (0)5 θ − ¯ α (2)3 α (2)3 α (0)1 θ + ¯ α (2)3 α (2)3 ¯ α (0)1 θ − ¯ α (2)4 α (0)5 θ + ¯ α (2)4 ¯ α (0)1 α (0)2 θ − ¯ α (2)4 α (2)3 ¯ α (0)1 θ +¯ α (2)4 α (2)4 ¯ α (0)1 θ + ¯ α (2)4 α (2)7 θ + ¯ α (2)6 α (0)2 θ + ¯ α (2)6 α (2)3 θ + ¯ α (2)8 α (0)1 θ − ¯ α (2)11 θ − α (2)3 α (0)4 θ + α (2)3 α (0)5 θ − α (2)3 α (2)3 α (0)1 θ + α (2)3 α (2)3 ¯ α (0)1 θ + α (2)3 ¯ α (2)3 α (0)1 θ − α (2)3 ¯ α (2)4 α (0)1 θ + α (2)3 ¯ α (2)4 ¯ α (0)1 θ + α (2)3 ¯ α (2)6 θ − α (2)4 α (0)5 θ + α (2)4 ¯ α (0)1 α (0)2 θ − α (2)4 α (2)3 ¯ α (0)1 θ + α (2)4 α (2)4 ¯ α (0)1 θ + α (2)4 α (2)7 θ − α (2)4 ¯ α (2)4 ¯ α (0)1 θ − α (2)4 ¯ α (2)6 θ + α (2)7 α (0)2 θ + α (2)7 α (2)3 θ − α (2)7 ¯ α (2)3 θ + α (2)7 ¯ α (2)4 θ − α (2)10 ¯ α (0)1 θ − ¯ α (2)3 ¯ α (2)3 α (0)1 θ − ¯ α (2)3 α (2)3 α (0)1 θ − ¯ α (2)4 α (0)4 θ + ¯ α (2)4 α (0)5 θ − ¯ α (2)4 α (2)3 α (0)1 θ + ¯ α (2)4 α (2)3 ¯ α (0)1 θ + ¯ α (2)4 ¯ α (2)3 α (0)1 θ − ¯ α (2)4 ¯ α (2)4 α (0)1 θ +¯ α (2)4 ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)4 ¯ α (2)6 θ + ¯ α (2)4 α (2)3 α (0)1 θ − ¯ α (2)4 α (2)4 α (0)1 θ + ¯ α (2)4 α (2)4 ¯ α (0)1 θ + ¯ α (2)4 α (2)7 θ + ¯ α (2)6 ¯ α (2)3 θ + ¯ α (2)6 α (2)3 θ + ¯ α (2)4 α (2)3 θ − α (0)1 ¯ α (2)6 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ − α (0)3 ¯ α (2)3 θ + α (0)6 θ − ¯ α (0)1 α (2)7 θ + ¯ α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ + α (2)3 α (2)3 θ + α (2)3 ¯ α (2)4 θ + α (2)3 α (0)2 θ − α (2)3 ¯ α (0)1 α (0)1 θ + α (2)4 ¯ α (0)1 α (0)1 θ + α (2)7 α (0)1 θ − α (2)7 ¯ α (0)1 θ + ¯ α (2)4 α (2)3 θ + ¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (0)2 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 α (2)3 θ − ¯ α (2)6 α (0)1 θ − α (0)1 α (2)7 θ + α (0)1 ¯ α (2)6 θ − α (0)1 α (0)1 α (2)3 θ + α (0)1 α (0)1 ¯ α (2)3 θ − α (0)1 α (0)1 ¯ α (2)4 θ + α (0)1 α (0)4 θ − α (0)1 α (0)5 θ + α (0)1 α (2)3 α (0)1 θ − α (0)1 α (2)3 ¯ α (0)1 θ − α (0)1 ¯ α (2)3 α (0)1 θ + α (0)1 ¯ α (2)4 α (0)1 θ − α (0)1 ¯ α (2)4 ¯ α (0)1 θ − α (0)1 ¯ α (2)6 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ + α (0)2 α (0)2 θ − α (0)2 ¯ α (0)1 α (0)1 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ − α (0)3 ¯ α (2)3 θ − α (0)4 α (0)1 θ + α (0)5 α (0)1 θ − α (0)5 ¯ α (0)1 θ + α (0)6 θ + ¯ α (0)1 α (2)7 θ + ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)5 θ − ¯ α (0)1 ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (0)1 α (2)4 θ − ¯ α (0)1 ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 ¯ α (0)1 α (0)2 θ + ¯ α (0)1 α (2)3 ¯ α (0)1 θ − ¯ α (0)1 α (2)4 ¯ α (0)1 θ − ¯ α (0)1 α (2)7 θ + ¯ α (0)1 ¯ α (2)4 ¯ α (0)1 θ +¯ α (0)1 ¯ α (2)6 θ − ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 α (2)4 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)1 α (0)2 θ − ¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)4 θ +¯ α (0)1 α (0)2 ¯ α (0)1 θ + ¯ α (0)1 α (0)4 θ − ¯ α (0)1 α (0)5 θ + α (0)7 θ + α (2)3 α (0)3 θ − α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (2)3 α (0)3 θ + ¯ α (2)4 α (0)3 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)6 α (0)1 θ − α (2)3 α (0)3 θ + α (2)4 α (0)3 θ − α (2)4 ¯ α (0)1 α (0)1 θ − α (2)7 α (0)1 θ − ¯ α (2)4 α (0)3 θ − α (0)1 ¯ α (2)3 α (0)1 θ − α (0)1 α (2)3 α (0)1 θ − α (0)1 ¯ α (2)4 α (0)1 θ − α (0)2 α (0)3 θ − α (0)5 α (0)1 θ − ¯ α (0)1 α (0)4 θ + ¯ α (0)1 α (0)5 θ − ¯ α (0)1 α (2)3 α (0)1 θ + ¯ α (0)1 α (2)3 ¯ α (0)1 θ + ¯ α (0)1 ¯ α (2)3 α (0)1 θ − ¯ α (0)1 ¯ α (2)4 α (0)1 θ +¯ α (0)1 ¯ α (2)4 ¯ α (0)1 θ + ¯ α (0)1 ¯ α (2)6 θ + ¯ α (0)1 α (2)3 α (0)1 θ − ¯ α (0)1 α (2)4 α (0)1 θ + ¯ α (0)1 α (2)4 ¯ α (0)1 θ + ¯ α (0)1 α (2)7 θ + ¯ α (0)1 ¯ α (2)4 α (0)1 θ + ¯ α (0)1 α (0)1 ¯ α (2)3 θ +¯ α (0)1 α (0)1 α (2)3 θ + ¯ α (0)1 α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 α (0)1 θ + ¯ α (0)1 α (0)5 θ − α (0)6 θ − ¯ α (0)1 α (0)4 θ + ¯ α (0)1 α (0)5 θ − α (2)3 α (0)2 θ + α (2)3 ¯ α (0)1 α (0)1 θ − α (2)4 ¯ α (0)1 α (0)1 θ − α (2)7 α (0)1 θ + α (2)7 ¯ α (0)1 θ − ¯ α (2)4 α (0)2 θ + ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)4 α (2)3 θ + ¯ α (2)6 α (0)1 θ − α (0)1 α (0)4 θ + α (0)1 α (0)5 θ − α (0)1 α (2)3 α (0)1 θ + α (0)1 α (2)3 ¯ α (0)1 θ + α (0)1 ¯ α (2)3 α (0)1 θ − α (0)1 ¯ α (2)4 α (0)1 θ + α (0)1 ¯ α (2)4 ¯ α (0)1 θ + α (0)1 ¯ α (2)6 θ − α (0)2 α (0)2 θ + α (0)2 ¯ α (0)1 α (0)1 θ − α (0)2 α (2)3 θ − α (0)2 ¯ α (2)4 θ + α (0)3 ¯ α (2)3 θ + α (0)4 α (0)1 θ − α (0)5 α (0)1 θ + α (0)5 ¯ α (0)1 θ − α (0)6 θ − ¯ α (0)1 α (0)5 θ + ¯ α (0)1 ¯ α (0)1 α (0)2 θ − ¯ α (0)1 α (2)3 ¯ α (0)1 θ + ¯ α (0)1 α (2)4 ¯ α (0)1 θ + ¯ α (0)1 α (2)7 θ − ¯ α (0)1 ¯ α (2)4 ¯ α (0)1 θ − ¯ α (0)1 ¯ α (2)6 θ + ¯ α (0)1 α (0)1 α (0)2 θ + ¯ α (0)1 α (0)1 α (2)3 θ − ¯ α (0)1 α (0)1 ¯ α (2)3 θ +¯ α (0)1 α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)2 ¯ α (0)1 θ − ¯ α (0)1 α (0)4 θ + ¯ α (0)1 α (0)5 θ − α (2)3 α (0)2 θ + α (2)3 ¯ α (0)1 α (0)1 θ − α (2)3 α (2)3 θ − α (2)3 ¯ α (2)4 θ + α (2)3 α (0)1 α (0)1 θ − α (2)3 α (0)2 θ − α (2)3 ¯ α (0)1 α (0)1 θ + α (2)3 ¯ α (0)1 ¯ α (0)1 θ + α (2)3 ¯ α (0)1 α (0)1 θ − α (2)4 ¯ α (0)1 ¯ α (0)1 θ − α (2)4 ¯ α (0)1 α (0)1 θ − α (2)7 ¯ α (0)1 θ − α (2)7 α (0)1 θ + α (2)7 ¯ α (0)1 θ + ¯ α (2)3 α (0)3 θ − ¯ α (2)3 ¯ α (0)1 α (0)1 θ − ¯ α (2)3 α (0)1 α (0)1 θ − ¯ α (2)4 α (0)2 θ + ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)4 α (2)3 θ − ¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (0)1 α (0)1 θ − ¯ α (2)4 α (0)2 θ − ¯ α (2)4 ¯ α (0)1 α (0)1 θ + ¯ α (2)4 ¯ α (0)1 ¯ α (0)1 θ + ¯ α (2)4 ¯ α (0)1 α (0)1 θ − ¯ α (2)4 α (2)3 θ − ¯ α (2)6 α (0)1 θ +¯ α (2)6 ¯ α (0)1 θ + ¯ α (2)6 α (0)1 θ + ¯ α (2)3 α (2)3 θ − ¯ α (2)4 α (2)3 θ + ¯ α (2)4 α (2)4 θ + α (2)3 α (2)3 θ + α (2)3 ¯ α (2)4 θ − α (2)4 α (2)3 θ + α (2)4 α (2)4 θ − α (2)4 ¯ α (2)4 θ − α (2)10 θ + ¯ α (2)4 α (2)3 θ + ¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (2)4 θ + α (0)2 α (2)3 θ + α (0)2 ¯ α (2)4 θ + α (0)2 α (2)4 θ + α (2)3 α (2)3 θ + α (2)3 ¯ α (2)4 θ + α (2)3 α (2)4 θ − ¯ α (2)3 α (2)3 θ + ¯ α (2)3 ¯ α (2)3 θ − ¯ α (2)3 ¯ α (2)4 θ + ¯ α (2)3 α (2)3 θ − ¯ α (2)3 α (2)4 θ + ¯ α (2)3 ¯ α (2)4 θ + ¯ α (2)3 α (0)2 θ + ¯ α (2)3 α (2)3 θ +¯ α (2)4 α (2)3 θ + ¯ α (2)4 ¯ α (2)4 θ + ¯ α (2)4 α (2)4 θ − ¯ α (2)8 θ + α (0)4 θ − ¯ α (0)1 α (0)3 θ + α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ − α (0)1 α (0)3 θ + α (0)2 α (0)1 θ +¯ α (0)1 α (0)3 θ − ¯ α (0)1 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (0)1 α (0)1 θ + α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ + α (0)1 α (0)3 θ − α (0)1 ¯ α (0)1 α (0)1 θ − α (0)1 α (0)1 α (0)1 θ + α (0)2 α (0)1 θ − ¯ α (0)1 α (0)2 θ + ¯ α (0)1 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)1 α (0)1 θ − ¯ α (0)1 α (0)2 θ − ¯ α (0)1 ¯ α (0)1 α (0)1 θ +¯ α (0)1 ¯ α (0)1 ¯ α (0)1 θ + ¯ α (0)1 ¯ α (0)1 α (0)1 θ − ¯ α (0)1 α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)1 α (0)1 θ + ¯ α (0)1 α (0)1 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 α (0)1 θ − ¯ α (0)1 α (0)2 θ − ¯ α (2)6 θ − α (2)7 θ − α (0)1 ¯ α (2)3 θ − α (0)1 α (2)3 θ − α (0)1 ¯ α (2)4 θ − α (0)5 θ − ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)3 θ − ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 α (2)4 θ + ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (0)2 θ − α (2)3 ¯ α (0)1 θ + α (2)4 ¯ α (0)1 θ + α (2)7 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ − α (0)1 α (2)3 θ + α (0)1 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 θ + α (0)1 α (2)3 θ − α (0)1 α (2)4 θ + α (0)1 ¯ α (2)4 θ + α (0)1 α (0)2 θ + α (0)1 α (2)3 θ − α (0)1 ¯ α (2)3 θ + α (0)1 ¯ α (2)4 θ − α (0)2 ¯ α (0)1 θ − α (0)4 θ + α (0)5 θ + ¯ α (0)1 α (2)3 θ + ¯ α (0)1 ¯ α (2)4 θ + ¯ α (0)1 α (2)4 θ + α (2)7 θ + α (0)1 α (2)3 θ + α (0)1 ¯ α (2)4 θ + α (0)5 θ − ¯ α (0)1 α (2)3 θ + ¯ α (0)1 α (2)4 θ − ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)2 θ + α (2)3 ¯ α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)6 θ − α (0)1 α (2)3 θ + α (0)1 α (2)4 θ − α (0)1 ¯ α (2)4 θ − α (0)1 α (0)2 θ − α (0)1 α (2)3 θ + α (0)1 ¯ α (2)3 θ − α (0)1 ¯ α (2)4 θ + α (0)2 ¯ α (0)1 θ + α (0)4 θ − α (0)5 θ − ¯ α (0)1 α (2)4 θ + α (2)3 ¯ α (0)1 θ + α (2)3 α (0)1 θ − α (2)3 ¯ α (0)1 θ + ¯ α (2)3 α (0)1 θ − ¯ α (2)3 ¯ α (0)1 θ − ¯ α (2)3 α (0)1 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)4 α (0)1 θ − ¯ α (2)4 ¯ α (0)1 θ − ¯ α (2)6 θ − α (0)4 θ + α (0)5 θ − α (2)3 α (0)1 θ + α (2)3 ¯ α (0)1 θ + ¯ α (2)3 α (0)1 θ − ¯ α (2)4 α (0)1 θ + ¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)6 θ + α (2)3 α (0)1 θ − α (2)4 α (0)1 θ + α (2)4 ¯ α (0)1 θ + α (2)7 θ +¯ α (2)4 α (0)1 θ + α (0)1 ¯ α (2)3 θ + α (0)1 α (2)3 θ + α (0)1 ¯ α (2)4 θ + α (0)2 α (0)1 θ + α (0)5 θ + ¯ α (0)1 α (0)2 θ + ¯ α (0)1 α (2)3 θ − ¯ α (0)1 ¯ α (2)3 θ + ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (2)3 θ + ¯ α (0)1 α (2)4 θ − ¯ α (0)1 ¯ α (2)4 θ − ¯ α (0)1 α (0)2 θ + α (2)3 α (0)1 θ + α (2)3 ¯ α (0)1 θ − α (2)4 ¯ α (0)1 θ − α (2)7 θ − ¯ α (2)3 α (0)1 θ + ¯ α (2)4 α (0)1 θ +¯ α (2)4 ¯ α (0)1 θ + ¯ α (2)6 θ + α (2)3 θ + ¯ α (2)4 θ + α (0)2 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)2 θ + ¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ + α (2)3 θ + ¯ α (2)4 θ + α (0)1 α (0)1 θ − α (0)1 ¯ α (0)1 θ − α (0)1 α (0)1 θ + α (0)2 θ + ¯ α (0)1 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ − α (0)3 θ − ¯ α (0)1 α (0)1 θ − α (0)1 α (0)1 θ + ¯ α (0)1 α (0)1 θ + ¯ α (0)1 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 θ − α (0)2 θ + ¯ α (0)1 α (0)1 θ − α (2)3 θ − ¯ α (2)4 θ + α (0)1 α (0)1 θ − α (0)2 θ − ¯ α (0)1 α (0)1 θ + ¯ α (0)1 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 θ − α (2)3 θ − ¯ α (2)4 θ − α (0)1 α (0)1 θ + α (0)1 ¯ α (0)1 θ + α (0)1 α (0)1 θ − α (0)2 θ − ¯ α (0)1 ¯ α (0)1 θ − ¯ α (0)1 α (0)1 θ + ¯ α (0)1 ¯ α (0)1 θ + ¯ α (0)1 α (0)1 θ − α (2)3 θ − ¯ α (2)4 θ + α (0)1 θ − ¯ α (0)1 θ + ¯ α (0)1 θ + α (0)1 θ − ¯ α (0)1 θ − α (0)1 θ + θ = 0 , ¯ α (0)1 + α (0)1 = 0 ,α (0)1 ¯ α (0)1 + α (0)3 = 0 ,α (0)2 ¯ α (0)1 + α (0)4 = 0 ,α (0)4 ¯ α (0)1 − α (0)5 ¯ α (0)1 + α (0)7 = 0 ,α (0)6 ¯ α (0)1 + α (0)8 = 0 , ¯ α (2)3 + α (2)3 = 0 , ¯ α (2)4 + α (2)4 = 0 , − α (1)2 + ¯ α (2)6 + α (2)7 = 0 , ¯ α (2)8 + α (2)3 ¯ α (2)3 − α (2)4 ¯ α (2)4 − α (2)10 = 0 ,α (2)3 ¯ α (2)4 − α (2)4 ¯ α (2)4 − α (2)10 = 0 , − α (1)4 + ¯ α (2)11 − α (2)3 ¯ α (2)6 + α (2)4 ¯ α (2)6 + α (2)7 ¯ α (2)3 − α (2)7 ¯ α (2)4 = 0 , ¯ α (2)13 + α (2)14 = 0 , ¯ α (2)15 + α (2)3 ¯ α (2)11 + α (2)15 = 0 ,α (4)1 + α (3)1 = 0 ,α (4)1 + ¯ α (3)2 = 0 , ¯ α (3)3 − ¯ α (3)5 = 0 , ¯ α (3)2 α (3)1 + ¯ α (3)4 − ¯ α (3)5 = 0 , ¯ α (3)3 α (3)1 − ¯ α (3)5 α (3)1 = 0 , − ¯ α (4)4 − ¯ α (3)5 α (3)1 + ¯ α (3)7 = 0 , ¯ α (4)3 − ¯ α (4)4 − ¯ α (3)5 α (3)1 + ¯ α (3)8 = 0 , ¯ α (4)5 − ¯ α (3)7 α (3)1 + ¯ α (3)8 α (3)1 + ¯ α (3)9 − ¯ α (3)10 = 0 , ¯ α (4)6 + α (4)1 ¯ α (4)3 + ¯ α (3)11 = 0 , ¯ α (4)7 − α (4)1 ¯ α (4)5 + ¯ α (3)9 α (3)1 − ¯ α (3)11 α (3)1 + ¯ α (3)12 − ¯ α (3)14 = 0 , ¯ α (4)7 − α (4)1 ¯ α (4)5 − ¯ α (3)11 α (3)1 + ¯ α (3)13 − ¯ α (3)14 = 0 , − ¯ α (4)8 − ¯ α (3)13 α (3)1 − ¯ α (3)16 = 0 , − ¯ α (4)9 − α (4)1 ¯ α (4)8 + ¯ α (3)17 = 0 , N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E − α (5)1 + ¯ α (4)3 = 0 , − ¯ α (5)1 + ¯ α (4)4 = 0 , − ¯ α (5)2 − ¯ α (4)4 α (4)1 + ¯ α (4)5 − ¯ α (4)6 = 0 ,α (5)1 ¯ α (5)1 + ¯ α (4)8 = 0 , ¯ α (5)3 + α (5)1 ¯ α (5)2 − ¯ α (4)8 α (4)1 + ¯ α (4)9 = 0 ,α (5)1 + ¯ α (5)1 = 0 , − ¯ α (5)2 α (5)1 + ¯ α (5)3 = 0 . Further, we apply formulas for θ , θ and θ (obtained in A.1), to the above equations and solve obtained system of equationsfor the remaining variables α (0)1 , α (0)2 , α (0)3 , α (0)4 , α (0)5 , α (0)6 , α (0)7 , α (0)8 , ¯ α (0)1 , α (1)2 , α (1)4 , α (2)3 , α (2)4 , α (2)7 , α (2)10 , α (2)13 , α (2)14 , α (2)15 , ¯ α (2)3 , ¯ α (2)4 , ¯ α (2)6 , ¯ α (2)8 , ¯ α (2)11 , ¯ α (2)13 , ¯ α (2)15 , α (3)1 , ¯ α (3)2 , ¯ α (3)3 , ¯ α (3)4 , ¯ α (3)5 , ¯ α (3)7 , ¯ α (3)8 , ¯ α (3)9 , ¯ α (3)10 , ¯ α (3)11 , ¯ α (3)12 , ¯ α (3)13 , ¯ α (3)14 , ¯ α (3)16 , ¯ α (3)17 , α (4)1 , ¯ α (4)3 , ¯ α (4)4 , ¯ α (4)5 , ¯ α (4)6 , ¯ α (4)7 , ¯ α (4)8 , ¯ α (4)9 , α (5)1 , ¯ α (5)1 , ¯ α (5)2 , ¯ α (5)3 . A.4.
