SUSY2: Supersymmetric Functions and Algebra of Supersymmetric Operators

REDUCE

20.61 SUSY2: Supersymmetric Functions and Algebra of Supersymmetric Operators

This package deals with supersymmetric functions and with the algebra of supersymmetric operators in the extended \(N=2\) as well as in the non-extended \(N=1\) supersymmetry (SuSy). It allows us to realize the SuSy algebra of diﬀerential operators, compute the gradients of given SuSy Hamiltonians and obtain the SuSy version of soliton equations using the SuSy Lax approach. There are also many additional procedures encountered in the SuSy soliton approach, as for example: conjugation of a given SuSy operator, computation of the general form of SuSy Hamiltonians (up to SuSy divergence equivalence), and checking the validity of the Jacobi identity for SuSy Hamiltonian operators.

Author: Ziemowit Popowicz

20.61.1 Introduction

\( \newcommand {\bos }{\mathbf {bos}} \newcommand {\fer }{\mathbf {fer}} \newcommand {\fun }{\mathbf {fun}} \newcommand {\gras }{\mathbf {gras}} \newcommand {\axp }{\mathbf {axp}} \newcommand {\der }{\mathbf {der}} \newcommand {\del }{\mathbf {del}} \newcommand {\pr }{\mathbf {pr}} \newcommand {\pg }{\mathbf {pg}} \newcommand {\d }{\mathbf {d}} \)The main idea of supersymmetry (SuSy) is to treat boson and fermion operators equally [1,2]. This has been realised by introducing so-called supermultiplets constructed from the boson and fermion operators and additionally from the Mayorana spinors. Such supermultiplets possess the proper transformation property under the transformation of the Lorentz group. At the moment we have no experimental conﬁrmation that supersymmetry appears in nature.

The idea of using supersymmetry for the generalization of the soliton equations [3–7] appeared almost in parallel to the usage of SuSy in the quantum ﬁeld theory. The ﬁrst results, concerning the construction of classical ﬁeld theories with fermionic and bosonic ﬁelds depending on time and one space variable can be found in [8–12]. In many cases, the addition of fermion ﬁelds does not guarantee that the ﬁnal theory becomes SuSy invariant and therefore this method was named the fermionic extension in order to distinguish it from the fully SuSy method.

In order to get a SuSy theory we have to add to a system of \(k\) bosonic equations \(kN\) fermion and \(k(N-1)\) boson ﬁelds \((k=1,2,\ldots ,\;N=1,2,\ldots )\) in such a way that the ﬁnal theory becomes SuSy invariant. From the soliton point of view we can distinguish two important classes of the supersymmetric equations: the non-extended \((N = 1)\) and extended \((N > 1)\) cases. Consideration of the extended case may imply new bosonic equations whose properties need further investigation. This may be viewed as a bonus, but this extended case is no more fundamental than the non-extended one. The problem of the supersymmetrization of the nonlinear partial diﬀerential equations has its own history, and at the moment we have no unique solution [13–40]. We can distinguish three diﬀerent methods of supersymmetrization: algebraic, geometric and direct.

In the ﬁrst two cases we are looking for the symmetry group of the given equation and then we replace this group by the corresponding SuSy group. As a ﬁnal product we are able to obtain a SuSy generalization of the given equation. The classiﬁcation into the algebraic or geometric approach is connected with the kind of symmetry which appears in the classical case. For example, if our classical equation could be described in terms of the geometrical object then the simple exchange of the classical symmetry group of this object with its SuSy partner justiﬁes the name geometric. In the algebraic case we are looking for the symmetry group of the equation without any reference to its geometrical origin. This strategy could be applied to the so-called hidden symmetry as for example in the case of the Toda lattice. These methods each have advantages and disadvantages. For example, sometimes we obtain the fermionic extensions. In the case of the extended supersymmetric Korteweg-de Vries equation we have three diﬀerent fully SuSy extensions; however only one of them ﬁts these two classiﬁcations.

In the direct approach we simply replace all objects which appear in the evolution equation by all possible combinations of the supermultiplets and their superderivatives so as to conserve the conformal dimensions. This is non-unique and yields many diﬀerent possibilities. However, the arbitrariness is reduced if we additionally investigate super-bi-hamiltonian structure or try to ﬁnd its supersymmetric Lax pair. In many cases this approach is successful.

The utilization of the above methods can be helped by symbolic computer algebraic and for this reason we developed the package SuSy2 in the symbolic language REDUCE [41].

