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[34], . . . , just doing part of the work on computations on di¬erential forms,
vector ¬elds, solutions of overdetermined systems of partial di¬erential equa-
tions, covering conditions, etc.
Since then, quite a number of programs has been constructed and it
seems that nowadays each individual researcher in this ¬eld of mathematical
physics uses his or her own developed software to do the required computa-
tions in more or less the most or almost most e¬cient way. An overview of
existing programs in all distinct related areas was recently given by Hereman
in his extensive paper [30].
In the following sections, we shall discuss in some detail a number of
types of computations which can be carried through on a computer sys-
tem. The basis of these programs has been constructed by Gragert [22],
Kersten [37], Gragert, Kersten and Martini [24, 25], Roelofs [85, 86], van
Bemmelen [9, 8] at the University of Twente, starting in 1979 with exterior
di¬erential forms, construction and solution of overdetermined systems of
partial di¬erential equations arising from symmetry computations, exten-
sion of the software to work in a graded setting, meaning supercalculus,

349
350 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

required for the interesting ¬eld of super and supersymmetric extensions of
classical di¬erential equations and at the end a completely new package,
being extremely suitable for classical as well as supersymmetrical systems,
together with packages for computation of covering structures of completely
integrable systems, and a package to handling the computations with to-
tal derivative operators. We should mention here too (super) Lie algebra
computations for covering structures by Gragert and Roelofs [23, 26].
We prefer to start in Section 1 with setting down the basic notions of the
graded or supercalculus, since classical di¬erential geometric computations
can be embedded in a very e¬ective way in this more general setting, which
will be done in Section 2.
In Section 3 we shall give an idea how the software concerning construc-
tion of solutions of overdetermined systems of partial di¬erential equations
works, and what the facilities are.
Finally we shall present in Subsection 3.2 a computer session concerning
the construction of higher symmetries of third order of the Burgers equation,
i.e., de¬ning functions involving derivatives (with respect to x up to order 3),
cf. Chapter 2.

1. Super (graded) calculus
We give here a concise exposition of super (or graded) calculus needed
for symbolic computations.
At ¬rst sight the introduction of graded calculus requires a completely
new set of de¬nitions and objects. It has been shown that locally a graded
manifold, or equivalently the algebra of functions de¬ned on it, is given as
C ∞ (U ) — Λ(n), where Λ(n) is the exterior algebra of n (odd) variables, [50].
Below we shall set down the notions involved in the graded calculus and
graded di¬erential geometry.
Thus we give a short review of the notions of graded di¬erential ge-
ometry as far as they are needed for implementation by means of software
procedures, i.e., graded commutative algebra, graded Lie algebra, graded man-
ifold, graded derivation, graded vector ¬eld, graded di¬erential form, exterior
di¬erentiation, inner di¬erentiation or contraction by a vector ¬eld, Lie de-
rivative along a vector ¬eld, etc.
The notions and notations have been taken from Kostant [50] and the
reader is referred to this paper for more details, compare with Chapter 6.
Throughout this section, the basic ¬eld is R or C and the grading will be
with respect to Z2 = {0, 1}.
1. A vector space V over R is a graded vector space if one has V0 and V1
subspaces of V , such that
V = V0 • V1 (8.1)
is a direct sum. Elements of V0 are called even, elements of V1 are
called odd. Elements of V0 or V1 are called homogeneous elements.
1. SUPER (GRADED) CALCULUS 351