Coefficients for an exemplary homomorphism.
Finally, we obtain the following solution of the above system ofequation α (0)1 = 2 θ − θ ,α (0)2 = − θ + 2 θ θ − θ − θ + θ + θ ,α (0)3 = θ − θ θ + 4 θ ,α (0)4 = θ − θ θ + 6 θ θ − θ + 2 θ θ − θ θ − θ θ − θ θ + 2 θ θ + 2 θ θ ,α (0)5 = − θ + 11 θ θ − θ θ + 10 θ − θ θ + 2 θ θ + 3 θ θ − θ θ − θ θ + θ − θ ,α (0)6 = 4 θ θ + θ − θ θ − θ θ − θ θ θ − θ θ θ + 12 θ θ θ − θ − θ − θ θ + θ θ + 33 θ θ − θ θ + 69 θ θ + 3 θ θ + θ θ + 5 θ θ + 11 θ θ − θ θ − θ θ + θ θ − θ θ − θ θ + θ θ + θ + 6 θ − θ + θ ,α (0)7 = − θ + 21 θ θ − θ θ + 62 θ θ − θ − θ θ + θ θ + 3 θ θ + 13 θ θ θ − θ θ θ − θ θ θ − θ θ + 6 θ θ + 10 θ θ + θ θ − θ θ − θ θ + 2 θ θ ,α (0)8 = − θ θ + 2 θ θ − θ θ θ + 8 θ θ θ + 13 θ θ θ + θ θ + 35 θ θ θ − θ θ θ − θ θ θ + 15 θ θ θ − θ + 136 θ θ − θ θ + 164 θ θ − θ θ + 10 θ θ + 22 θ θ − θ θ + 12 θ θ − θ θ + 2 θ θ − θ θ + 2 θ θ + θ θ − θ θ + θ θ + 13 θ θ − θ θ − θ θ + θ θ + θ θ − θ θ θ + 3 θ θ θ − θ θ θ + 6 θ θ θ + 6 θ θ θ − θ θ θ − θ θ θ − θ θ − θ θ − θ θ + 6 θ − θ θ θ , ¯ α (0)1 = − θ + θ ,α (1)2 = θ − θ θ + 2 θ + θ θ − θ θ − θ θ − θ θ + 3 θ θ + θ θ − θ + θ ,α (1)4 = θ θ + θ θ − θ θ + 10 θ θ − θ θ θ + 18 θ θ θ − θ θ − θ θ θ − θ θ − θ θ θ + 36 θ θ θ + 46 θ θ θ − θ θ θ + 25 θ − θ θ + 122 θ θ − θ θ + 22 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ + 28 θ θ + 4 θ θ + θ θ + θ θ − θ θ − θ θ + 8 θ θ − θ θ + θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ − θ θ + θ θ − θ θ + θ θ − θ θ − θ θ + θ θ + 6 θ θ θ − θ θ θ + 34 θ θ θ + 22 θ θ θ − θ θ θ − θ θ θ + 42 θ θ + 18 θ θ + 16 θ θ − θ + 79 θ θ θ − θ + θ + θ ,α (2)3 = θ − θ θ + 4 θ + θ − θ ,α (2)4 = − θ θ + 2 θ − θ + θ ,α (2)7 = 3 θ − θ θ + 11 θ θ − θ + 4 θ θ − θ θ − θ θ − θ θ + 5 θ θ + 4 θ θ + θ ,α (2)10 = 2 θ θ − θ θ + 8 θ θ − θ + θ θ − θ θ + 2 θ θ θ − θ θ θ − θ θ + 2 θ θ + 2 θ − θ θ − θ θ + θ + θ θ ,α (2)13 = 12 θ − θ − θ θ − θ θ − θ θ + 201 θ θ θ + 221 θ θ θ − θ θ θ + 95 θ θ θ − θ θ θ − θ θ θ + 61 θ θ θ + 30 θ θ θ − θ θ θ + 51 θ θ θ + 172 θ θ θ + 92 θ θ θ + 60 θ θ θ + 2 θ θ + θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ − θ + 18 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 24 θ θ θ + 20 θ θ θ − θ θ θ + θ θ θ + θ θ θ + 75 θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ + 35 θ θ θ − θ θ θ + 28 θ θ θ + 4 θ θ − θ θ − θ θ + 3 θ θ + θ θ − θ θ − θ θ + 3 θ θ + 8 θ θ − θ θ + 8 θ θ − θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + 54 θ θ θ + 2 θ θ + θ θ − θ θ + 16 θ θ − θ θ + θ θ + 3 θ θ − θ θ θ θ − θ θ θ θ + 134 θ θ θ θ + θ θ + 421 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 546 θ θ θ + 487 θ θ θ + 730 θ θ θ − θ θ θ − θ θ θ − θ θ − θ θ + 35 θ θ + 220 θ θ − θ θ + 980 θ θ − θ θ + 445 θ θ − θ θ + 62 θ θ + 36 θ θ − θ θ + 136 θ θ + 150 θ θ + 15 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ − θ θ − θ θ − θ − θ − θ − θ − θ θ ,α (2)14 = − θ − θ + 2 θ θ + θ θ − θ θ + 4 θ θ + θ θ θ + θ θ θ − θ θ θ + θ θ + θ θ θ + θ θ θ + 3 θ θ θ + 28 θ θ θ + θ θ θ − θ θ θ − θ θ θ + 50 θ θ θ + 27 θ θ θ + 21 θ θ θ − θ θ − θ θ − θ θ + θ θ − θ + θ θ − θ θ θ − θ θ θ + 3 θ θ θ + 2 θ θ θ + θ θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ + θ θ − θ θ − θ θ + θ θ + 3 θ θ + θ θ − θ θ − θ θ + θ θ − θ θ + θ θ − θ θ + θ θ + θ θ θ − θ θ θ − θ θ θ + θ θ θ − θ θ θ − θ θ − θ θ − θ θ − θ θ + θ θ + θ θ − θ θ − θ θ θ θ − θ θ θ θ + 44 θ θ θ θ + θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + 125 θ θ θ + 83 θ θ θ − θ θ θ − θ θ θ + θ − θ θ + θ θ − θ θ − θ θ + θ + 53 θ θ − θ θ + θ θ − θ θ + 99 θ θ − θ θ + θ θ + θ θ − θ θ + 18 θ θ + 17 θ θ + θ θ − θ θ − θ θ − θ θ + θ θ − θ θ − θ θ − θ θ − θ − θ − θ − θ + 4 θ θ , α (2)15 = − θ θ − θ θ θ − θ θ + θ θ − θ θ − θ θ − θ θ θ + θ θ θ + θ θ − θ θ θ − θ θ θ − θ θ θ θ + 68 θ θ θ θ − θ θ θ + 64 θ θ θ + θ θ θ + θ θ θ + 4 θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ + θ θ θ − θ θ θ θ − θ θ θ θ + θ θ θ θ + θ θ θ − θ θ − θ θ + θ θ θ + θ + θ θ + θ θ θ + 9 θ θ θ − θ + θ θ θ − θ θ θ + θ θ − θ θ θ θ − θ θ θ θ + 86 θ θ θ θ + θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ θ θ + 725 θ θ θ + 558 θ θ θ + 1003 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 485 θ θ θ + 496 θ θ θ − θ θ − θ θ θ − θ θ θ + 160 θ θ θ + 115 θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ − θ θ θ + θ θ θ + θ θ θ θ + θ θ θ θ − θ θ θ θ − θ θ θ − θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + 35 θ θ + 14 θ θ − θ θ − θ θ + 13 θ θ − θ θ θ − θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ + 12 θ θ θ + 2 θ θ θ + θ θ θ + θ θ θ + 3 θ θ θ − θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + 3 θ θ θ + 7 θ θ θ + 2 θ θ θ − θ θ − θ θ − θ θ + θ θ − θ θ + θ θ − θ θ − θ θ θ + 4 θ θ θ + 7 θ θ θ + 2 θ θ θ + 4 θ θ θ + 12 θ θ θ + 5 θ θ θ − θ θ θ + θ θ θ − θ θ θ θ + θ θ θ − θ θ − θ θ − θ θ θ + θ θ θ + θ θ θ + θ θ − θ θ + θ θ − θ θ + θ θ − θ θ + θ θ − θ θ − θ θ − θ θ + 165 θ θ + θ θ + θ θ + θ θ + 2 θ θ + θ θ θ θ − θ θ θ θ + θ θ − θ θ − θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + 2 θ θ − θ θ θ + θ θ + θ θ − θ θ + 27 θ θ + 3 θ θ − θ θ − θ θ + θ θ − θ θ − θ θ − θ θ − θ θ − θ θ + θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ θ + 37 θ θ θ θ + 31 θ θ θ θ − θ θ θ − θ θ θ
