Abstract
We present a description of computational program (written in Maple) for calculation of fluxbrane polynomials corresponding to classical simple Lie algebras. These polynomials define certain special solutions to open Toda chain equations.
On calculation of fluxbrane polynomials corresponding to classical series of Lie algebras
A. A. Golubtsova^{1}^{1}1 and V. D. Ivashchuk^{2}^{2}footnotemark: 2^{1}^{1}footnotetext:
(a) Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Str., Moscow 119361, Russia
(b) Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 MiklukhoMaklaya Str., Moscow 117198, Russia
1 Introduction
In this paper we deal with a set of equations
(1.1) 
with the following boundary conditions imposed:
(1.2) 
. Here the functions are defined on the interval , for all and is the Cartan matrix for some finite dimensional simple Lie algebra of rank ( for all ).
The functions appear as moduli functions of generalized fluxbrane solutions obtained in [1]. Parameters are proportional to brane charge density squared and , where is a radial parameter. The boundary condition (1.2) guarantees the absence of singularity (in the metric) for . For fluxbrane solutions see [1], [3][11] and references therein. (The more general classes of solutions were described in [12, 13]). The simplest “fluxbrane” solution is a wellknown Melvin solution [2] describing the gravitational field of a flux tube. The Melvin solution corresponds to the Lie algebra of rank .
It was conjectured in [1] that eqs. (1.1), (1.2) have polynomial solutions
(1.3) 
where are constants, . Here and
(1.4) 
, where . Integers are components of the socalled twice dual Weyl vector in the basis of simple roots [14]. It was pointed in [1] that the conjecture on polynomial structure of (suggested originally for semisimple Lie algebras) may be verified for and Lie algebras along a line as it was done for blackbrane polynomials from [15] (see also [13]). In [1] certain examples of fluxbrane solutions corresponding to Lie algebras and were presented.
The substitution of (1.3) into (1.1) gives an infinite chain of relations on parameters and . The first relation in this chain
(1.5) 
, corresponds to term in the decomposition of (1.1).
Special solutions. We note that for a special choice of parameters: , , the polynomials have the following simple form [1]
(1.6) 
. This relation may be considered as nice tool for verification of general solutions obtained by either analytical or computer calculations.
Remark: open Toda chains. It should be also noted that a set of polynomials define a special solution to the open Toda chain equations [16, 17, 18, 19] corresponding to the Lie algebra
(1.7) 
where ,
(1.8) 
and .
In this paper we suggest a computational program for calculations of polynomials corresponding to classical series of simple Lie algebras.
2 Cartan matrices for classical simple Lie algebras
Here we list, for convenience, the Cartan matrices for all classical simple Lie algebras and inverse Cartan matrices as well.
In summary [14], there are four classical infinite series of simple Lie algebras, which are denoted by
(2.1) 
In all cases the subscript denotes the rank of the algebra. The algebras in the infinite series of simple Lie algebras are called the classical (Lie) algebras. They are isomorphic to the matrix algebras
(2.2) 
series. Let be Cartan matrix for the Lie algebra , . The Cartan matrices for series have the following form
(2.3) 
This matrix is described graphically by the Dynkin diagram pictured on Fig. 1.
[5pt] Fig. 1. Dynkin diagram for Lie algebra
For , if nodes and are connected by a line on the diagram and otherwise. Using the relation for the inverse matrix (see Sect.7.5 in [14])
(2.4) 
we may rewrite (1.4) as follows
(2.5) 
.
and series. For series we have the following Cartan matrices
(2.6) 
while for series the Cartan matrices read as follows
(2.7) 
Dynkin diagrams for these cases are pictured on Fig. 2.
[5pt] Fig. 2. Dynkin diagrams for and Lie algebras, respectively
In these cases we have the following formulas for inverse matricies :
(2.8) 
and relation (1.4) takes the form
(2.9) 
for and series, respectively, .
series. For series the Cartan matrices read
(2.10) 
We have the following Dynkin diagram for this case (Fig. 3):
[5pt] Fig. 3. Dynkin diagram for Lie algebra
For the simple Lie algebras of type , , all roots have the same length, and any two nodes of Dynkin diagram are connected by at most one line. In the other cases there are roots of two different lengths, the length of the long roots being times the length of the short roots for , , respectively.