If v ∈ Vi , i = 0, 1, then i is called the degree of v, i.e.,
|v| = i, i = 0, 1, or i ∈ Z2 . (8.2)
The notation |v| is used for homogeneous elements only.
2. A graded algebra B is a graded vector space B = B0 • B1 with a
multiplication such that
Bi · Bj ‚ Bi+j , i, j ∈ Z2 . (8.3)
3. A graded algebra B is called graded commutative if for any two ho-
mogeneous elements x, y ∈ B we have
xy = (’1)|x||y| yx. (8.4)
4. A graded space V is a left module over the graded algebra B, if V is
a left module in the usual sense and
Bi · Vj ‚ Vi+j , i, j ∈ Z2 ; (8.5)
right modules are de¬ned similarly.
5. If V is a left module over the graded commutative algebra B, then V
inherits a right module structure, where we de¬ne
def
v · b = (’1)|v||b| b · v, v ∈ V, b ∈ B. (8.6)
Similarly, a left module structure is determined by a right module
structure.
6. A graded vector space g = g0 • g1 , together with a bilinear operation
[·, ·] on g such that [x, y] ∈ g|x|+|y| is called a graded Lie algebra if
[x, y] = ’(’1)|x||y| [y, x],
(’1)|x||z| [x, [y, z]] + (’1)|z||y| [z, [x, y]] + (’1)|y||x| [y, [z, x]] = 0, (8.7)
where the last equality is called the graded Jacobi identity.
If V is a graded vector space, then End(V ) has the structure of a
graded Lie algebra de¬ned by
[±, β] = ±β ’ (’1)|±||β| β±, ±, β ∈ End(V ). (8.8)
7. If B is a graded algebra, an operator h ∈ Endi (B) is called a graded
derivation of B if
h(xy) = h(x)y + (’1)|i||x| xh(y). (8.9)
An operator h ∈ End(B) is a derivation if its homogeneous compo-
nents are so.
The graded vector space of derivations of B, denoted by Der(B),
is a graded Lie subalgebra of End(B). Equality (8.9) is called graded
Leibniz rule. If B is a graded commutative algebra then Der(B) is a
left B-module: if ζ ∈ Der(B), f, g ∈ B, then f ζ ∈ Der(B), where
(f ζ)g = f (ζg). (8.10)
352 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

8. The local picture of a graded manifold is an open neighborhood U ‚
Rm together with the graded commutative algebra
C ∞ (U ) — Λ(n), (8.11)
where Λ(n) is the antisymmetric (exterior) algebra on n elements
|si | = 1, si sj = ’sj si ,
s1 , . . . , sn , i, j = 1, . . . , n. (8.12)
The pair (m|n) is called the dimension of the graded manifold at
hand. A particular element f ∈ C ∞ (U ) — Λ(n) is represented as
f= fµ s µ , (8.13)
µ

where µ is a multi-index: µ ∈ Mn = {µ = (µ1 , . . . , µn ) | µi ∈ N, 1 ¤
µ1 ¤ µ2 · · · ¤ µk ¤ n},
fµ ∈ C ∞ (U ).
s µ = s µ1 · s µ2 · . . . · s µk , (8.14)
9. Graded vector ¬elds on a graded manifold (U, C ∞ (U ) — Λ(n)) are
introduced as graded derivations of the algebra C ∞ (U ) — Λ(n). They
constitute a left C ∞ (U ) — Λ(n)-module. Locally, a graded vector ¬eld
V is represented as
m n
‚ ‚
V= fi + gj , (8.15)
‚ri ‚sj
i=1 j=1

where fi , gj ∈ C ∞ (U ) — Λ(n), and ri , i = 1, . . . , m, are local coordi-
nates in U ‚ Rm .
The derivations ‚/‚ri , i = 1, . . . , m, are even, while the deriva-
tions ‚/‚sj , j = 1, . . . , n, are odd. They satisfy the relations
‚sj
‚rk ‚rk ‚sl
= δik , = 0, = 0, = δjl (8.16)
‚ri ‚ri ‚sj ‚sj
for all i, k = 1, . . . , m, j, l = 1, . . . , n.
10. A graded di¬erential k-form is introduced as k-linear mapping β on
Der(C ∞ (U ) — Λ(n)) which has to satisfy the identities
Pl’1
ζ1 , . . . , f ζl , . . . , ζk | β = (’1)|f | |ζi |
f ζ 1 , . . . , ζk | β (8.17)
i=1


and

ζ1 , . . . , ζj , ζj+1 . . . , ζk | β
= (’1)1+|ζj ||ζj+1 | ζ1 , . . . , ζj , ζj+1 , . . . , ζk | β , (8.18)
for all ζi ∈ Der(C ∞ (U ) — Λ(n)) and f ∈ C ∞ (U ) — Λ(n). The set of
k-forms is denoted by „¦k (U ).
Remark 8.1. Actually we have to write „¦k (U, C ∞ (U ) — Λ(n)),
but we made our choice for the abbreviated „¦k (U ).
1. SUPER (GRADED) CALCULUS 353