14 12 θ θ − θ θ − θ θ − θ θ − θ θ + 2 θ θ + 2 θ θ θ + 5 θ θ θ − θ θ θ + 2 θ θ θ + 5 θ θ θ + θ θ + θ θ − θ θ − θ θ + θ θ + θ θ − θ θ + 29 θ θ − θ θ + θ θ θ − θ θ θ + θ θ θ + 72 θ θ θ + θ θ θ − θ θ θ − θ θ θ + θ θ θ − θ θ θ − θ θ θ + θ θ + θ + 7 θ θ θ + θ θ − θ θ , ¯ α (2)3 = − θ + 4 θ θ − θ − θ + θ , ¯ α (2)4 = 2 θ θ − θ + θ − θ , ¯ α (2)6 = − θ + 7 θ θ − θ θ + 7 θ − θ θ + 2 θ θ + θ θ + 5 θ θ − θ θ − θ θ − θ − θ + θ , ¯ α (2)8 = θ − θ θ + 14 θ θ − θ θ + 8 θ + 3 θ θ − θ θ − θ θ − θ θ θ + 4 θ θ θ + 6 θ θ θ + 10 θ θ − θ θ − θ θ + 2 θ − θ θ − θ θ + θ θ + θ , ¯ α (2)11 = θ θ + θ θ − θ θ + 7 θ θ − θ θ θ + 14 θ θ θ − θ θ − θ θ θ − θ θ − θ θ θ + 28 θ θ θ + 42 θ θ θ − θ θ θ + 29 θ − θ θ + 118 θ θ − θ θ + 20 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ + 22 θ θ + 6 θ θ + θ θ + θ θ − θ θ − θ θ + 7 θ θ − θ θ + θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ − θ θ + θ θ − θ θ + θ θ + θ θ − θ θ − θ θ + θ θ + 4 θ θ θ + 31 θ θ θ + 19 θ θ θ − θ θ θ − θ θ θ + 37 θ θ + 16 θ θ + 14 θ θ − θ + 68 θ θ θ − θ + θ + θ , ¯ α (2)13 = 13 θ + θ − θ θ − θ θ + θ θ − θ θ − θ θ θ − θ θ θ + 35 θ θ θ − θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + 12 θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ + θ θ + θ θ − θ θ + θ − θ θ + 2 θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ + θ θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ + 18 θ θ θ − θ θ − θ θ + θ θ + θ θ − θ θ − θ θ − θ θ + θ θ + θ θ − θ θ + θ θ − θ θ + θ θ − θ θ − θ θ θ + 4 θ θ θ + 4 θ θ θ − θ θ θ + θ θ θ + θ θ + θ θ + θ θ + θ θ − θ θ − θ θ + θ θ + 67 θ θ θ θ + 65 θ θ θ θ − θ θ θ θ − θ θ − θ θ θ + 64 θ θ θ + 72 θ θ θ + 133 θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ + θ θ − θ θ + θ θ + θ θ − θ − θ θ + θ θ − θ θ + θ θ − θ θ + θ θ − θ θ − θ θ + 16 θ θ − θ θ − θ θ − θ θ + 24 θ θ + 11 θ θ + 8 θ θ − θ θ + 36 θ θ + θ θ + θ θ + 8 θ + 18 θ + θ + θ − θ θ , ¯ α (2)15 = 13 θ θ − θ θ θ + θ θ − θ θ + θ θ + θ θ + 3 θ θ θ − θ θ θ − θ θ + θ θ θ − θ θ θ − θ θ θ θ − θ θ θ θ + 73 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 10 θ θ + 14 θ θ − θ θ θ + 18 θ θ θ + θ θ θ + 21 θ θ θ + θ θ θ − θ θ θ − θ θ − θ θ θ + θ θ θ θ + θ θ θ θ − θ θ θ θ + 2 θ θ θ − θ θ θ + θ θ + θ θ − θ θ θ − θ − θ θ − θ θ θ − θ θ θ + 30 θ − θ θ θ + θ θ θ − θ θ + θ θ θ θ + θ θ θ θ − θ θ θ θ + θ θ θ + 4 θ θ θ − θ θ θ − θ θ θ + 1309 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 1119 θ θ θ + 1248 θ θ θ + 983 θ θ θ − θ θ θ − θ θ θ + 5 θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ + 23 θ θ θ + 473 θ θ θ + 230 θ θ θ + 201 θ θ θ − θ θ θ + 65 θ θ θ − θ θ θ − θ θ θ θ − θ θ θ θ + 380 θ θ θ θ + θ θ θ + θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ θ − θ θ + θ θ + θ θ − θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + 2 θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + 6 θ θ θ + θ θ θ + θ θ θ − θ θ θ + θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + 14 θ θ + 14 θ θ + 2 θ θ − θ θ + θ θ − θ θ + θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + 18 θ θ θ − θ θ θ + θ θ θ − θ θ θ + θ θ θ θ + θ θ θ + θ θ + 2 θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ + 932 θ θ − θ θ + 2315 θ θ − θ θ + 804 θ θ − θ θ + θ θ + θ θ + θ θ − θ θ − θ θ − θ θ − θ θ − θ θ − θ θ θ θ + θ θ θ θ − θ θ + θ θ + θ θ + 4 θ θ θ − θ θ θ + 2 θ θ θ + θ θ θ − θ θ + 4 θ θ θ − θ θ + θ θ − θ θ − θ θ − θ θ + θ θ − θ θ + θ θ + θ θ + θ θ + θ θ + θ θ + θ θ − θ θ − θ θ θ + θ θ θ + θ θ θ + 57 θ θ θ θ − θ θ θ θ − θ θ θ θ − θ θ θ + 3 θ θ θ + θ θ − θ θ + θ θ − θ θ + θ θ − θ θ + 8 θ θ θ + 5 θ θ θ + 6 θ θ θ − θ θ θ − θ θ θ − θ θ − θ θ + θ θ − θ θ + θ θ + θ θ + 56 θ θ + 21 θ θ + θ θ − θ θ θ + 283 θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + 7 θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ θ − θ − θ θ θ − θ θ + θ θ , N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E α (3)1 = θ , ¯ α (3)2 = θ , ¯ α (3)3 = − θ + 2 θ θ − θ − θ + θ + θ , ¯ α (3)4 = − θ + 2 θ θ − θ − θ + θ + θ , ¯ α (3)5 = − θ + 2 θ θ − θ − θ + θ + θ , ¯ α (3)7 = − θ θ + 6 θ θ − θ − θ θ + 2 θ θ + 2 θ θ + θ + θ θ − θ θ − θ θ − θ + θ , ¯ α (3)8 = 2 θ − θ θ + 10 θ θ − θ + 2 θ θ − θ θ − θ θ − θ θ + 3 θ θ + 3 θ θ − θ + θ , ¯ α (3)9 = − θ θ − θ + 5 θ θ + 6 θ θ − θ θ θ − θ θ θ + 29 θ θ θ − θ − θ − θ θ + θ θ + 26 θ θ − θ θ + 25 θ θ + 9 θ θ + 5 θ θ + 9 θ θ + 10 θ θ − θ θ − θ θ + 3 θ θ − θ θ − θ θ + θ θ − θ − θ + θ + θ , ¯ α (3)10 = 3 θ − θ θ + 28 θ θ − θ θ + 10 θ + 3 θ θ − θ θ − θ θ − θ θ θ + 9 θ θ θ + 7 θ θ θ + 9 θ θ − θ θ − θ θ + 2 θ θ − θ θ + θ θ − θ θ + θ , ¯ α (3)11 = θ θ − θ θ − θ θ θ − θ θ θ + 6 θ θ θ − θ − θ − θ θ − θ θ + 10 θ θ − θ θ + 8 θ θ + 3 θ θ + θ θ + 3 θ θ + 4 θ θ − θ θ − θ θ + 2 θ θ − θ θ + θ θ + θ , ¯ α (3)12 = 4 θ θ − θ θ + 9 θ θ − θ θ θ + 19 θ θ θ − θ θ − θ θ θ − θ θ − θ θ θ + 4 θ θ θ + 27 θ θ θ − θ θ θ + 3 θ − θ θ + 41 θ θ − θ θ + θ θ + 3 θ θ + 2 θ θ − θ θ − θ θ − θ θ + 27 θ θ + θ θ − θ θ + θ θ + θ θ − θ θ + 8 θ θ + θ θ − θ θ + θ θ − θ θ − θ θ − θ θ + 15 θ θ − θ θ − θ θ + θ θ + 6 θ θ − θ θ θ + 8 θ θ θ − θ θ θ + 21 θ θ θ + 15 θ θ θ − θ θ θ − θ θ θ + 24 θ θ + 11 θ θ + 11 θ θ − θ + 55 θ θ θ + θ + θ , ¯ α (3)13 = 4 θ θ − θ θ + 7 θ θ − θ θ θ + 17 θ θ θ − θ θ − θ θ θ − θ θ − θ θ θ − θ θ θ + 13 θ θ θ − θ θ θ − θ − θ θ + 2 θ θ + 4 θ θ − θ θ + 12 θ θ + 12 θ θ − θ θ − θ θ − θ θ + 22 θ θ + θ θ − θ θ + θ θ + θ θ − θ θ + 6 θ θ + θ θ + θ θ − θ θ + θ θ − θ θ − θ θ − θ θ + θ θ − θ θ − θ θ + θ θ + θ θ − θ θ θ + 11 θ θ θ − θ θ θ + 21 θ θ θ + 15 θ θ θ − θ θ θ − θ θ θ + 17 θ θ + 11 θ θ + 11 θ θ − θ + 42 θ θ θ + θ + θ , ¯ α (3)14 = − θ + 7 θ θ − θ θ + 60 θ θ − θ θ + 22 θ − θ θ + 2 θ θ + 10 θ θ θ + 7 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 17 θ θ θ + 7 θ θ + 5 θ θ − θ θ + θ θ − θ θ − θ θ θ − θ θ θ + 6 θ θ θ − θ θ + 4 θ θ θ + 2 θ θ θ − θ θ − θ θ + 4 θ θ + 6 θ θ − θ θ + 3 θ θ − θ θ θ − θ θ θ + θ θ θ + 2 θ θ − θ θ − θ θ − θ θ + θ θ + θ θ − θ θ + θ θ + θ , ¯ α (3)16 = − θ + 28 θ θ θ + 28 θ θ θ − θ θ θ + θ θ θ + θ θ θ + θ θ θ + 2 θ θ θ + 2 θ θ θ − θ θ θ − θ θ θ + 17 θ θ θ + 9 θ θ θ + 6 θ θ θ − θ θ − θ − θ θ θ + θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ θ + θ θ θ + θ θ + θ θ + 2 θ θ − θ θ − θ θ θ − θ θ θ + θ θ θ − θ θ θ + θ θ − θ θ + θ θ − θ θ θ θ − θ θ θ θ + 14 θ θ θ θ + 28 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 26 θ θ θ + 45 θ θ θ + 23 θ θ θ − θ θ θ − θ θ θ − θ θ − θ θ − θ θ + 12 θ θ − θ θ + 45 θ θ − θ θ + 20 θ θ − θ θ + 2 θ θ + 2 θ θ + 12 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ + 13 θ θ − θ θ − θ θ − θ θ − θ − θ , ¯ α (3)17 = − θ θ − θ θ θ − θ θ θ + 4 θ θ − θ θ θ − θ θ θ − θ θ θ θ + 26 θ θ θ θ + 12 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 22 θ θ + 14 θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ θ + 2 θ θ θ θ + 17 θ θ θ θ − θ θ θ + θ θ − θ θ − θ + θ θ − θ θ θ + 2 θ θ θ + 14 θ − θ θ θ + 8 θ θ θ − θ θ + 190 θ θ θ θ + 164 θ θ θ θ − θ θ θ θ + 14 θ θ θ − θ θ θ + 2 θ θ θ + 8 θ θ θ + 524 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 480 θ θ θ + 426 θ θ θ + 506 θ θ θ − θ θ θ − θ θ θ + 106 θ θ θ + 106 θ θ θ − θ θ θ − θ θ θ + 30 θ θ θ + 156 θ θ θ + 80 θ θ θ + 74 θ θ θ − θ θ θ + 3 θ θ θ + 4 θ θ θ − θ θ θ θ − θ θ θ θ + 138 θ θ θ θ + 2 θ θ θ − θ θ θ + 172 θ θ θ + 136 θ θ θ − θ θ − θ θ − θ θ − θ θ − θ θ θ − θ θ θ − θ θ θ + 18 θ θ θ − θ θ θ + 6 θ θ θ + 5 θ θ θ + 16 θ θ θ + 22 θ θ θ + 20 θ θ θ − θ θ θ + 2 θ θ θ − θ θ θ − θ θ θ + 4 θ θ θ + 2 θ θ θ + 6 θ θ + 2 θ θ − θ θ θ + 2 θ θ θ + 2 θ θ θ + 32 θ θ θ − θ θ θ + 30 θ θ θ θ + 8 θ θ θ + 42 θ θ − θ θ θ − θ θ θ − θ θ + 348 θ θ − θ θ + 892 θ θ − θ θ + 354 θ θ − θ θ + 90 θ θ + 108 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ − θ θ θ θ + 36 θ θ θ θ + 24 θ θ − θ θ − θ θ − θ θ θ − θ θ θ + 2 θ θ θ + 5 θ θ − θ θ + 10 θ θ − θ θ − θ θ + 4 θ θ + 2 θ θ − θ θ + 6 θ θ + θ θ + 2 θ θ − θ θ θ + 6 θ θ θ + 6 θ θ θ + 8 θ θ θ θ − θ θ θ θ − θ θ θ θ − θ θ θ − θ θ θ + 6 θ θ θ + 40 θ θ + 36 θ θ + 12 θ θ − θ θ θ + 103 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 34 θ θ θ + 3 θ θ θ + θ θ θ + 2 θ θ θ + 2 θ θ θ − θ θ − θ θ θ − θ θ ,α (4)1 = − θ , ¯ α (4)3 = − θ + 3 θ θ − θ θ + 3 θ − θ θ + θ θ + θ θ + θ θ − θ θ − θ θ + θ − θ ¯ α (4)4 = θ − θ θ + 4 θ θ − θ + θ θ − θ θ − θ θ − θ θ + θ θ + θ θ − θ + θ , ¯ α (4)5 = θ θ + 2 θ − θ θ − θ θ + 26 θ θ θ + 22 θ θ θ − θ θ θ + 18 θ + 10 θ + θ θ − θ θ − θ θ + 70 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ + 16 θ θ + 29 θ θ − θ θ + 3 θ θ + θ θ − θ θ + 2 θ + 5 θ − θ − θ + θ , ¯ α (4)6 = 3 θ − θ θ + 17 θ θ − θ θ + 6 θ + 3 θ θ − θ θ − θ θ − θ θ θ + 5 θ θ θ + 7 θ θ θ + 5 θ θ − θ θ − θ θ + θ θ + 3 θ θ − θ θ + 3 θ θ − θ θ + θ θ − θ θ − θ , ¯ α (4)7 = − θ θ + 6 θ θ − θ θ + 10 θ θ θ − θ θ θ + 27 θ θ θ + 16 θ θ + 43 θ θ θ − θ θ θ − θ θ θ + 34 θ θ θ + 3 θ + 38 θ θ − θ θ + θ θ − θ θ + 11 θ θ + 3 θ θ + θ θ + θ θ + 12 θ θ − θ θ − θ θ − θ θ + 3 θ θ − θ θ + θ θ − θ θ + 2 θ θ − θ θ + 2 θ θ − θ θ + θ θ − θ θ + 4 θ θ + θ θ − θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ + 29 θ θ θ + 25 θ θ θ − θ θ − θ θ − θ θ + 5 θ − θ θ θ − θ , ¯ α (4)8 = (cid:0) θ − θ θ + 8 θ θ − θ + 2 θ θ − θ θ − θ θ − θ θ + 2 θ θ + 2 θ θ − θ + θ (cid:1) , ¯ α (4)9 = − θ θ − θ θ θ − θ θ θ + 4 θ θ − θ θ θ − θ θ θ − θ θ θ θ + 26 θ θ θ θ + 12 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 22 θ θ + 14 θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ − θ θ θ θ + 3 θ θ θ θ + 16 θ θ θ θ − θ θ θ + θ θ − θ θ − θ + θ θ − θ θ θ + θ θ θ + 14 θ − θ θ θ + 8 θ θ θ − θ θ + 188 θ θ θ θ + 162 θ θ θ θ − θ θ θ θ + 14 θ θ θ − θ θ θ + 2 θ θ θ + 8 θ θ θ + 516 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 466 θ θ θ + 412 θ θ θ + 492 θ θ θ − θ θ θ − θ θ θ + 104 θ θ θ + 104 θ θ θ − θ θ θ − θ θ θ + 30 θ θ θ + 154 θ θ θ + 78 θ θ θ + 72 θ θ θ − θ θ θ + 6 θ θ θ + 3 θ θ θ − θ θ θ θ − θ θ θ θ + 134 θ θ θ θ + 2 θ θ θ − θ θ θ + 170 θ θ θ + 134 θ θ θ − θ θ − θ θ − θ θ − θ θ − θ θ θ − θ θ θ − θ θ θ + 18 θ θ θ − θ θ θ + 6 θ θ θ + 5 θ θ θ + 16 θ θ θ + 22 θ θ θ + 20 θ θ θ − θ θ θ + 2 θ θ θ − θ θ θ − θ θ θ + 4 θ θ θ + 2 θ θ θ + 6 θ θ + 2 θ θ − θ θ θ + 2 θ θ θ + 2 θ θ θ + 32 θ θ θ − θ θ θ + 29 θ θ θ θ + 8 θ θ θ + 39 θ θ − θ θ θ − θ θ θ − θ θ + 342 θ θ − θ θ + 862 θ θ − θ θ + 330 θ θ − θ θ + 84 θ θ + 102 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ − θ θ θ θ + 35 θ θ θ θ + 27 θ θ − θ θ − θ θ − θ θ θ − θ θ θ + 2 θ θ θ + 5 θ θ − θ θ + 10 θ θ − θ θ − θ θ + 4 θ θ + 2 θ θ − θ θ + θ θ + θ θ + 2 θ θ − θ θ θ + 6 θ θ θ + 6 θ θ θ + 8 θ θ θ θ − θ θ θ θ − θ θ θ θ − θ θ θ − θ θ θ + 6 θ θ θ + 40 θ θ + 36 θ θ + 12 θ θ − θ θ θ + 100 θ θ θ − θ θ θ − θ θ θ − θ θ θ + 33 θ θ θ + 3 θ θ θ + θ θ θ + 2 θ θ θ + 2 θ θ θ − θ θ − θ θ θ − θ θ ,α (5)1 = − θ + 3 θ θ − θ θ + 3 θ − θ θ + θ θ + θ θ + θ θ − θ θ − θ θ + θ − θ , ¯ α (5)1 = θ − θ θ + 4 θ θ − θ + θ θ − θ θ − θ θ − θ θ + θ θ + θ θ − θ + θ , ¯ α (5)2 = 2 θ θ + 2 θ − θ θ − θ θ + 20 θ θ θ + 14 θ θ θ − θ θ θ + 9 θ + 7 θ + θ θ − θ θ − θ θ + 50 θ θ − θ θ − θ θ − θ θ − θ θ − θ θ + 13 θ θ + 23 θ θ − θ θ + 3 θ θ + θ θ − θ θ + 2 θ + 5 θ − θ + θ , ¯ α (5)3 = − (cid:0) θ − θ θ + 8 θ θ − θ + 2 θ θ − θ θ − θ θ − θ θ + 2 θ θ + 2 θ θ − θ + θ (cid:1)(cid:0) θ − θ θ + 100 θ θ − θ θ + 18 θ + 26 θ θ − θ θ − θ θ − θ θ θ + 40 θ θ θ + 28 θ θ θ + 46 θ θ − θ θ − θ θ + 5 θ θ + 5 θ θ − θ θ − θ θ − θ θ + 6 θ θ + 10 θ − θ θ − θ θ + 4 θ + 4 θ θ + 4 θ − θ + 2 θ (cid:1) . Therefore, by Corollary 2.2, there exists a K -algebra isomorphism ϕ : P f ( E ) → P ( E ) defined on arrows by formulas fromA.2, with the coefficients α ( i ) j i , ¯ α ( i ) j i ∈ K , for i = 0 , . . . , j = 1 , . . . , j i , j = 11 , j = 5 , j = 15 , j = 17 , j = 9 , j = 3, takingvalues defined in A.3 and computed in A.4. Consequently, the algebras P ( E ) and P f ( E ) are isomorphic. Appendix B. Calculations for type E In this appendix we present the sketch of the construction of a K -algebra isomorphism ϕ : P f ( E ) → P ( E ) for a givenadmissible deforming element f according to the algorithm described in the main part of the article. This appendix is organizedas follows. In B.1 we describe the equations for the coefficients of f indicated by the admissibility condition of f . In B.2 wepresent the general form of the homomorphisms from P f ( E ) to P ( E ) (in particular we present the chosen basis elements of P ( E )). In B.3 we describe the chosen method of simplifying the system of equations for the coefficients of an isomorphismfrom P f ( E ) to P ( E ). B.4 is devoted to considerations on solving the obtained system of equations and analysing the obtainedsolution.We recall that considered case is much more complicated than the case of type E (described in A) and the full formulasobtained in the calculations are to long to be presented in the article. Hence we only provide here suitable choices in thesubsequent steps of the algorithm and describe the general shape of obtained formulas without presenting their full form (werefer for them to [4]) nor going into details.We refer to the main part of the article for description of applied algorithms as well as for theoretical background and morereferences.B.1. Equations derived from the admissibility condition.