We have implemented and ordered the superfunctions in our program, extensively using the concept of “noncom operator” in order to implement the supersymmetric integro-diﬀerential operators. The program is meant to perform the symbolic calculations using either fully supersymmetric supermultiplets or the component version of our supersymmetry. We have constructed 25 diﬀerent commands to allow us to compute almost all objects encountered in the supersymmetrization procedure of the soliton equation.

20.61.2 Supersymmetry

The basic object in the supersymmetric analysis is the superﬁeld and the supersymmetric derivative. The superﬁelds are the superfermions or the superbosons [1]. These ﬁelds, in the case of extended \(N=2\) supersymmetry, depend, in addition to \(x\) and \(t\), upon two anticommuting variables, \(\theta _{1}\) and \(\theta _{2}\) (such that \(\theta _{2}\theta _{1} = - \theta _{1}\theta _{2},\;\theta _{1}^{2} = \theta _{2}^{2} = 0\)). Their Taylor expansion with respect to \(\theta _{1},\theta _{2}\) is

\begin{equation*} b(x,t,\theta _{1},\theta _{2}):=w+\theta _{1}\zeta _{1}+ \theta _{2}\zeta _{2}+\theta _{2}\theta _{1}u \end{equation*}

for superbosons and

\begin{equation*} f(x,t,\theta _{1},\theta _{2}):=\zeta _{1}+\theta _{1}w+ \theta _{2}u+\theta _{2}\theta _{1}\zeta _{2} \end{equation*}

for superfermions, where \(w\) and \(u\) are classical (commuting) functions depending on \(x\) and \(t\), and \(\zeta _{1}\) and \(\zeta _{2}\) are odd Grassmann-valued functions depending on \(x\) and \(t\).

On the set of these superfunctions we can deﬁne the usual derivative and the superderivative. Usually, we encounter two diﬀerent realizations of the superderivative: the ﬁrst we call “traditional” and the second “chiral”.

The traditional realization can be deﬁned by introducing two superderivatives \(D_{1}\) and \(D_{2}\):

\begin{align*} D_{1} &= \partial _{\theta _{1}}+\theta _{1}\partial ,\\ D_{2} &= \partial _{\theta _{2}}+\theta _{2}\partial , \end{align*}

with the properties:

\begin{gather*} D_{1} D_{1} = D_{2} D_{2} = \partial ,\\ D_{1} D_{2} + D_{2} D_{1} = 0. \end{gather*}

The chiral realization is deﬁned by

\begin{align*} D_{1} &= \partial _{\theta _{1}} - \frac {1}{2}\theta _{2}\partial ,\\ D_{2} &= \partial _{\theta _{2}} - \frac {1}{2}\theta _{1}\partial , \end{align*}

with the properties:

\begin{gather*} D_{1} D_{1} = D_{2} D_{2} = 0,\\ D_{1} D_{2} + D_{2} D_{1} = -\partial . \end{gather*}

Below we shall use the names “traditional”, “chiral” or “chiral1” algebras to denote the kind of commutation relations assumed for the superderivatives. The chiral1 algebras possess, additionally to the chiral algebra, the commutator of \(D_{1}\) and \(D_{2}\) deﬁned by

\begin{equation*} D_{3} = D_{1} D_{2} -D_{2} D_{1}. \end{equation*}

In the SUSY2 package we have implemented the superfunctions and the algebra of superderivatives. Moreover, we have deﬁned many additional procedures which are useful in the supersymmetrization of the classical nonlinear system of partial diﬀerential equation. Diﬀerent applications of this package to physical problems can be found in the papers [34–38].

20.61.3 Superfunctions

In this manual entry, REDUCE procedure (function) and let-rule names are usually displayed in a bold font, while all other input and output is usually displayed as normal typeset mathematics. The value returned by a procedure (function) is indicated by the notation

\begin{equation*} \mathbf {function}(\mathit {arguments}) \Rightarrow \mathit {function~value}. \end{equation*}

However, REDUCE input without corresponding output and REDUCE command names are usually displayed in a typewriter font.

Superfunctions are represented in this package by

\begin{equation}\label {susy2-eqn1} \bos (f,0,0) \end{equation}

for superbosons and

\begin{equation*} \fer (g,0,0) \end{equation*}

for superfermions.

