On calculation of fluxbrane polynomials corresponding to classical series of Lie algebras

On calculation of fluxbrane p olynomials corresp onding to classical series of Lie algebras A. A. Golubtso v a 1 ,b and V. D. Iv ashc h uk 2 ,a,b (a) Center for Gr avitation and F undamental Metr olo gy, VNIIMS, 46 Ozyornaya Str., Mosc ow 1193 61, Ru ssia (b) Institute of Gr a vitation and Cosmolo gy, Pe oples ’ F riendship University of Russia, 6 Miklukho-Maklaya Str., Mosc ow 117198, Russia Abstract W e pr esen t a description of computational program (written in Maple) for calcu- lation of fluxbrane p olynomials corresp ond ing to classical simple Lie algebras. These p olynomials define certain sp ecial solutions to op en T o d a c hain equations. 1 In tro duction In this p ap er we deal with a set of equations d dz  z H s d dz H s  = P s r Y s ′ =1 H − A ss ′ s ′ , (1.1) with the follo w ing b oundary conditions imp osed: H s (+0) = 1 , (1.2) s = 1 , ..., r . Here the fu nctions H s ( z ) > 0 are defined on the in terv al (0 , + ∞ ) , P s > 0 for all s and ( A ss ′ ) is th e Cartan matrix for some fin ite dimensional simple L ie algebra G of rank r ( A ss = 2 for all s ). The functions H s ( z ) > 0 app ear as mo du li functions of generalize d fluxbr ane solutions ob- tained in [1]. Parameters P s are prop ortional to brane c harge dens it y squared Q 2 s and z = ρ 2 , where ρ is a radial p arameter. The b oundary condition (1 .2) guaran tees the absence o f singularit y (in the metric) for ρ = +0 . F or fluxb rane solutions see [1], [3]-[11] and references therein. (The more g eneral classes o f solutions were describ ed in [12, 1 3]). The simplest “fluxb rane” solution is a w ell-kno wn Melvin solution [2] describing the gra vitational field of a flux tub e. The Melvin solution corresp onds to the Lie algebra A 1 = sl (2) of rank 1 . It was conjectured in [1] that eqs. (1.1), (1.2) ha ve p olynomial solutions H s ( z ) = 1 + n s X k =1 P ( k ) s z k , (1.3) where P ( k ) s are constan ts, k = 1 , . . . , n s . Here P ( n s ) s 6 = 0 and n s = 2 r X s ′ =1 A ss ′ . (1.4) 1 siedhe@gmail.co m 1 rusgs@phys.msu.ru 1 s = 1 , ..., r , where ( A ss ′ ) = ( A ss ′ ) − 1 . Intege rs n s are comp onents of th e so-call ed t wice dual W eyl v ector in the basis of s imple ro ots [14]. It w as p ointed in [1] that the conjecture on p olynomial structure of H s (suggested originally for s emisimple Lie algebras) ma y b e v erifi ed for A n and C n Lie alge bras along a line as i t w as done for b lac k-brane p olynomials from [15] (see also [13]). In [1] certain examples of fluxbr ane solutio ns corresp onding to Lie algebras A 1 ⊕ . . . ⊕ A 1 and A 2 w ere present ed. The substitution of (1.3 ) in to (1.1) giv es an infinite c h ain of r elations on paramete rs P ( k ) s and P s . The first relation in this c hain P s = P (1) s , (1.5) s = 1 , ..., r , co rresp onds to z 0 -term in th e decomp osition of (1.1). Sp ecial solutions. W e note that for a sp ecial choice of P s parameters: P s = n s P , P > 0 , the p olynomials ha v e the follo wing simple form [1 ] H s ( z ) = (1 + P z ) n s , (1.6) s = 1 , ..., r . This