We carry out the steps described in Section 5. First we computea base of R ( E ) (see Algorithm 1 for details). We take as a basis the set { , x, y, xy, yx, yy, xyx, xyy, yxy, yyx, xyxy, xyyx, yxyx,yxyy, yyxy, xyxyx, xyxyy, xyyxy, yxyxy, yxyyx, yyxyx, xyxyxy, xyxyyx, xyyxyx, yxyxyx, yxyxyy, yxyyxy, xyxyxyx, xyxyxyy,xyxyyxy, yxyxyxy, yxyxyyx, yxyyxyx, xyxyxyxy, xyxyxyyx, xyxyyxyx, yxyxyxyx, yxyxyxyy, yxyxyyxy, xyxyxyxyx, xyxyxyxyy,xyxyxyyxy, yxyxyxyxy,yxyxyxyyx, yxyxyyxyx, xyxyxyxyxy,xyxyxyxyyx,xyxyxyyxyx, yxyxyxyxyx,yxyxyxyxyy,xyxyxyxyxyx,xyxyxyxyxyy,yxyxyxyxyxy,yxyxyxyxyyx,xyxyxyxyxyxy,xyxyxyxyxyyx,yxyxyxyxyxyx,xyxyxyxyxyxyx,yxyxyxyxyxyxy,xyxyxyxyxyxyxy } . Hence we know that each element f ∈ rad R ( E ) is of the form f ( x,y ) = θ xy + θ yx + θ yy + θ xyx + θ xyy + θ yxy + θ yyx + θ xyxy + θ xyyx + θ yxyx + θ yxyy + θ yyxy + θ xyxyx + θ xyxyy + θ xyyxy + θ yxyxy + θ yxyyx + θ yyxyx + θ xyxyxy + θ xyxyyx + θ xyyxyx + θ yxyxyx + θ yxyxyy + θ yxyyxy + θ xyxyxyx + θ xyxyxyy + θ xyxyyxy + θ yxyxyxy + θ yxyxyyx + θ yxyyxyx + θ xyxyxyxy + θ xyxyxyyx + θ xyxyyxyx + θ yxyxyxyx + θ yxyxyxyy + θ yxyxyyxy + θ xyxyxyxyx + θ xyxyxyxyy + θ xyxyxyyxy + θ yxyxyxyxy + θ yxyxyxyyx + θ yxyxyyxyx + θ xyxyxyxyxy + θ xyxyxyxyyx + θ xyxyxyyxyx + θ yxyxyxyxyx + θ yxyxyxyxyy + θ xyxyxyxyxyx + θ xyxyxyxyxyy + θ yxyxyxyxyxy + θ yxyxyxyxyyx + θ xyxyxyxyxyxy + θ xyxyxyxyxyyx + θ yxyxyxyxyxyx + θ xyxyxyxyxyxyx + θ yxyxyxyxyxyxy + θ xyxyxyxyxyxyxy N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E for some θ , . . . , θ ∈ K . We fix these coefficients. Further, we compute ( x + y + f ( x, y )) (see Sections 5 and 6 for details).We obtain the formula of the form (cid:0) x + y + f ( x, y ) (cid:1) = ( θ + θ − θ ) xyxyxy + ( θ + θ − θ ) xyxyyx + ( θ + θ − θ ) xyyxyx + ( θ + θ − θ ) yxyxyx + (3 θ − θ + θ − θ + 2 θ + θ θ − θ θ + 2 θ − θ θ − θ ) xyxyxyx + (3 θ − θ + θ − θ + 2 θ + θ θ − θ θ + 2 θ − θ θ − θ ) yxyxyxy + (3 θ − θ + 3 θ − θ − θ + 6 θ θ − θ θ + 2 θ θ − θ θ + 3 θ θ + θ θ − θ θ − θ θ − θ θ + 3 θ + θ θ − θ θ + 2 θ θ − θ θ θ − θ θ + θ ) xyxyxyxy + . . . (where appear 30 basis elements of R ( E )). See [4, e8-adm-power.txt ] for its full form. Hence f satisfy the admissibility condi-tion if and only if the corresponding 30 equalities (starting with “ θ + θ − θ = 0”) are satisfied. See [4, e8-adm-equations.txt ]for full list of these equations.We recall that following [5, Theorem] there is a non-isomorphic deformation of P ( E ) in characteristic 2. Hence we mayassume that K is of characteristic different from 2. Then, applying Algorithm 2 to these equations with the following chosensequence of coefficients θ , θ , θ , θ , θ , θ , θ , θ , θ , θ , we obtain the required set of 10 independent equations: θ = 2 θ − θ ,θ = − θ + 6 θ θ − θ − θ + 2 θ + 2 θ ,θ = θ θ − θ θ + θ − θ θ + θ θ + θ θ − θ θ − θ − θ + ( θ + θ ) ,θ = − θ + 38 θ θ − θ θ + 46 θ θ − θ − θ θ + 10 θ θ + 14 θ θ + 54 θ θ θ − θ θ θ − θ θ θ − θ θ + 8 θ θ + 16 θ θ − θ θ θ θ − θ θ + 4 θ θ − θ θ + 2 θ θ − θ + 18 θ θ + 18 θ θ − θ − θ θ − θ + 3 θ − θ + 3 θ , ... θ = 4 θ θ θ − θ θ − θ θ θ − θ θ θ + 5 θ θ θ + θ θ θ + 2 θ θ θ + θ θ θ − θ θ θ − θ θ θ − θ θ − θ θ + 49 θ θ θ θ θ − θ θ θ θ θ − θ θ θ θ θ + 746 θ θ θ θ θ + 109 θ θ θ θ θ + 111 θ θ θ θ θ + . . . See [4, e8-substitutions.txt ] for list of these equations (we note that only the last of them alone would take about 8–10 pagesof the article, so it is not possible to show all them here in their full form).B.2.
General form of homomorphism.
Let f be a given admissible element of the structure described in B.1. In order toconstruct an isomorphism ϕ : P f ( E ) → P ( E ) we want to find the coefficients satisfying the assumptions of Corollary 2.2. Westart with computing the base of P ( E ), according to Algorithm 3. In particular, for each arrow α ∈ Q we compute a basis of e s ( α ) P ( E ) e t ( α ) . Then we conclude that our constructed isomorphism should be given by the following formulas ϕ ( a ) = a + α (0)1 a ¯ a a + α (0)2 a ¯ a a ¯ a a + α (0)3 a ¯ a a ¯ a a + α (0)4 a ¯ a a ¯ a a ¯ a a + α (0)5 a ¯ a a ¯ a a ¯ a a + α (0)6 a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)7 a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)8 a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)9 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)10 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)11 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)12 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)13 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)14 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)15 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)16 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)17 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)18 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)19 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)20 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)21 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)22 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)23 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)24 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)25 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)26 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)27 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)28 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (0)29 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ϕ (¯ a ) = ¯ a + ¯ α (0)1 ¯ a a ¯ a + ¯ α (0)2 ¯ a a ¯ a a ¯ a + ¯ α (0)3 ¯ a a ¯ a a ¯ a + ¯ α (0)4 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)5 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)6 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)7 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)8 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)9 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)10 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)11 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)12 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)13 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)14 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)15 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)16 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)17 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)18 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)19 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)20 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)21 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)22 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)23 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)24 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)25 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)26 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)27 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)28 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (0)29 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ϕ ( a ) = a + α (1)1 a a ¯ a a ¯ a + α (1)2 a a ¯ a a ¯ a a ¯ a + α (1)3 a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)4 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)5 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)6 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)7 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)8 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)9 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)10 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)11 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)12 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + α (1)13 a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ϕ (¯ a ) = ¯ a + ¯ α (1)1 a ¯ a a ¯ a ¯ a + ¯ α (1)2 a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)3 a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)4 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)5 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)6 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)7 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)8 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)9 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)10 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)11 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)12 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a + ¯ α (1)13 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ¯ a ϕ ( a ) = a + α (2)1 a ¯ a a + α (2)2 a ¯ a a + α (2)3 a ¯ a a ¯ a a + α (2)4 a ¯ a a ¯ a a + α (2)5 a ¯ a a ¯ a a ¯ a a + α (2)6 a ¯ a a ¯ a a ¯ a a + α (2)7 a ¯ a a ¯ a a ¯ a a + α (2)8 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)9 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)10 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)11 a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)12 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)13 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)14 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)15 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)16 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)17 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)18 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)19 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)20 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)21 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)22 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)23 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)24 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)25 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)26 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)27 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)28 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)29 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)30 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)31 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)32 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)33 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)34 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)35 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)36 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)37 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)38 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + α (2)39 a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ϕ (¯ a ) = ¯ a + ¯ α (2)1 ¯ a a ¯ a + ¯ α (2)2 ¯ a a ¯ a + ¯ α (2)3 ¯ a a ¯ a a ¯ a + ¯ α (2)4 ¯ a a ¯ a a ¯ a + ¯ α (2)5 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)6 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)7 ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)8 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)9 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)10 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)11 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)12 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)13 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)14 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)15 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)16 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)17 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)18 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)19 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)20 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)21 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)22 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)23 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)24 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)25 