The set „¦k (U ) has the structure of a right C ∞ (U ) — Λ(n)-module
by
ζ1 , . . . , ζk | βf = ζ1 , . . . , ζk | β f. (8.19)
We also set „¦0 (U ) = C ∞ (U ) — Λ(n) and „¦(U ) = •∞ „¦k (U ).
k=0
Moreover „¦(U ) can be given a structure of a bigraded (Z+ , Z2 )-com-
mutative algebra, that is, if βi ∈ „¦ki (U )ji , i = 1, 2, then
β1 β2 ∈ „¦k1 +k2 (U )j1 +j2 (8.20)
and
β1 β2 = (’1)k1 k2 +j1 j2 β2 β1 . (8.21)
For the general de¬nition of β1 β2 see [50].
11. One de¬nes the exterior derivative (or de Rham di¬erential )
d : „¦0 (U ) ’ „¦1 (U ), f ’ df, (8.22)
by the condition
ζ | df = ζf (8.23)
for ζ ∈ Der(C ∞ (U ) — Λ(n)) and f ∈ „¦0 (U ) = C ∞ (U ) — Λ(n). By
[50] and the de¬nition of β1 β2 ,
dri , i = 1, . . . , m, dsj , j = 1, . . . , n, (8.24)
de¬ned by
‚ ‚
| dri = δik , | dri = 0,
‚rk ‚sj
‚ ‚
| dsl = 0, | dsl = δjl , (8.25)
‚rk ‚sj
generate „¦(U ) and any β ∈ „¦(U ) can be uniquely written as
drµ dsν fµ,ν ,
β= (8.26)
µ,ν

where
1 ¤ µ1 ¤ . . . ¤ µk ¤ n,
µ = (µ1 , . . . , µk ), l(µ) = k,
νi ∈ N = Z+ \ {0},
ν = (ν1 , . . . , νn ),
n
fµ,ν ∈ C ∞ (U ) — Λ(n).
|ν| = νi , (8.27)
i=1
Note in particular that by (8.21),
dri drj = ’drd ri , dri dsj = ’dsj dri , dsj dsk = dsk dsj , (8.28)
and by consequence
dsj . . . dsj = 0. (8.29)
k times
354 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

By means of (8.22) and (8.23), the operator d : „¦0 (U ) ’ „¦1 (U ) has
the following explicit representation
m n
‚f ‚f
df = dri + dsj . (8.30)
‚ri ‚sj
i=1 j=1

12. Since „¦(U ) is a (Z+ , Z2 )-bigraded commutative algebra, the algebra
End(„¦(U )) is bigraded too and if u ∈ End(„¦(U )) is of bidegree (b, j) ∈
(Z+ , Z2 ), then
u(„¦a (U )i ) ∈ „¦a+b (U )i+j . (8.31)
Now, an element u ∈ End(„¦(U )) of bidegree (b, j) is a bigraded deriva-
tion of „¦(U ), if for any ± ∈ „¦a (U )i , β ∈ „¦(U ) one has the Leibniz
rule
u(±β) = u(±)β + (’1)ab+ij ±u(β). (8.32)
There exists a unique derivation, the exterior di¬erentiation,
d : „¦(U ) ’ „¦(U ) (8.33)

of bidegree (1, 0), such that d is de¬ned by (8.22), (8.30), and
„¦0 (U )

d2 = 0. (8.34)
If β ∈ „¦(U ),

drµ dsν fµ,ν ,
β= (8.35)
µ,ν

then
(’1)l(µ)+|ν| drµ dsν dfµ,ν .
dβ = (8.36)
µ,ν

Other familiar operations on ordinary manifolds have their counter-
parts in the graded case too.
13. If ζ ∈ Der(C ∞ (U ) — Λ(n)), inner di¬erentiation by ζ, or contraction
by ζ, iζ is de¬ned by
Pb
|ζ| |ζi |
ζ1 , . . . , ζb | iζ β = (’1) ζ, ζ1 , . . . , ζb | β (8.37)
i=1