The ﬁrst argument denotes the name of the given superobject, the second denotes the value of the SuSy derivative, and the last denotes the value of the usual derivative. The \(\bos \) and \(\fer \) objects are declared as noncom operators in REDUCE. The ﬁrst argument can take an arbitrary value but with the following restrictions:

\begin{gather*} \bos (0,m,n) = 0, \\ \fer (0,m,n) = 0, \end{gather*}

for all values of \(m,n\).

The program has the capability to compute the coordinates of arbitrary SuSy expressions, using expansions in powers of \(\theta \). We have here four commands:

1.: In order to have the given expression in components use
\begin{equation*} \mathbf {fpart}(\mathit {expression}). \end{equation*}
The output is in the form of a list, in which the ﬁrst element is the zero-order term in \(\theta \), the second is the ﬁrst-order term in \(\theta _{1}\), the third is the ﬁrst-order term in \(\theta _{2}\) and the fourth is the term in \(\theta _{2}\theta _{1}\). For example, the superfunction \eqref {susy2-eqn1} has the representation
\begin{equation*} \mathbf {fpart}(\bos (f,0,0)) \Rightarrow \{ \fun (f_{0},0), \gras (\mathit {ff}_{1},0), \gras (\mathit {ff}_{2},0),\fun (f_{1},0) \}, \end{equation*}
where \(\fun \) denotes the classical function and \(\gras \) denotes the Grassmann function. The ﬁrst argument in \(\fun \) or \(\gras \) denotes the name of the given object, while the second denotes the usual derivative.
2.: In order to have the bosonic sector only, in which all odd Grassmann functions disappear, use
\begin{equation*} \mathbf {bpart}(\mathit {expression}). \end{equation*}

Example:
\begin{equation*} \mathbf {bpart}(\fer (g,0,0)) \Rightarrow \{0, \fun (g_{0},0), \fun (g_{1},0), 0\} \end{equation*}
3.: In order to have the given coordinates use
\begin{equation*} \mathbf {bf\_part}(\mathit {expression},n), \end{equation*}
where \(n=0,1,2,3\).
Example:
\begin{equation*} \mathbf {bf\_part}(\bos (f,0,0),3) \Rightarrow \fun (f_{1},0) \end{equation*}
4.: In order to have the given coordinates in the bosonic sector use
\begin{equation*} \mathbf {b\_part}(\mathit {expression},n), \end{equation*}
where \(n=0,1,2,3\).
Example:
\begin{equation*} \mathbf {b\_part}(\fer (g,0,0),1) \Rightarrow \fun (g_{0},0) \end{equation*}

Notice that the program switches on factoring of \(\fer ,\bos ,\gras ,\fun \). If you remove this factoring then many commands give wrong results (for example the commands \(\mathbf {lyst}\), \(\mathbf {lyst1}\) and \(\mathbf {lyst2}\)).

20.61.4 The Inverse and Exponentials of Superfunctions

In addition to our deﬁnitions of the superfunctions we can also deﬁne the inverse and exponential of the superboson.

The inverse of the given \(\bos \) function (not to be confused with the “inverse function” encountered in the usual analysis) is deﬁned as

\begin{equation*} \bos (f,n,m,-1), \end{equation*}

for arbitrary \(f,n,m\) with the property \(\bos (f,n,m,-1)\,\bos (f,n,m,1)=1\). The object \(\bos (f,n,m,k)\), in general, denotes the \(k\)-th power of the \(\bos (f,n,m)\) superfunction. If we use the command let inverse then three-index \(\bos \) objects are transformed into four-index \(\bos \) objects.

The exponential of the superboson function is

\begin{equation*} \axp (\bos (f,0,0)). \end{equation*}

It is also possible to use \(\axp (f)\), but then we should specify what is \(f\).

We have the following representation in components for the inverse and \(\axp \) superfunctions:

\begin{multline*} \mathbf {fpart}(\bos (f,0,0,-1)) = \{ \fun (f_{0},0,-1), -\fun (f_{0},0,-1)\,\gras (\mathit {ff}_{1},0), \\ -\fun (f_{0},0,-1)\,\gras (\mathit {ff}_{2},0), - \fun (f_{0},0,-2)\,\fun (f_{1},0,1) \\ + 2\,\fun (f_{0},0,-3)\,\gras (\mathit {ff}_{1},0)\,\gras (\mathit {ff}_{2},0) \}, \end{multline*}