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)26 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)27 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)28 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)29 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)30 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)31 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)32 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)33 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)34 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)35 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)36 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)37 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)38 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a + ¯ α (2)39 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a ϕ ( a ) = a + α (3)1 ¯ a a a + α (3)2 ¯ a a a + α (3)3 ¯ a a ¯ a a a + α (3)4 ¯ a a ¯ a a a + α (3)5 ¯ a a ¯ a a a + α (3)6 ¯ a a ¯ a a ¯ a a a + α (3)7 ¯ a a ¯ a a ¯ a a a + α (3)8 ¯ a a ¯ a a ¯ a a a + α (3)9 ¯ a a ¯ a a ¯ a a a + α (3)10 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)11 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)12 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)13 ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)14 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)15 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)16 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)17 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)18 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)19 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)20 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)21 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)22 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)23 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)24 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)25 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)26 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)27 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)28 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)29 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)30 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)31 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)32 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)33 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)34 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)35 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)36 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)37 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)38 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)39 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)40 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)41 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)42 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)43 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)44 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)45 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E + α (3)46 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + α (3)47 ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a ϕ (¯ a ) = ¯ a + ¯ α (3)1 ¯ a ¯ a a + ¯ α (3)2 ¯ a ¯ a a + ¯ α (3)3 ¯ a ¯ a a ¯ a a + ¯ α (3)4 ¯ a ¯ a a ¯ a a + ¯ α (3)5 ¯ a ¯ a a ¯ a a + ¯ α (3)6 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)7 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)8 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)9 ¯ a ¯ a a ¯ a a ¯ a a + ¯ α (3)10 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)11 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)12 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)13 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)14 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)15 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)16 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)17 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)18 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)19 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)20 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)21 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)22 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)23 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)24 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)25 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)26 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)27 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)28 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)29 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)30 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)31 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)32 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)33 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)34 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)35 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)36 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)37 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)38 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)39 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)40 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)41 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)42 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)43 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)44 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)45 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)46 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a + ¯ α (3)47 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ϕ ( a ) = a + α (4)1 ¯ a a a + α (4)2 ¯ a ¯ a a a a + α (4)3 ¯ a ¯ a a a a + α (4)4 ¯ a ¯ a a ¯ a a a a + α (4)5 ¯ a ¯ a a ¯ a a a a + α (4)6 ¯ a ¯ a a ¯ a a ¯ a a a a + α (4)7 ¯ a ¯ a a ¯ a a ¯ a a a a + α (4)8 ¯ a ¯ a a ¯ a a ¯ a a a a + α (4)9 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)10 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)11 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)12 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)13 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)14 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)15 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)16 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)17 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)18 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)19 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)20 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)21 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)22 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)23 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)24 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)25 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)26 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)27 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)28 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + α (4)29 ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a ϕ (¯ a ) = ¯ a + ¯ α (4)1 ¯ a ¯ a a + ¯ α (4)2 ¯ a ¯ a ¯ a a a + ¯ α (4)3 ¯ a ¯ a ¯ a a a + ¯ α (4)4 ¯ a ¯ a ¯ a a ¯ a a a + ¯ α (4)5 ¯ a ¯ a ¯ a a ¯ a a a + ¯ α (4)6 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a + ¯ α (4)7 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a + ¯ α (4)8 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a + ¯ α (4)9 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)10 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)11 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)12 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)13 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)14 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)15 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)16 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)17 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)18 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)19 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)20 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)21 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)22 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)23 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)24 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)25 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)26 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)27 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)28 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a + ¯ α (4)29 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a ϕ ( a ) = a + α (5)1 ¯ a a a + α (5)2 ¯ a ¯ a ¯ a a a a a + α (5)3 ¯ a ¯ a ¯ a a ¯ a a a a a + α (5)4 ¯ a ¯ a ¯ a a ¯ a a a a a + α (5)5 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a a a + α (5)6 ¯ a ¯ a ¯ a a ¯ a a ¯ a a a a a + α (5)7 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)8 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)9 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)10 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)11 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)12 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)13 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)14 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + α (5)15 ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a ϕ (¯ a ) = ¯ a + ¯ α (5)1 ¯ a ¯ a a + ¯ α (5)2 ¯ a ¯ a ¯ a ¯ a a a a + ¯ α (5)3 ¯ a ¯ a ¯ a ¯ a a ¯ a a a a + ¯ α (5)4 ¯ a ¯ a ¯ a ¯ a a ¯ a a a a + ¯ α (5)5 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)6 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)7 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)8 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)9 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)10 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)11 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)12 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)13 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)14 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a + ¯ α (5)15 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a ϕ ( a ) = a + α (6)1 ¯ a ¯ a ¯ a ¯ a a a a a a + α (6)2 ¯ a ¯ a ¯ a ¯ a a ¯ a a a a a a + α (6)3 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a a + α (6)4 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a a + α (6)5 ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a a ϕ (¯ a ) = ¯ a + ¯ α (6)1 ¯ a ¯ a ¯ a ¯ a ¯ a a a a a + ¯ α (6)2 ¯ a ¯ a ¯ a ¯ a ¯ a a ¯ a a a a a + ¯ α (6)3 ¯ a ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + ¯ α (6)4 ¯ a ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a + ¯ α (6)5 ¯ a ¯ a ¯ a ¯ a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a ¯ a a a a a with coefficients α ( i ) j i , ¯ α ( i ) j i ∈ K , for i = 0 , . . . , j = 1 , . . . , j i , j = 29 , j = 13 , j = 39 , j = 47 , j = 29 , j = 15 , j = 5. B.3.
Simplified system of equations.