relation ma y b e considered as nice to ol for verificat ion of general solutions obtained by either analytical or computer calculations. Remark: op en T o da c hains. It should b e also note d that a set of p olynomials H s define a sp ecial solution to the op en T o da c hain equations [16, 17, 18, 19] corresp onding to the Lie algebra G d 2 q s du 2 = − B s exp( r X s ′ =1 A ss ′ q s ′ ) , (1.7) where B s = 4 P s , H s = exp( − q s ( u ) − n s u ) , (1.8) s = 1 , ..., r and z = e − 2 u . In this pap er w e suggest a computational p r ogram for ca lculations of p olynomials corresp ondin g to classical series of simp le Lie algebras. 2 Cartan matrices for classical simple Lie algeb ras Here we list, for conv enience, the Cartan matrices f or all classical simple Lie algebras and in v erse Cartan matrices as w ell. In s ummary [14], there are four classical infinite series of simple Lie algebras, w hic h are d enoted b y A r ( r ≥ 1) , B r ( r ≥ 3) , C r ( r ≥ 2) , D r ( r ≥ 4) . (2.1) In all cases th e su b script d enotes the r an k of the algebra. Th e alge br as in the infinite series of simp le Lie algebras are called the classical (Lie) algebras. They are isomorphic to the m atrix algebras A r ∼ = sl( r + 1) , B r ∼ = so(2 r + 1) , C r ∼ = sp( r ) , D r ∼ = so(2 r ) . (2.2) 2 A r series. Let A = ( A ss ′ ) b e r × r Cartan m atrix for the Lie algebra A r = sl( r + 1) , r ≥ 1 . The Cartan matrices for A r -series hav e the follo wing form ( A ss ′ ) =         2 − 1 0 . . . 0 0 − 1 2 − 1 . . . 0 0 0 − 1 2 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 2 − 1 0 0 0 . . . − 1 2         (2.3) This matrix is describ ed graph ically by the Dynkin d iagram p ictured on Fig. 1. ✉ ✉ ✉ ✉ ✉ 1 2 3 . . . r − 1 r Fig. 1. Dynkin diagr am for A r Lie algebr a F or s 6 = s ′ , A ss ′ = − 1 if no des s and s ′ are connected b y a lin e on the d iagram and A ss ′ = 0 otherwise. Using the r elation for th e inv erse matrix A − 1 = ( A ss ′ ) (see S ect.7.5 in [14]) A ss ′ = 1 r + 1 min( s, s ′ )[ r + 1 − max( s, s ′ )] (2.4) w e may rewrite (1.4) as follo ws n s = s ( r + 1 − s ) , (2.5) s = 1 , . . . , r . B r and C r series. F or B r -series we h a v e the follo wing Cartan matrices ( A ss ′ ) =         2 − 1 0 . . . 0 0 − 1 2 − 1 . . . 0 0 0 − 1 2 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 2 − 2 0 0 0 . . . − 1 2         (2.6) while for C r -series the Cartan matrices read as follo ws ( A ss ′ ) =         2 − 1 0 . . . 0 0 − 1 2 − 1 . . . 0 0 0 − 1 2 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 2 − 1 0 0 0 . . . − 2 2         (2.7) Dynkin diagrams for these cases are pictured on Fig. 2. ✉ ✉ ✉ ✉ ✉ 1 2 3 . . . r − 1 r > ✉ ✉ ✉ ✉ ✉ 1 2 3 . . . r − 1 r < Fig. 2. Dynkin diagr ams for B r and C r Lie algebr as, r esp e ctively 3 In these cases w e ha v e the follo wing formula s f or inv erse matricies A − 1 = ( A ss ′ ) : A ss ′ = ( min( s, s ′ ) for s 6 = r , 1 2 s ′ for s = r , A ss ′ = ( min( s, s ′ ) for s ′ 6 = r , 1 2 s for s ′ = r (2.8) and relation (1.4 ) tak es the form n s =  s (2 r + 1 − s ) for s 6 = r, r 2 ( r + 1) for s = r ; n s = s (2 r − s ) , (2.9) for B r and C r series, resp ectiv ely , s = 1 , . . . , r . D r series. F or D r -series the Cartan matrices read ( A ss ′ ) =           2 − 1 0 . . . 