3 Computing of fluxbrane polynomials
At the moment the problem of finding all coefficients in polynomials analytically is looking as too complicated one. Thereby it is essential to create a program, calculating all the coefficients required.
The algorithm of finding coefficients is the following one:

we substitute the polynomials into the set of differential equations (1.1) and reduce the differential equations to a set of algebraic equations for (expanding the equations into degrees of the variable );

the derived system of algebraic equations is solved in terms of the first coefficients .
We choose the symbolic computational system Maple v.11.01. for
implementation of the discussed algorithm. The standard Maple
packages LinearAlgebra
and PolynomialTools
are
used for working with matrices and polynomials appropriately. So
let us start in the following way:
> with(LinearAlgebra): > with(PolynomialTools):
At the next step we need to specify the dimension of the Cartan matrix:
>S:=3:
Thus, the dimension of the Cartan matrix is 3. This variable () also determines the twice dual Weyl vector dimension and the number of the differential equations. A variable for the Lie algebra is entered similarly.
>algn:=bn:
Consequently, by default the program calculates the fluxbrane
polynomials for the Lie algebra with the Cartan matrix,
the size of which is 3 . We use the standard Maple function
Matrix
to declare the Cartan matrix.
>A:=Matrix(S,S):
It is more convenient to fill the Cartan matrix with the help of the separate procedure, which constructs matrix elements depending on the Lie algebras. Let us consider the procedure. There are three callable variables in it: a Lie algebra, the matrix size and the Cartan matrix itself.
>AlgLie:=proc(algn,S,CartA:=Matrix(S,S))
The matrix elements are constructed in compliance with Dynkin diagrams. Here we consider the construction of the matrix elements for the simple Lie algebras , , and . The following local variables are essential
local i,mu,nu; i := 0; mu := 0; nu:=0; mu := S1; nu:=S2;
where i
is an iteration variable and mu
,nu
are variables for subscripting in , and
algebras. By default all elements of the matrix are zeros. Thereby
it is necessary to fill only the elements different from null
according to the Dynkin diagrams. Initially we fill in the
main diagonal of the matrix, because it is identical to the Cartan
matrix of any algebra.
for i to S do CartA[i, i] := 2 end do;
Then, using the conditional operator, we consequently take under consideration the condition on the matrix size and the fact of belonging to the Lie algebra. We begin with the algebra.
if (S>=1) and algn=an then end if;
The elements of the secondary diagonals for algebra equal according to the Dynkin diagram. Firstly, we fill in the upper secondary diagonal, and then it is mirrored one. These actions are performed down inside the previous operator.
for i to S1 do CartA[i, i+1] := 1 end do; for i to S1 do CartA[i+1, i] := CartA[i, i+1] end do;
The similar actions are run with the conditional operator for the , in agreement 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]:=CartA[i+1,i] end do; end if;
But under the Dynkin diagrams for the Lie algebras and , certain elements are different from for the secondary diagonals. So, let us supplement the preceding lines with the following matrix element
CartA[mu,S]:=2;
In much the same way we have for the Lie algebra
if (S>=2) and algn=cn then for i to S1 do CartA[i, i+1] := 1 end do; for i to S2 do CartA[i+1, i] := CartA[i, i+1] end do; CartA[S, mu] := 2; end if;
For the Lie algebra certain nondiagonal elements of the Cartan matrix differ from , while some elements in the secondary diagonals are equal to . According to the Dynkin diagram for this algebra the elements of the Cartan matrix are defined in the following way
if (S>=4) and algn=dn then for i to S2 do CartA[i, i+1] := 1 end do; for i to S2 do CartA[i+1, i] := CartA[i, i+1] end do; CartA[S, nu] := CartA[mu, nu]; CartA[nu, S] := CartA[S, nu]; end if;
We return the variable CartA
and do not forget to set
closing end
at the end of the procedure. Thus, the Cartan
matrix is filled, depending on the chosen algebra, by a call of
this procedure with the appropriate parameters.