for ζ, ζ1 , . . . , ζb ∈ Der(C ∞ (U ) — Λ(n)) and β ∈ „¦b+1 (U ). Moreover
iζ : „¦(U ) ’ „¦(U ), β ∈ „¦b+1 (U ), iζ β ∈ „¦b (U ), is a derivation of
bidegree (’1, |ζ|).
Bigraded derivations on „¦(U ) can be shown to constitute a bi-
graded Lie algebra Der „¦(U ) by the following Lie bracket. If u1 , u2 ∈
Der „¦(U ) of bidegree (bi , bj ), i = 1, 2, then
[u1 , u2 ] = u1 u2 ’ (’1)b1 b2 +j1 j2 u2 u1 ∈ Der „¦(U ). (8.38)
2. CLASSICAL DIFFERENTIAL GEOMETRY 355

14. From (8.38) we have that Lie derivative by the vector ¬eld ζ de¬ned
by
Lζ = diζ + iζ d (8.39)

is a derivation of „¦(U ) of bidegree (0, |ζ|).
The fact that exterior di¬erentiation d, inner di¬erentiation by ζ,
iζ , and Lie derivative by ζ, Lζ are derivations, has been used to imple-
ment them on the computer system starting from the representation
of vector ¬elds and di¬erential forms (8.15) and (8.35).
15. If one has a graded manifold (U, C ∞ (U ) — Λ(n)), the exterior deriv-
ative is easy to be represented as an odd vector ¬eld in the following
way
m n
‚ ‚
dri § dsj §
d= + , (8.40)
‚ri ‚sj
i=1 j=1

where now the initial system has been augmented by n even variables
ds1 , . . . , dsn and m odd variables dr1 , . . . , drm . The implementation
of the supercalculus package is based on the theorem proved in [50]
that locally a supermanifold, or a graded manifold, is represented as
U, C ∞ (U ) — Λ(n), U ‚ Rn , from which it is now easy to construct
the di¬erential geometric operations.
Suppose we have a supermanifold of dimension (m|n). Local
variables are given by (r, s) = (ri , sj ), i = 1, . . . , m, j = 1, . . . , n.
Associated to these coordinates, we have (dri , dsj ), i = 1, . . . , m,
j = 1, . . . , n. We have to note that dri , i = 1, . . . , m, are odd while
dsj , j = 1, . . . , n, are even.
So the exterior algebra is
C ∞ (Rm ) — R[ds] — Λ(n) — Λ(m), (8.41)

where in (8.41) a speci¬c element is given by

dsk1 . . . dskm drµ1 . . . drµr fk,µ
f= (8.42)
m
1

while in (8.42) ki ≥ 0, i = 1, . . . , m, 1 ¤ µ1 < · · · < µr ¤ n, while
fk,µ ∈ C ∞ (Rm ) — Λ(n).


2. Classical di¬erential geometry
We shall describe here how classical di¬erential geometric objects are
realised in the graded setting of the previous section. We start at a super-
algebra A on n even elements, r1 , . . . , rn , and n odd elements s1 , . . . , sn ,
i.e.,
A = C ∞ (Rn ) — Λ(n), (8.43)
356 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

where Λ(n) is the exterior algebra on n elements, s1 , . . . , sn . A particular
element f ∈ A = C ∞ (Rn ) — Λ(n) is represented as
f= fµ s µ , (8.44)
µ