0 0 − 1 2 − 1 . . . 0 0 0 − 1 2 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 2 − 1 − 1 0 0 0 . . . − 1 2 0 0 0 0 . . . − 1 0 2           (2.10) W e h a v e the follo wing Dynkin d iagram for this case (Fig. 3): ✉ ✉ ✉ ✉ 1 2 3 . . . r − 2 ✉ ✉   ❅ ❅ r r − 1 Fig. 3. Dynkin diagr am f or D r Lie algebr a and formula f or the inv erse matrix A − 1 = ( A ss ′ ) [14]: A ss ′ =                  min( s, s ′ ) for s, s ′ / ∈ { r , r − 1 } , 1 2 s for s / ∈ { r, r − 1 } , s ′ ∈ { r , r − 1 } , 1 2 s ′ for s ∈ { r , r − 1 } , s ′ / ∈ { r , r − 1 } , 1 4 r for s = s ′ = r or s = s ′ = r − 1 , 1 4 ( r − 2) for s = r, s ′ = r − 1 or vice versa. (2.11) The relation (1.4 ) in this case reads n s =  s (2 r − 1 − s ) for s / ∈ { r , r − 1 } , r 2 ( r − 1) for s ∈ { r , r − 1 } , (2.12) s = 1 , . . . , r . F or the simple Lie algebras of typ e A r , D r , all ro ots ha v e th e same length, and any t w o n o des of Dynkin diagram are connected b y at most one line. In the other cases there are ro ots of tw o differen t lengths, the length of the long ro ots b eing √ 2 times the length of the short ro ots for B r , C r , resp ectiv ely . 4 3 Computing of flu xb rane p olyn omials A t the momen t the pr ob lem of findin g all co efficien ts P ( k ) s in p olynomials H s analyticall y is lo oking as to o complicated one. Thereby it is essent ial to create a program, calculating all the co efficients required. The algorithm of find ing co efficient s P ( k ) s is the follo wing one: • we substitute the p olynomials H s in to the set of d ifferential equations (1.1 ) and redu ce the differen tial equations to a set of algebraic equations for P ( k ) s (expanding th e equations into degrees of the v ariable z ); • th e d eriv ed system of algebraic equations is solv ed in terms of the first co efficients P (1) s = P s . W e c ho ose the symb olic computational system Maple v.11.01. for implementa tion of the d is- cussed algorithm. T he standard Maple p ack ages LinearAlgeb ra and Pol ynomialT ools are used for wo rking with matrices an d p olynomials appropriately . So let us start in the follo wing wa y: > with(Linea rAlgebra ): > with(Polyn omialToo ls): A t the next step w e n eed to sp ecify the d imension of the Cartan matrix: >S:=3: Th us, the dimension of th e C artan matrix is 3. This v ariable ( S ) also determines the twice dual W eyl v ector dimension and the num b er of the differenti al equations. A v ariable for the Lie algebra is entered similarly . >algn:=b n: Consequent ly , b y d efault the p rogram calculates the fluxb r ane p olynomials for the Lie algebra B n with the Cartan matrix, the size of whic h is 3 . W e use the standard Maple function Matrix to declare the Cartan matrix. >A:=Matr ix(S,S): It is more conv enient to fill th e Cartan matrix with the help of th e separate pro cedu re, which constructs matrix elemen ts dep ending on th e Lie algebras. Let us consider the p ro cedure. There are three callable v ariables in it: a Lie algebra, the matrix size and