\begin{multline*} \mathbf {fpart}(\axp (f)) = \{ \mathbf {axx}(\mathbf {bf\_part}(f,0)), \mathbf {axx}(\mathbf {bf\_part}(f,0))\,\mathbf {bf\_part}(f,1), \\ \mathbf {axx}(\mathbf {bf\_part}(f,0))\,\mathbf {bf\_part}(f,2), \mathbf {axx}(\mathbf {bf\_part}(f,0)) \\ (\mathbf {bf\_part}(f,3)+2\,\mathbf {bf\_part}(f,1)\,\mathbf {bf\_part}(f,2)) \}, \end{multline*}

where \(\mathbf {axx}(f)\) denotes the exponentiation of the given classical function while \(\fun (f,m,n)\) denotes the \(n\)-th power of the function \(\fun (f,m)\).

20.61.5 Ordering

The three diﬀerent superfunctions \(\fer ,\bos ,\axp \) are ordered among themselves as

\begin{gather*} \fer (f,n,m)\,\bos (h,j,k)\,\axp (g), \\ \fer (f,n,m)\,\bos (h,j,k,l)\,\axp (g), \end{gather*}

independently of the arguments. The superfunctions \(\bos \) and \(\axp \) commute with themselves, while the superfunctions \(\fer \) anticommute with themselves. For these superfunctions we introduce the following ordering.

The \(\bos \) objects with three and four arguments are ordered as follows: the ﬁrst argument anti-lexicographically, the second and third by decreasing order of natural numbers; the last (fourth) is not ordered because
\begin{equation*} \bos (f,n,m,k)*\bos (f,n,m,l) \Rightarrow \bos (f,n,m,k+l) \end{equation*}
The anticommuting \(\fer \) objects are ordered as follows: the ﬁrst argument anti-lexicographically, the second and third by decreasing order of natural numbers.
Example:
\begin{equation*} \fer (f,n,m)*\fer (g,k,l) \Rightarrow -\fer (g,k,l)\,\fer (f,n,m) \end{equation*}
for arbitrary \(n,m,k,l\).
\begin{equation*} \fer (f,n,m)*\fer (f,n,m) \Rightarrow 0 \end{equation*}
for arbitrary \(f,n,m\).
\begin{eqnarray*} \bos (f,2,3,7)*\bos (a,0,3)*\bos (f,2,3,-7) & \Rightarrow & \bos (a,0,3), \\ \bos (f,2,3,2)*\bos (z,0,3,2)*\bos (f,2,3,-2) & \Rightarrow & \bos (z,0,3,2). \end{eqnarray*}
For all exponential functions we have
\begin{equation*} \axp (f)*\axp (g) \Rightarrow \axp (f+g). \end{equation*}

20.61.6 (Super)Diﬀerential Operators

We have implemented three diﬀerent realizations of the supersymmetric derivatives. In order to select the traditional realization declare let trad. In order to select the chiral or chiral1 algebra declare let chiral or let chiral1. By default we have the traditional algebra.

We have introduced three diﬀerent types of SuSy operators which act on the superfunctions and are considered as noncommuting operators in REDUCE.

For the usual diﬀerentiation we introduced two types of operators:

right diﬀerentations
\begin{equation*} \d (1)*\bos (f,0,0) \Rightarrow \bos (f,0,1) + \bos (f,0,0)\,\d (1); \end{equation*}
left diﬀerentations
\begin{equation*} \fer (f,0,0)*\d (2) \Rightarrow -\fer (f,0,1) + \d (2)\,\fer (f,0,0). \end{equation*}

This example illustrates that the third argument in the \(\bos \) and \(\fer \) objects can take an arbitrary integer value.

We denote SuSy derivatives as \(\der \) and \(\del \), which represent the right and left operations respectively, and are one-argument operators. The action of these objects on the superfunctions depends on the choice of the supersymmetric algebra.