where µ is a multi-index µ ∈ Mn = {µ = (µ1 , . . . , µk ) | µi ∈ N, 1 ¤ µ1 ¤
· · · ¤ µk ¤ n} and
fµ ∈ C ∞ (Rn ),
s µ = s µ1 s µ2 . . . s µk , (8.45)
where we in e¬ect formally assume:
si = dri , i = 1, . . . , n. (8.46)
1. Functions are represented as elements of the algebra A0 = C ∞ (Rn ).
2. Derivations of A0 can be identi¬ed with vector ¬elds
‚ ‚
+ · · · + Vn
V = V1 , (8.47)
‚r1 ‚rn
where Vi ∈ C ∞ (Rn ), i = 1, . . . , n.
3. Di¬erential forms are just speci¬c elements of A.
4. Exterior derivative is a derivation of A which is odd and can be
represented as the vector ¬eld
‚ ‚
+ · · · + drn
d = dr1 , dri = si . (8.48)
‚r1 ‚rn
5. Contraction by a V , where V is given by (8.47), can be represented
as an odd derivation of A by
‚ ‚
+ · · · + Vn
V ±= V1 (±). (8.49)
‚s1 ‚sn
6. The Lie derivative by V can be easily implemented by the formula
LV (±) = V d(±) + d(V ±). (8.50)

3. Overdetermined systems of PDE
In construction of classical and higher symmetries, nonlocal symmetries
and deformations or recursion operators, one is always left with an overde-
termined system of partial di¬erential equations for a number of so-called
generating functions (or sections). The ¬nal result is obtained as the general
solution to this resulting system.
In Section 3.1 we shall describe how by the procedure which is called
here solve equation, written in the symbolic language LISP, one is able
to solve the major part of the construction of the general solution of the
overdetermined system of partial di¬erential equations resulting from the
symmetry condition (2.29) on p. 72 or the deformation condition (6.42) on
p. 266.
It should be noted that each speci¬c equation or system of equations
arising from mathematical physics has its own speci¬cs, e.g., the sine-Gordon
3. OVERDETERMINED SYSTEMS OF PDE 357

equation is not polynomial but involves the sine function, similar to the
Harry Dym equations, where radicals are involved.
In Subsection 3.2 we discuss, as an application, symmmetries of the
Burgers equation, while ¬nally in Subection 3.3 we shall devote some words
to the polynomial and graded cases.

3.1. General case. Starting at the symmetry condition (2.29), one
arrives at an overdetermined system of homogeneous linear partial di¬eren-
tial equations for the generating functions Fi , i = 1, . . . , m. First of all,
one notes that in case one deals with a di¬erential equation1 E k ‚ J k (x, u),
x = (x1 , . . . , xn ), u = (u1 , . . . , um ), then the r-th prolongation E k+r is always
polynomial with respect to the higher jet variables in the ¬bre E k+r ’ E k .
The symmetry condition (2.29) is also polynomial with respect to these
variables, cf. Subsection 3.2. So the overdetermined system of partial di¬er-
ential equations can always be splitted with respect to the highest variables
leading to a new system of equations.
These equations are stored in the computer system memory as right-
hand sides of operators equ(1), . . . , equ(te), where the variable te stands
for the Total Number of Equations involved.
If at a certain stage, the computer system constructs new expressions
which have to vanish in order to generate the general solution to the system
of equations (for instance, the derivative of an equation is a consequence,
which might be easier to solve). These new equations are added to the
system as equ(te + 1), . . . and the value of te is adjusted automatically to
the new situation.
In the construction of solutions to the system of equations we distinguish
between a number of di¬erent cases:
1. CASE A: A partial di¬erential equation is of a polynomial type in
one (or more) of the variables, the functions F— appearing in this
equation are independent of this (or these) variable(s). By conse-
quence, each of the coe¬cients of the polynomial has to be zero, and
the partial di¬erential equation decomposes into some new additional
and smaller equations.
Example 8.1. The partal di¬erential equation is
equ(.) :=x2 (F1 )x2 + x1 F2 , (8.51)
1

where in (8.51) the functions F1 , F2 are independent of x1 .
By consequence, the coe¬cients of the polynomial in x1 have to
be zero, i.e., (F1 )x2 and F2 . So equation (8.51) is equivalent to the
system
equ(.):=(F1 )x2 ,
(8.52)
equ(.):=F2

1
We use the notation J k (x, u) as a synonim for J k (π), where π : (x, u) ’ (x).
358 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

2. CASE B: The partial di¬erential equations equ(.) represents a de-
rivative of a function F— . In general

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