th e Cartan matrix itself. >AlgLie: =proc(al gn,S,CartA: =Matrix(S,S)) The matrix elemen ts are constructed in compliance with Dyn k in diagrams. Here we consider the constru ction of the matrix elemen ts for th e simple Lie algebras A n , B n , C n and D n . Th e follo w ing lo cal v ariables are essentia l local i,mu,nu; i := 0; mu := 0; nu:=0; mu := S-1; nu:=S-2; 5 where i is an iteratio n v ariable and mu , nu are v ariables for sub scripting in B n , C n and D n algebras. By default all element s of the m atrix are zeros. Thereby it is necessary to fi ll only the elemen ts different from n ull according to the Dyn kin diagrams. In itially w e fill in the main d iagonal of the matrix, b ecause it is identica l to the Cartan matrix of an y algebra. for i to S do CartA[i, i] := 2 end do; Then, using the conditional op erator, we consequen tly tak e under consideration the condition on the m atrix size and the fact of b elonging to the Lie algebra. W e b egin w ith the A n algebra. if (S>=1) and algn=an then end if; The elemen ts of the secondary diagonals for A n algebra equal − 1 according to the Dynkin diagram. Firstly , we fill in the up p er secondary diagonal, and then it is mirrored one. These actio ns are p erformed do wn in side the pr evious op erator. for i to S-1 do CartA[i , i+1] := -1 end do; for i to S-1 do CartA[i +1, i] := Cart A[i, i+1] end do; The similar actions are run w ith the conditional op erator for the B n , in agreemen t with the Dynkin diagram for this algebra. if (S>=3) and algn=bn then for i from 1 to (S - 1 ) do CartA[i+1 ,i]:=-1 end do; for i from 1 to (S - 2) do CartA[i,i+ 1]:=Cart A[i+1,i] end do; end if; But un der the Dynkin diagrams for the Lie algebras B n and C n , certain elemen ts are differen t from − 1 for the secondary diagonals. So, let us s u pplement the preceding lines with the f ollo w ing matrix elemen t CartA[mu ,S]:=-2; In muc h the same wa y we ha v e for the Lie algebra C n if (S>=2) and algn=cn then for i to S-1 do CartA[i , i+1] := -1 end do; for i to S-2 do CartA[i +1, i] := Cart A[i, i+1] end do; CartA[S, mu] := -2; end if; F or the Lie algebra D n certain non-diagonal elemen ts of the C artan matrix differ f r om z er o , while some elements in the secondary diagonals are equal to z er o . According to th e Dynkin diagram for th is algebra the elements of the Cartan matrix are defined in the follo wing w a y if (S>=4) and algn=dn then for i to S-2 do CartA[i , i+1] := -1 end do; for i to S-2 do CartA[i +1, i] := Cart A[i, i+1] end do; CartA[S, nu] := CartA[m u, nu]; CartA[nu , S] := Ca rtA[S, nu]; end if; 6 W e return the v ariable CartA and do not forget to set closing end at the end of the pro cedure. Th us, the Cartan matrix is filled, dep en ding on th e c hosen algebra, by a call of this p ro cedure with the appr opriate parameters. >AlgLie( algn, S, A); A :=   2 − 1 0 − 1 2 − 2 0 − 1 2   F urther we declare the twice dual W eyl vecto r b y means of the standard Maple pr o cedure Vector . >n := Vector[ro w](1 .. S): The elemen ts of the inv erse Cartan matrix are necessary f or calculating the t wice dual W eyl v ector’s comp onents. The