Explicitly, we have for the traditional algebra:

: right SuSy derivative
\begin{eqnarray*} \der (1)*\bos (f,0,0) &\Rightarrow & \fer (f,1,0)+\bos (f,0,0)\,\der (1), \\ \der (2)*\fer (g,0,0) &\Rightarrow & \bos (g,2,0)-\fer (g,0,0)\,\der (2), \\ \der (1)*\fer (f,2,0) &\Rightarrow & \bos (f,3,0)-\fer (f,2,0)\,\der (1), \\ \der (2)*\bos (f,3,0) &\Rightarrow & -\fer (f,1,1)+\bos (f,3,0)\,\der (2), \\ \der (1)*\bos (f,0,0,-1) &\Rightarrow & -\fer (f,1,0)\,\bos (f,0,0,-2) + \mbox {}\\ && \bos (f,0,0,-1)\,\der (1), \\ \der (2)*\axp (\bos (f,0,0)) &\Rightarrow & \fer (f,2,0)\,\axp (\bos (f,0,0)) + \mbox {}\\ && \axp (\bos (f,0,0))\,\der (2). \end{eqnarray*}
: left SuSy derivative
\begin{eqnarray*} \bos (f,0,0)*\mathbf {\del (1)} &\Rightarrow & -\fer (f,1,0)+\del (1)\,\bos (f,0,0), \\ \fer (g,0,0)*\mathbf {\del (2)} &\Rightarrow & \bos (g,2,0)-\del (2)\,\fer (g,0,0), \\ \fer (f,2,0)*\del (2) &\Rightarrow & \bos (f,3,0)-\del (1)\,\fer (f,2,0), \\ \bos (f,3,0)*\del (2) &\Rightarrow & \fer (f,1,1)+\del (2)\,\bos (f,3,0), \\ \bos (f,0,0,-1)*\del (1) &\Rightarrow & \fer (f,1,0)\,\bos (f,0,0,-2) + \mbox {}\\ && \del (1)\,\bos (f,0,0,-1), \\ \axp (\bos (f,0,0))*\del (2) &\Rightarrow & -\fer (f,2,0)\,\axp (\bos (f,0,0)) + \mbox {}\\ && \del (2)\,\axp (\bos (f,0,0)). \end{eqnarray*}

These examples illustrate that the second argument in the \(\fer \) and \(\bos \) objects can take values 0, 1, 2, 3 only with the following meaning: 0 – no SuSy derivatives, 1 – ﬁrst SuSy derivative, 2 – second SuSy derivative, 3 – ﬁrst and second SuSy derivative.

Using the results above we obtain

\begin{multline*} \der (1)*\der (2)*\bos (f,0,0) \Rightarrow \\ \bos (f,3,0) + \bos (f,0,0)\,\der (1)\,\der (2) + \mbox {} \\ \fer (f,1,0)\,\der (2) - \fer (f,2,0)\,\der (1). \end{multline*}

For the chiral representation, the meaning of the second argument in the \(\bos \) or \(\fer \) object is the same as for the traditional representation while the actions of SuSy operators on the superfunctions are diﬀerent. For example, we have

\begin{eqnarray*} \der (1)*\fer (f,1,0) &\Rightarrow & -\fer (f,1,0)\,\der (1), \\ \der (1)*\fer (f,2,0) &\Rightarrow & \bos (g,3,0) - \fer (f,2,0)\,\der (1), \\ \der (2)*\bos (g,3,0) &\Rightarrow & -\fer (g,2,1) + \bos (g,3,0)\,\der (2), \\ \bos (g,2,0)*\del (2) &\Rightarrow & \del (2)\,\bos (g,2,). \end{eqnarray*}

For the chiral1 representation we have a diﬀerent meaning of the second argument in the \(\bos \) and \(\fer \) objects: the values \(0,1,2\) for this second argument denote the values of the SuSy derivatives while 3 denotes the value of the commutator. Explicitly, we have

\begin{eqnarray*} \der (3)*\bos (f,0,0) &\Rightarrow & \bos (f,3,0) + 2\,\fer (f,1,0,0)\,\der (2) \\ && \mbox {} - 2\,\fer (f,2,0)\,\der (1) + \bos (f,0,0)\,\der (3), \\ \der (1)*\fer (f,2,0) &\Rightarrow & (\bos (f,3,0)-\bos (f,0,1))/2 - \fer (f,2,0)\,\der (1). \end{eqnarray*}