In order to obtain the simplified system of equations (see Section 8 for details) wewill assume that in the above definition of ϕ we have α (2)1 = 1 and that the following coefficients are equal zero: α (0)2 , α (0)6 , α (0)10 , α (0)11 , α (0)16 , α (0)22 , α (0)24 , α (0)26 , α (0)27 , α (0)28 , α (0)29 , ¯ α (0)4 , ¯ α (0)12 , ¯ α (0)13 , ¯ α (0)16 , ¯ α (0)18 , ¯ α (0)21 , ¯ α (0)22 , ¯ α (0)23 , ¯ α (0)25 , α (1)6 , α (1)8 , ¯ α (1)2 , ¯ α (1)3 , ¯ α (1)4 , ¯ α (1)5 , ¯ α (1)6 , ¯ α (1)7 , ¯ α (1)8 , ¯ α (1)9 , ¯ α (1)10 , ¯ α (1)11 , ¯ α (1)12 , α (2)15 , α (2)16 , α (2)29 , α (2)30 , α (2)31 , α (2)32 , α (2)35 , α (2)36 , α (2)39 , ¯ α (2)3 , ¯ α (2)4 , ¯ α (2)5 , ¯ α (2)6 , ¯ α (2)7 , ¯ α (2)8 , ¯ α (2)9 , ¯ α (2)10 , ¯ α (2)11 , ¯ α (2)12 , ¯ α (2)13 , ¯ α (2)14 , ¯ α (2)15 , ¯ α (2)16 , ¯ α (2)17 , ¯ α (2)18 , ¯ α (2)19 , ¯ α (2)20 , ¯ α (2)21 , ¯ α (2)22 , ¯ α (2)23 , ¯ α (2)24 , ¯ α (2)25 , ¯ α (2)26 , ¯ α (2)27 , ¯ α (2)28 , ¯ α (2)29 , ¯ α (2)30 , ¯ α (2)31 , ¯ α (2)32 , ¯ α (2)33 , ¯ α (2)34 , ¯ α (2)35 , ¯ α (2)36 , ¯ α (2)37 , ¯ α (2)38 , α (3)2 , α (3)3 , α (3)4 , α (3)5 , α (3)6 , α (3)7 , α (3)8 , α (3)9 , α (3)10 , α (3)11 , α (3)12 , α (3)13 , α (3)14 , α (3)15 , α (3)16 ,α (3)17 , α (3)18 , α (3)19 , α (3)20 , α (3)21 , α (3)22 , α (3)23 , α (3)24 , α (3)25 , α (3)26 , α (3)27 , α (3)28 , α (3)29 , α (3)30 , α (3)31 , α (3)32 , α (3)33 , α (3)34 , α (3)35 , α (3)36 , α (3)37 , α (3)38 , α (3)39 ,α (3)40 , α (3)41 , α (3)42 , α (3)43 , α (3)44 , α (3)45 , α (3)46 , α (3)47 , ¯ α (3)1 , ¯ α (3)10 , ¯ α (3)17 , ¯ α (3)45 , ¯ α (3)47 , α (4)2 , α (4)3 , α (4)4 , α (4)5 , α (4)6 , α (4)7 , α (4)8 , α (4)9 , α (4)10 , α (4)11 ,α (4)12 , α (4)13 , α (4)14 , α (4)15 , α (4)16 , α (4)17 , α (4)18 , α (4)19 , α (4)20 , α (4)21 , α (4)22 , α (4)23 , α (4)24 , α (4)25 , α (4)26 , α (4)27 , α (4)28 , ¯ α (4)1 , ¯ α (4)4 , ¯ α (4)5 , ¯ α (4)6 , ¯ α (4)10 , ¯ α (4)17 , ¯ α (4)19 , ¯ α (4)27 , α (5)2 , α (5)3 , α (5)4 , α (5)5 , α (5)6 , α (5)7 , α (5)8 , α (5)9 , α (5)10 , α (5)11 , α (5)12 , α (5)13 , α (5)14 , ¯ α (5)1 , ¯ α (5)2 , ¯ α (5)8 , ¯ α (5)15 , α (6)2 , α (6)3 , α (6)4 , ¯ α (6)5 . Further, from the relations of P ( E ) (see Section 7) we obtain the set of 170 equations [4, e8-equations-reduced.txt ]. Then,we apply to it the set of 10 formulas for θ , θ , θ , θ , θ , θ , θ , θ , θ , θ described in B.1 and solve it (see the nextsection for details) for the remaining variables α (0)1 , α (0)3 , α (0)4 , α (0)5 , α (0)7 , α (0)8 , α (0)9 , α (0)12 , α (0)13 , α (0)14 , α (0)15 , α (0)17 , α (0)18 , α (0)19 , α (0)20 , α (0)21 , α (0)23 , α (0)25 , ¯ α (0)1 , ¯ α (0)2 , ¯ α (0)3 , ¯ α (0)5 , ¯ α (0)6 , ¯ α (0)7 , ¯ α (0)8 , ¯ α (0)9 , ¯ α (0)10 , ¯ α (0)11 , ¯ α (0)14 , ¯ α (0)15 , ¯ α (0)17 , ¯ α (0)19 , ¯ α (0)20 , ¯ α (0)24 , ¯ α (0)26 , ¯ α (0)27 , ¯ α (0)28 , ¯ α (0)29 , α (1)1 , α (1)2 , α (1)3 , α (1)4 , α (1)5 , α (1)7 , α (1)9 , α (1)10 , α (1)11 , α (1)12 ,α (1)13 , ¯ α (1)1 , ¯ α (1)13 , α (2)2 , α (2)3 , α (2)4 , α (2)5 , α (2)6 , α (2)7 , α (2)8 , α (2)9 , α (2)10 , α (2)11 , α (2)12 , α (2)13 , α (2)14 , α (2)17 , α (2)18 , α (2)19 , α (2)20 , α (2)21 , α (2)22 , α (2)23 , α (2)24 ,α (2)25 , α (2)26 , α (2)27 , α (2)28 , α (2)33 , α (2)34 , α (2)37 , α (2)38 , ¯ α (2)1 , ¯ α (2)2 , ¯ α (2)39 , α (3)1 , ¯ α (3)2 , ¯ α (3)3 , ¯ α (3)4 , ¯ α (3)5 , ¯ α (3)6 , ¯ α (3)7 , ¯ α (3)8 , ¯ α (3)9 , ¯ α (3)11 , ¯ α (3)12 , ¯ α (3)13 , ¯ α (3)14 , ¯ α (3)15 , ¯ α (3)16 , ¯ α (3)18 , ¯ α (3)19 , ¯ α (3)20 , ¯ α (3)21 , ¯ α (3)22 , ¯ α (3)23 , ¯ α (3)24 , ¯ α (3)25 , ¯ α (3)26 , ¯ α (3)27 , ¯ α (3)28 , ¯ α (3)29 , ¯ α (3)30 , ¯ α (3)31 , ¯ α (3)32 , ¯ α (3)33 , ¯ α (3)34 , ¯ α (3)35 , ¯ α (3)36 , ¯ α (3)37 , ¯ α (3)38 , ¯ α (3)39 , ¯ α (3)40 , ¯ α (3)41 , ¯ α (3)42 , ¯ α (3)43 , ¯ α (3)44 , ¯ α (3)46 , α (4)1 , α (4)29 , ¯ α (4)2 , ¯ α (4)3 , ¯ α (4)7 , ¯ α (4)8 , ¯ α (4)9 , ¯ α (4)11 , ¯ α (4)12 , ¯ α (4)13 , ¯ α (4)14 , ¯ α (4)15 , ¯ α (4)16 , ¯ α (4)18 , ¯ α (4)20 , ¯ α (4)21 , ¯ α (4)22 , ¯ α (4)23 , ¯ α (4)24 , ¯ α (4)25 , ¯ α (4)26 , ¯ α (4)28 , ¯ α (4)29 , α (5)1 , α (5)15 , ¯ α (5)3 , ¯ α (5)4 , ¯ α (5)5 , ¯ α (5)6 , ¯ α (5)7 , ¯ α (5)9 , ¯ α (5)10 , ¯ α (5)11 , ¯ α (5)12 , ¯ α (5)13 , ¯ α (5)14 , α (6)1 , α (6)5 , ¯ α (6)1 , ¯ α (6)2 , ¯ α (6)3 , ¯ α (6)4 . We note that we need to perform above simplification in order to be able to solve the obtained system of equation withinan “acceptable” period of time. It let us reduce the number of equations from 178 to 170, the complexity of these equations,and the number of variables from 354 to 168 (see [4, e8-equations-full.txt and e8-equations-reduced.txt ] for details).Unfortunately, the obtained set of equations is still to large to be presented here (we note that its size is about one thousandtimes bigger than size of substitutions considered in B.1).B.4.
On solving the obtained set of equations.
It was already mentioned in Section 8, that system of equations obtainedin B.3 is to complicated to be solved manually, and natural approach is to use some dedicated software package to solve it. Inour case we decided to use Maple 2016 (which was later replaced by Maple 2019).We may divide the process into following three steps:(1) solving set of solutions;(2) evaluate (expand) the obtained result;(3) verify if it satisfy the assumptions.Invocation of procedures for these steps are listed in the Maple Text file [4, commands.txt ]. Below we briefly describe thesesteps.We start with defining the sequence of substitutions, set of variables and system of equations. Then we invoke procedure solve (). In the result we obtain sequence of formulas describing the solutions. In particular it contains 369930 polynomials whichroots are used (in the form of the construction “
RootOf ()”) in formulas for values of the required coefficients α ( i ) j i . Moreover,all denominators from the quotients appearing in these formulas belong to the set { , , , , , } . We recall that from thegeneral assumption the computations are performed over an algebraically closed field of characteristic different from 2. Hencewe know that we perform the calculations in right characteristic and that we can find roots of all these polynomials.It suffices to find only one solution, so in the next step we invoke the procedure allvalues () on the result from the abovestep and take the first element of the obtained sequence of solutions. Chosen solution contain the formulas for all variables.These formulas contain 369930 square roots of expressions build from fixed invariables θ i . All denominators from the quotientsappearing in these formulas again belong to the set { , , , , , } . We note that in order to list the denominators we mayuse some simple recursive procedure (see [4, denominators.txt ] for an example). We may also use some external tools to parseand analyse the formulas saved in “raw” form to the text files. Summing up, we conclude that this system of equations hasa solution (for arbitrary algebraically closed field K of characteristic different from 2). Hence, following Corollary 2.2, thereexist an isomorphism ϕ : P f ( E ) → P ( E ). defined on arrows by formulas from B.2, with the coefficients α ( i ) j i , ¯ α ( i ) j i ∈ K of thesolution of this set of equations. Hence algebras P ( E ) and P f ( E ) are isomorphic.We end this section with some notes regarding complexity of the problem of solving the obtained set of equations.We note that solving the simplified system of equations, described in B.3, has a relatively low memory and computationalcomplexity. Maple 2016 on a computer equipped with an Intel Xeon processor E5-1620 v2 (4 cores, 3.70 Ghz) and 128 GBRAM memory solves it in about 12 hours and these computations utilize about 10.2 GB RAM. The size of the file with a singlesolution saved in a “raw” Maple Text format is about 260 MB, so it is also “relatively” small. N ALGORITHM FOR THE PERIODICITY OF DEFORMED PREPROJECTIVE ALGEBRAS OF DYNKIN TYPES E , E AND E On the other hand, the process of selecting the right variables to be fixed with zero values has higher memory and timerequirements. Verifying (on the above mentioned machine) if the set of equations with 150–160 coefficients fixed with 0 value hasa solution, usually takes 4–5 weeks, and utilize about 64GB RAM. Moreover, in the case of positive answer, the data obtainedto the analysis are significantly larger (above 1 GB) and requires external tools for further analysis (just parsing the obtainedterm (representing the obtained solution) to be printed in text mode in Maple text console could take several days (about twodays in the case of 1.7 GB file) and showing the solution in graphical interface would require to increase the limit of the termsof the output above reasonable value).We note that the choices presented in B.3 are expected to not be optimal. Clearly, it can result in obtaining a morecomplicated solution then necessary. Solving the set of equations without any simplification could give us some hits on choosingright variables, but it is very complicated problem (see note below). We also note that we fix the coefficient α (2)1 with value 1in order to avoid considering cases and avoid denominators different from positive integers (containing expressions with θ i , for i ∈ { , . . . , } ). This construction also seems to be a little artificial and it is expected to not be necessary for a better choiceof variables.Finally, we note that solving the non-simplified set of equations (without fixing any coefficients) has even higher requirements.Such a computation was also launched on the above mentioned computer. But the computation was interrupted after 2 monthbecause of rapidly increasing number of reallocated sectors on the hard disc designated for swap. It was also started on a morepowerful computer – with double Intel Xeon processors E5-2650L v2 (20 cores, 1.90 GHz). Unfortunately, the calculations wereaccidentally terminated after 5 months (the Maple process has allocated 420 GB RAM in that moment) due to the licenseserver upgrade. These calculation were started again and are in progress (for about half a year) at the moment of preparingthis article. References [1] I. Assem, D. Simson, and A. Skowro´nski.
Elements of the representation theory of associative algebras. Vol. 1 , volume 65 of
LondonMathematical Society Student Texts . Cambridge University Press, Cambridge, 2006.[2] J. Bia lkowski. Deformed preprojective algebras of Dynkin type D n . in preparations.[3] J. Bia lkowski. Deformed preprojective algebras of Dynkin type E . Comm. Algebra , 47(4):1568–1577, 2019.[4] J. Bia lkowski. On solving equations for deformed preprojective algebras of Dynkin type E Colloq. Math. , 156(2):165–197, 2019.[6] J. Bia lkowski. Deformed mesh algebras of Dynkin type F . J. Algebra Appl. , 19(3), 2020. in press, DOI 10.1142/S0219498820500498.[7] K. Erdmann and A. Skowro´nski. Periodic algebras. In
Trends in representation theory of algebras and related topics , EMS Ser. Congr. Rep.,pages 201–251. Eur. Math. Soc., Z¨urich, 2008.[8] D. Simson. Mesh algorithms for solving principal Diophantine equations, sand-glass tubes and tori of roots.
Fund. Inform. , 109(4):425–462,2011.[9] D. Simson. Mesh geometries of root orbits of integral quadratic forms.
J. Pure Appl. Algebra , 215(1):13–34, 2011.[10] D. Simson. Tame-wild dichotomy of Birkhoff type problems for nilpotent linear operators.
J. Algebra , 424:254–293, 2015.[11] D. Simson. Representation-finite Birkhoff type problems for nilpotent linear operators.
J. Pure Appl. Algebra , 222(8):2181–2198, 2018.[12] D. Simson. A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices.
Linear Algebra Appl. , 586:190–238,2020.[13] D. Simson and A. Skowro´nski.
Elements of the representation theory of associative algebras. Vol. 2 , volume 71 of
London MathematicalSociety Student Texts . Cambridge University Press, Cambridge, 2007.[14] D. Simson and A. Skowro´nski.
Elements of the representation theory of associative algebras. Vol. 3 , volume 72 of
London MathematicalSociety Student Texts . Cambridge University Press, Cambridge, 2007.[15] A. Skowro´nski and K. Yamagata.
Frobenius algebras. I . EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z¨urich,2011.(Jerzy Bia lkowski). EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z¨urich,2011.(Jerzy Bia lkowski)