matrix inv ersion is done b y means of the standard Maple pro cedur e MatrixIn verse . >A1 := MatrixIn verse(A) ; A 1 :=       1 1 1 1 2 2 1 2 1 3 2       W e use the standard Maple p ro cedure add for calculating the t wice dual W eyl v ector’s comp o- nen ts. >for i to S do n[i] := 2*add(A1[i, j], j = 1 .. S) end do: The coefficien ts P ( k ) s are represen ted b y a matrix. Th e num b er of ro ws of this matrix is the num b er of the differen tial equations (that is S ), and the num b er of columns is the maximal comp onent of the t wice dual W eyl ve ctor. But the twice du al W eyl v ector was set by means of the pro cedur e Vector and th e vec tor must b e con v erted into the list to fi nd the maximal comp onent of it using the standard Maple pro cedur e "max". Th is action is p erformed by means of the stand ard Maple pro cedure co nvert . >maxel := max(c onvert(n , list)[]) : No w the matrix of th e co efficien ts P ( k ) s can b e declared. >P := array(1 .. S, 1 .. maxel): Let us declare the matrix of the p olynomials by means of the p ro cedur e Vect or . >H := Vector[ro w](1 .. S): Eac h element of this matrix is d efined according to the hypothesis in the f ollo wing wa y: >for i to S do H[i] := 1+add(P[i, k]*z^k, k = 1 .. n[i]) end do: 7 It’s necessary to conv ert the elemen ts of the matrices H and A int o the ind exed v ariables for correct calculations. >for i to S do for j to S do a[i, j] := A[i, j] end do end do: >for i to S do h[i] := H[i] end do: Let us en ter one more indexed v ariable c i,v for con v enience. >for i to S do for v to S do c[i, v] := h[v]^(-a [i, v]) end do end do: W e repr esen t the set of equ ations as a matrix b y the us e of the pro cedure Vecto r . Now the system of the differenti al equ ations can b e defin ed using the standard Maple pr o cedure diff . >equal := Vector[ro w](1 .. S): >for i to S do equal[i] := diff (z*(dif f(H[i],z ))/H[i],z)-P[i,1]*(product(c[i,m], m = 1. .S)) end do: The p ro cedure p roduct is the standard Maple pr o cedure for a pro duct. F urther w e enter t w o more matrices for simplifi ed equations and numerato rs of these equations using the p r o cedure Vector . > simequal := Vector[ro w](1 .. S): > newequal := Vector[ro w](1 .. S): The fir st of these matrices is filled, collecting by degrees eac h of the equations by means of the standard pr o cedures. The eleme nts of the second matrix are turned out b y selection of the n umerators fr om the s im p lified equations. > for i to S do simequal[i ] := simpli fy(comb ine(valu e(equal[i]), po wer)) end do: > for i to S do newequal[i ] := numer( simequa l[i]) end do: It’s necessa ry to find out the degrees of th e n u merators to collec t the coefficien ts P ( k ) s at v arious degrees of th e v ariable z . S o let us describ e a matrix, whic h elemen ts are the degrees of the v ariable z . The d egrees are calculated by the s tand ard Maple pro cedure degre e . > maxcoeff := Vector[ro w](1 .. S): > for i to S do maxcoeff[i ] := degree (newequ al[i], z) end do: W e define a t wo- dimensional table (the standard Maple structure of d ata) for the sys tem of algebraic equations