>AlgLie(algn, S, A);
Further we declare the twice dual Weyl vector by means of the
standard Maple procedure Vector
.
>n := Vector[row](1 .. S):
The elements of the inverse Cartan matrix are necessary for
calculating the twice dual Weyl vector’s components. The matrix
inversion is done by means of the standard Maple procedure
MatrixInverse
.
>A1 := MatrixInverse(A);
We use the standard Maple procedure add
for calculating the
twice dual Weyl vector’s components.
>for i to S do n[i] := 2*add(A1[i, j], j = 1 .. S) end do:
The coefficients are represented by a matrix.
The number of rows of this matrix is the number of the
differential equations (that is ), and the number of columns
is the maximal component of the twice dual Weyl vector. But the
twice dual Weyl vector was set by means of the procedure
Vector
and the vector must be converted into the list to
find the maximal component of it using the standard Maple
procedure "max".
This action is performed by means of the
standard Maple procedure convert
.
>maxel := max(convert(n, list)[]):
Now the matrix of the coefficients can be declared.
>P := array(1 .. S, 1 .. maxel):
Let us declare the matrix of the polynomials by means of the
procedure Vector
.
>H := Vector[row](1 .. S):
Each element of this matrix is defined according to the hypothesis in the following way:
>for i to S do H[i] := 1+add(P[i, k]*z^k, k = 1 .. n[i]) end do:
It’s necessary to convert the elements of the matrices and into the indexed variables 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 enter one more indexed variable for convenience.
>for i to S do for v to S do c[i, v] := h[v]^(a[i, v]) end do end do:
We represent the set of equations as a matrix by the use of the
procedure Vector
. Now the system of the differential
equations can be defined using the standard Maple procedure
diff
.
>equal := Vector[row](1 .. S): >for i to S do equal[i]:= diff(z*(diff(H[i],z))/H[i],z)P[i,1]*(product(c[i,m], m = 1..S)) end do:
The procedure product
is the standard Maple procedure for a
product. Further we enter two more matrices for simplified
equations and numerators of these equations using the procedure
Vector
.
> simequal := Vector[row](1 .. S): > newequal := Vector[row](1 .. S):
The first of these matrices is filled, collecting by degrees each of the equations by means of the standard procedures. The elements of the second matrix are turned out by selection of the numerators from the simplified equations.
> for i to S do simequal[i] := simplify(combine(value(equal[i]), power)) end do: > for i to S do newequal[i] := numer(simequal[i]) end do:
It’s necessary to find out the degrees of the numerators to
collect the coefficients at various degrees of the
variable . So let us describe a matrix, which elements are the
degrees of the variable . The degrees are calculated by the
standard Maple procedure degree
.
> maxcoeff := Vector[row](1 .. S): > for i to S do maxcoeff[i] := degree(newequal[i], z) end do:
We define a twodimensional table (the standard Maple structure of data) for the system of algebraic equations and fill the table’s elements in the following way:
> coefflist := table(): > for i to S do for c from 0 to maxcoeff[i] do coefflist[i, c] := coeff(newequal[i], z, c) = 0 end do end do:
The system should be converted into the list to solve it by means
of the standard Maple procedure "solve".
This action allows
us to apply the function solve
.
> Sys := convert(coefflist, list): > sol := solve(Sys):
But the form of the answer is inconvenient. Thus substituting the answer into the polynomial form, we get
> trans := {seq(seq(P[i,j] = P[i,j], i = 1..S), j = 1..maxel)}: > sol := simplify(map2(subs, trans, sol)): > P1 := map2(subs, sol, evalm(P)): > for i to S do H[i]:= 1+add(P1[i,k]*z^k, k = 1..n[i]) end do;
It should be noted that throughout the program we use a slightly different notation for the first coefficients, i.e.
(3.1) 
4 Examples of polynomials
Here we present certain examples of polynomials corresponding to the Lie algebras , , and .
4.1 polynomials, .
case. The simplest example occurs in the case of the Lie algebra . We get [1]
(4.1) 
case. For the Lie algebra with the Cartan matrix
case. The polynomials for the case read as follows
(4.5)  
(4.6)  
(4.7) 
4.2 polynomials
For the Lie algebra we get the following polynomials
(4.8)  
(4.9)  
(4.10)  
4.3 polynomials
For the Lie algebra with the Cartan matrix
(4.11) 
we get from (1.4) and .
For polynomials we obtain in agreement with [11]
(4.12)  
(4.13) 
4.4 polynomials
For the Lie algebra we find the following set of polynomials
(4.14)  
(4.15)  
(4.16)  
(4.17)  
5 Some relations between polynomials
Let us denote the set of polynomials corresponding to a set of parameters , …, as following
(5.1) 
, where is the Cartan matrix corresponding to a (semi)simple Lie algebra .
5.1 polynomials from ones
The set of polynomials corresponding to the Lie algebra may be obtained from the set of polynomials corresponding to the Lie algebra according to the following relations
(5.2) 
, i.e. the parameters are identified symmetrically w.r.t. . See Dynkin diagrams on Figs. 12. Relation (5.2) may be verified using the program from the Section 3. (For the case see formulas from the previous section.)
5.2 polynomials from ones
The set polynomials corresponding to the Lie algebra may be obtained from the set of polynomials corresponding to the Lie algebra according to the following relation
(5.3) 
, i.e. the parameters and are identified. See Dynkin diagrams on Figs. 23. Relation (5.3) may be verified using the program from the Section 3. (For the case see formulas from the previous section.)
5.3 Reduction formulas
Here we denote the Cartan matrix as follows: , where is the related Dynkin graph. Let be a node of . Let us denote by a Dynkin graph (corresponding to a certain semisimple Lie algebra) that is obtained from by erasing all lines that have endpoints at . It may be verified (e.g. by using the program) that the following reduction formulae are valid
(5.4) 
. Moreover,
(5.5) 
This means that setting we reduce the set of polynomials by replacing the the Cartan matrix by the Cartan matrix . In this case the polynomial corresponds to subalgebra (depicted by the node ) and the parameter . ^{3}^{3}3The analytical proof of the relations (5.2)(5.4) will be given in a separate publication.
As an example of reduction formulas we present the following relations
(5.6) 
, for with appropriate restrictions on (see (2.1)). In writing relation (5.4) we use the numbering of nodes in agreement with the Dynkin diagrams depicted on Figs. 13.
The reduction formulas (5.5) for polynomials with are depicted on Fig. 4. The reduced polynomials are coinciding with those corresponding to semisimple Lie algebra .
[5pt] Fig. 4. Dynkin diagram for semisimple Lie algebra describing the set of polynomials with
6 Conclusions
Here we have presented a description of computational program (written in Maple) for calculation of fluxbrane polynomials related to classical simple Lie algebras. (Generalization to semisimple Lie algebras is a straightforward one.) This program gives by product a verification of the conjecture suggested previously in [1]. The polynomials considered above define special solutions to open Toda chain equations corresponding to simple Lie algebras that may be of interest for certain applications of Toda chains.
We have also considered (without proof) certain relations between polynomials, e.g. socalled reduction formulas. These relations tells us that the most important is the calculation of polynomials, since all other polynomials (e.g. , and ones) may be obtained from series of polynomials by using certain reduction formulas.
A calculation of polynomials corresponding to exceptional Lie algebras (i.e. , , , and ) will be considered in a separate publications. (The polynomials were obtained earlier in [11].)
Acknowledgments
This work was supported in part by the Russian Foundation for Basic Research grant Nr. and by a grant of People Friendship University (NPK MU).
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