an d fill the table’s elemen ts in the follo wing wa y: > coefflist := table(): > for i to S do for c from 0 to maxcoef f[i] do coefflis t[i, c] := coe ff(neweq ual[i], z, c) = 0 end do end do: 8 The system should b e conv erted in to the list to solv e it b y means of the standard Maple pro cedur e "so lve". Th is action allo ws us to apply the fu nction solve . > Sys := convert(co efflist, list) : > sol := solve(Sys) : But the form of th e answ er is inconv enient. Thus su bstituting the answer int o th e p olynomial form, we get > trans := {seq(seq (P[i,j] = P[i, j], i = 1..S), j = 1..maxel)}: > sol := simplify(m ap2(subs , trans, sol)): > P1 := map2(su bs, sol, evalm (P)): > for i to S do H[i]:= 1+add(P1[i, k]*z^k, k = 1..n[ i]) end do; H 1 := 1 + P 1 , 1 z + 1 4 P 1 , 1 P 2 , 1 z 2 + 1 18 P 1 , 1 P 2 , 1 P 3 , 1 z 3 + 1 144 P 1 , 1 P 2 , 1 P 2 3 , 1 z 4 + 1 3600 P 1 , 1 P 2 2 , 1 P 2 3 , 1 z 5 + 1 12960 0 P 2 1 , 1 P 2 2 , 1 P 2 3 , 1 z 6 , H 2 := 1 + P 2 , 1 z + ( 1 4 P 1 , 1 P 2 , 1 + 1 2 P 2 , 1 P 3 , 1 ) z 2 + ( 1 9 P 2 , 1 P 2 3 , 1 + 2 9 P 1 , 1 P 2 , 1 P 3 , 1 ) z 3 + ( 1 144 P 2 2 , 1 P 2 3 , 1 + 1 72 P 1 , 1 P 2 2 , 1 P 3 , 1 + 1 16 P 1 , 1 P 2 , 1 P 2 3 , 1 ) z 4 + 7 600 P 1 , 1 P 2 2 , 1 P 2 3 , 1 z 5 + ( 1 1600 P 1 , 1 P 3 2 , 1 P 2 3 , 1 + 1 5184 P 2 1 , 1 P 2 2 , 1 P 2 3 , 1 + 1 2592 P 1 , 1 P 2 2 , 1 P 3 3 , 1 ) z 6 +( 1 16200 P 1 , 1 P 3 2 , 1 P 3 3 , 1 + 1 32400 P 2 1 , 1 P 3 2 , 1 P 2 3 , 1 ) z 7 + ( 1 51840 0 P 1 , 1 P 3 2 , 1 P 4 3 , 1 + 1 25920 0 P 2 1 , 1 P 3 2 , 1 P 3 3 , 1 ) z 8 + 1 46656 00 P 2 1 , 1 P 3 2 , 1 P 4 3 , 1 z 9 + 1 46656 0000 P 2 1 , 1 P 4 2 , 1 P 4 3 , 1 z 10 , H 3 := 1+ P 3 , 1 z + 1 4 P 2 , 1 P 3 , 1 z 2 +( 1 36 P 1 , 1 P 2 , 1 P 3 , 1 + 1 36 P 2 , 1 P 2 3 , 1 ) z 3 + 1 144 P 1 , 1 P 2 , 1 P 2 3 , 1 z 4 + 1 3600 P 1 , 1 P 2 2 , 1 P 2 3 , 1 z 5 + 1 12960 0 P 1 , 1 P 2 2 , 1 P 3 3 , 1 z 6 . It should b e noted that throughout the program we u se a sligh tly different notation for the first co efficient s, i.e. P s, 1 = P s . (3.1) 4 Examples of p ol yn omials Here w e p resen t certain examples of p olynomials corresp onding to the Lie algebras A 1 , A 2 , A 3 , B 3 , C 2 and D 4 . 4.1 A r -p olynomials, r = 1 , 2 , 3 . A 1 -case. The simplest example o ccurs in the case of the Lie algebra A 1 = sl (2) . W e get [1] H 1 ( z ) = 1 + P 1 z . (4.1) A 2 -case. F or the Lie algebra A 2 = sl (3) with the C artan matrix ( A ss ′ ) =  2 − 1 − 1 2  (4.2) 9 w e h a v e [1] n 1 = n 2 = 2 and H 1 = 1 + P 1 z + 1 4 P 1 P 2 z 2 , (4.3) H 2 = 1 + P 2 z + 1 4 P 1 P 2 z 2 , (4.4) s = 1 , 2 . A 3 -case. The p olynomials for the A 3 -case read as f ollo w s H 1 = 1 + P 1 z + 1 4 P 1 P 2 z 2 + 1 36 P 1 P 2 P 3 z 3 , (4.5) H 2 = 1 + P 2 z +  1 4 P 1 P 2 + 1 4 P 2 P 3  z 2 + 1 9 P 1 P 2 P 3 z 3 (4.6) + 1 144 P 1 P 2 2 P 3 z 4 , H 3 = 1 + P 3 z + 1 4 P 2 P 3 z 2 + 1 36 P 1 P 2 P 3 z 3 . (4.7) 4.2 B 3 -p olynomials F or the Lie algebra B 3 w e get the follo wing p olynomials H 1 = 1 + P 1 z + 1 4 P 1 P 2 z 2 + 1 18 P 1 P 2 P 3 z 3 + 1 144 P 1 P 2 P 2 3 z 4 + 1 3600 P 1 P 2 2 P 2 3 z 5 (4.8) + 1 12960 0 P 2 1 P 2 2 P 2 3 z 6 , H 2 = 1 + P 2 z +  1 4 P 1 P 2 + 1 2 P 2 P 3  z 2 +  1 9 P 2 P 2 3 + 2 9 P 1 P 2 P 3  z 3 +  1 144 P 2 2 P 2 3 (4.9) + 1 72 P 1 P 2 2 P 3 + 1 16 P 1 P 2 P 2 3  z 4 + 7 600 P 1 P 2 2 P 2 3 z 5 +  1 1600 P 1 P 3 2 P 2 3 + 1 5184 P 2 1 P 2 2 P 2 3 + 1 2592 P 1 P 2 2 P 3 3 ) z 6 +  1 16200 P 1 P 3 2 P 3 3 + 1 32400 P 2 1 P 3 2 P 2 3  z 7 +  1 51840 0 P 1 P 3 2 P 4 3 + 1 25920 0 P 2 1 P 3 2 P 3 3  z 8 + 1 46656 00 P 2 1 P 3 2 P 4 3 z 9 + 1 46656 0000 P 2 1 P 4 2 P 4 3 z 10 , H 3 = 1 + P 3 z + 1 4 P 2 P 3 z 2 +  1 36 P 1 P 2 P 3 + 1 36 P 2 P 2 3  z 3 + 1 144 P 1 P 2 P 2 3 z 4 (4.10) + 1 3600 P 1 P 2 2 P 2 3 z 5 + 1 12960 0 P 1 P 2 2 P 3 3 z 6 . 4.3 C 2 -p olynomials F or the Lie algebra C 2 = so (5) with the Cartan matrix ( A ss ′ ) =  2 − 1 − 2 2  (4.11) w e get from (1.4) n 1 = 3 and n 2 = 4 . 10 F or C 2 -p olynomials we obtain in agreemen t with [11] H 1 = 1 + P 1 z + 1 4 P 1 P 2 z 2 + 1 36 P 2 1 P 2 z 3 , (4.12) H 2 = 1 + P 2 z + 1 2 P 1 P 2 z 2 + 1 9 P 2 1 P 2 z 3 + 1 144 P 2 1 P 2 2 z 4 . (4.13) 4.4 D 4 -p olynomials F or the Lie algebra D 4 w e fi nd the follo wing set of p olynomials H 1 = 1 + P 1 z + 1 4 P 1 P 2 z 2 +  1 36 P 1 P 2 P 3 + 1 36 P 1 P 2 P 4  z 3 + 1 144 P 1 P 2 P 3 P 4 z 4 (4.14) + 1 3600 P 1 P 2 2 P 3 P 4 z 5 + 1 12960 0 P 2 1 P 2 2 P 3 P 4 z 6 , H 2 = 1 + P 2 z +  1 4 P 1 P 2 + 1 4 P 2 P 3 + 1 4 P 2 P 4  z 2 +  1 9 P 1 P 2 P 3 + 1 9 P 1 P 2 P 4 (4.15) + 1 9 P 2 P 3 P 4  z 3 +  1 144 P 1 P 2 2 P 3 + 1 144 P 1 P 2 2 P 4 + 1 144 P 2 2 P 3 P 4 + 1 16 P 1 P 2 P 3 P 4  z 4 + 7 600 P 1 P 2 2 P 3 P 4 z 5 +  1 1600 P 1 P 3 2 P 3 P 4 + 1 5184 P 1 P 2 2 P 2 3 P 4 + 1 5184 P 2 1 P 2 2 P 3 P 4 + 1 5184 P 1 P 2 2 P 3 P 2 4  z 6 +  1 32400 P 2 1 P 3 2 P 3 P 4 + 1 32400 P 1 P 3 2 P 3 P 2 4 + 1 32400 P 1 P 3 2 P 2 3 P 4  z 7 +  1 51840 0 P 2 1 P 3 2 P 3 P 2 4 + 1 51840 0 P 2 1 P 3 2 P 2 3 P 4 + 1 51840 0 P 1 P 3 2 P 2 3 P 2 4  z 8 + 1 46656 00 P 2 1 P 3 2 P 2 3 P 2 4 z 9 + 1 46656 000 P 2 1 P 4 2 P 2 3 P 2 4 z 10 , H 3 = 1 + P 3 z + 1 4 P 2 P 3 z 2 +  1 36 P 1 P 2 P 3 + 1 36 P 2 P 3 P 4  z 3 + 1 144 P 1 P 2 P 3 P 4 z 4 (4.16) + 1 3600 P 1 P 2 2 P 3 P 4 z 5 + 1 12960 0 P 1 P 2 2 P 2 3 P 4 z 6 , H 4 = 1 + P 4 z + 1 4 P 2 P 4 z 2 +  1 36 P 1 P 2 P 4 + 1 36 P 2 P 3 P 4  z 3 + 1 144 P 1 P 2 P 3 P 4 z 4 (4.17) + 1 3600 P 1 P 2 2 P 3 P 4 z 5 + 1 12960 0 P 1 P 2 2 P 3 P 2 4 z 6 . 5 Some relations b etw een p ol ynomials Let us denote the set of p olynomials corresp onding to a set of parameters P 1 > 0 , ..., P r > 0 as follo w ing H s = H s ( z , P 1 , ..., P r ; A ) , (5.1) s = 1 , . . . , r , wh ere A = A [ G ] is the Cartan matrix corresp onding to a (semi)simple Lie algebra G . 11 5.1 C n +1 -p olynomials from A 2 n +1 -ones The set of p olynomials corresp onding to the L ie algebra C n +1 ma y b e ob tained from the set of p olynomials corr esp onding to the Lie algebra A 2 n +1 according to the follo wing relations H s ( z , P 1 , ..., P n +1 ; A [ C n +1 ]) = H s ( z , P 1 , ..., P n +1 , P n +2 = P n , ..., P 2 n +1 = P 1 ; A [ A 2 n +1 ]) , (5.2) s = 1 , . . . , n + 1 , i.e. the parameters P 1 , ..., P n +1 , P n +2 , ..., P 2 n +1 are id en tified symmetricall y w.r.t. P n +1 . See Dynkin d iagrams on Figs. 1-2. Relation (5.2) ma y b e verified u sing the program from the Section 3. (F or the case n = 1 see formulas from the pr evious section.) 5.2 B n -p olynomials from D n +1 -ones The set p olynomials corresp ondin g to th e Lie algebra B n ma y b e obtained from th e set of p oly- nomials corresp onding to the Lie algebra D n +1 according to the follo wing relation H s ( z , P 1 , ..., P n ; A [ B n ]) = H s ( z , P 1 , ..., P n , P n +1 = P n ; A [ D n +1 ]) , (5.3) s = 1 , . . . , n , i.e. th e parameters P n and P n +1 are iden tified. See Dynkin d iagrams on Figs. 2-3. Relation (5.3) ma y b e ve rified using the program from the S ection 3. (F or th e case n = 3 see form ulas fr om the pr evious section.) 5.3 Reduction form ulas Here we d enote the Cartan matrix as follo ws: A = A Γ , wh ere Γ is th e related Dyn kin graph. Let i b e a no de of Γ . Let us denote b y Γ i a Dynkin graph (corresp onding to a certain semi-simp le Lie algebra) that is obtained from Γ by erasing all lines that hav e endp oin ts at i . It ma y b e verified (e.g. b y usin g the p r ogram) th at the follo w ing reduction formulae are v alid H s ( z , P 1 , ..., P i = 0 , ..., P r ; A Γ ) = H s ( z , P 1 , ..., P i = 0 , ..., P r ; A Γ i ) , (5.4) s = 1 , . . . , r . Moreo ve r, H i ( z , P 1 , ..., P i = 0 , ..., P r ; A Γ ) = 1 . (5.5) This means th at setting P i = 0 w e reduce the set of p olynomials by rep lacing the the Cartan matrix A Γ b y the Cartan matrix A Γ i . In this case the p olynomial H i = 1 corresp onds to A 1 - subalgebra (depicted by the no de i ) and the parameter P i = 0 . 3 . As an example of r eduction formulas we pr esen t the f ollo w ing relations H s ( z , P 1 , ..., P n , P n +1 = 0; A [ G ]) = H s ( z , P 1 , ..., P n ; A [ A n ]) , (5.6) s = 1 , . . . , n , for G = A n +1 , B n +1 , C n +1 , D n +1 with appropr iate restrictions on n (see (2 .1 )). In writing relatio n (5.4) we use the num b erin g of no des in agreemen t with the Dynkin diagrams depicted on Figs. 1-3. The reduction formulas (5.5) for A 5 -p olynomials with P 3 = 0 are dep icted on Fig. 4. T he r e- duced p olynomials are coinciding with those corresp onding to semisimple Lie algebra A 2 L A 1 L A 2 . ✉ ✉ ✉ ✉ ✉ 1 2 3 4 5 Fig. 4. Dynkin diagr am for semisimple Lie algebr a A 2 L A 1 L A 2 describing the set of A 5 -p olynomials with P 3 = 0 3 The analytica l pr o of of the relations (5.2)-(5.4) will be given in a sepa rate publica tion 12 6 Conclus ions Here w e h a v e presented a description of computational p rogram (written in Maple) for calculation of flu xbrane p olynomials related to classical simple L ie algebras. (Generalizatio n to semisimple Lie algebras is a straigh tforw ard one.) This program giv es by pro duct a verificat ion of the conjecture suggested p reviously in [1]. The p olynomials considered ab o ve defin e sp ecial solutions to open T o da c hain equations corresp onding to simple Lie algebras that ma y b e of in terest for certain applications of T o da c hains. W e ha ve also considered (without pro of ) certain relations b et w een p olynomials, e.g. so-called reduction form ulas. 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