The supersymmetric operators are always ordered in the case of traditional algebra as

\begin{eqnarray*} \der (2)*\der (1) &\Rightarrow & -\der (1)\,\der (2),\\ \del (2)*\del (1) &\Rightarrow & -\del (1)\,\del (2), \\ \der (1)*\del (1) &\Rightarrow & \d (1), \\ \der (1)*\del (2) &\Rightarrow & -\del (2)\,\der (1); \end{eqnarray*}

for the chiral algebra we have

\begin{eqnarray*} \der (2)*\der (1) &\Rightarrow & -\d (1) - \der (1)\,\der (2),\\ \del (2)*\del (1) &\Rightarrow & -\d (1) - \del (1)\,\del (2), \\ \der (1)*\del (1) &\Rightarrow & 0, \\ \der (1)*\del (2) &\Rightarrow & -\d (1) - \del (2)\,\der (1); \end{eqnarray*}

while for chiral1 additionally we have

\begin{eqnarray*} \der (3)*\der (1) &\Rightarrow & -\der (1)\,\d (1), \\ \der (1)*\der (3) &\Rightarrow & \der (1)\,\d (1), \\ \der (3)*\der (2) &\Rightarrow & \der (2)\,\d (1), \\ \der (2)*\der (3) &\Rightarrow & -\der (2)\,\d (1). \end{eqnarray*}

Please notice that if we would like to have the components of some \(\bos (f,3,0,-1)\) superfunction in the chiral representation then new objects appear:

\begin{equation*} \mathbf {b\_part}(\bos (f,3,0,-1), 1) \Rightarrow \fun (f1,0,f0,1,-1), \end{equation*}

We should consider the ﬁve-argument object \(\fun \) as

\begin{equation*} \fun (f,n,g,m,-k) \Rightarrow (\fun (f,n)-\fun (g,m)/2)^{-k}. \end{equation*}

Similar interpretation is valid for other commands containing objects like \(\bos (f,3,n,-k)\).

20.61.7 Action of the Operators

In order to have the value of the action of the given operator on some superfunction we introduce two operators \(\pr \) and \(\pg \). The operator

\begin{equation*} \pr (n,\mathit {expression}) \end{equation*}

where \(n=0,1,2,3\) denotes the value itself of the action of the SuSy derivatives on the given expression. For \(n=0\) there is no SuSy derivative, \(n=1\) corresponds to \(\der (1)\), \(n=2\) to \(\der (2)\), and \(n=3\) to \(\der (1)*\der (2)\).

Example:

\begin{gather*} \pr (1,\bos (f,0,0)) \Rightarrow \fer (f,1,0), \\ \pr (3,\fer (g,0,0)) \Rightarrow \fer (f,3,0). \end{gather*}

For the usual derivative we reserve the command

\begin{equation*} \pg (n,\mathit {expression}) \end{equation*}

where \(n=0,1,2,\ldots \) denotes the value of the usual derivative on the expression.

Example:

\begin{equation*} \pg (2,\bos (f,0,0)) \Rightarrow \bos (f,0,2) \end{equation*}

20.61.8 Supersymmetric Integration

There is one command \(\mathbf {s\_int}(\mathit {number},\mathit {expression},\mathit {list})\) only. This allows us to compute the supersymmetric integral of arbitrary polynomial expressions constructed from \(\fer \) and \(\bos \) objects. It is valid in the traditional representation of supersymmetry. The argument \(number\) takes the following values: \(0\) corresponds to the usual \(x\) integration, \(1\) or \(2\) to integration over the ﬁrst or second supersymmetric argument, while \(3\) corresponds to integration over both the ﬁrst and second arguments. The argument \(list\) is a list of the names of the superfunctions over which we would like to integrate. The output of this command is in the form of the integrated part and non-integrated part. The non-integrated part is denoted by \(\del (-\mathit {number})\) for \(\mathit {number} = 1,2,3\) and by \(\d (-3)\) for \(\mathit {number} = 0\).

Example:

\begin{equation*} \mathbf {s\_int}(0,2*\bos (f,0,1)*\bos (f,0,1),\{f\}) \Rightarrow \bos (f,0,0)^{2}, \end{equation*}

\begin{equation*} \mathbf {s\_int}(1,2*\fer (f,1,0)*\bos (f,0,0),\{f\}) \Rightarrow \bos (f,0,0)^{2}, \end{equation*}

\begin{multline*} \mathbf {s\_int}(3,\bos (f,3,0)*\bos (g,0,0)+\bos (f,0,0)*\bos (g,3,0),\{f,g\}) \Rightarrow \\ \bos (f,0,0)\,\bos (g,0,0) - \mbox {} \\ \del (-3)\,\big ( \fer (f,1,0)\,\fer (g,2,0) - \fer (f,2,0)\,\bos (g,1,0) \big ). \end{multline*}

20.61.9 Integration Operators

We introduced four diﬀerent types of integration operators: right and left denoted by \(\d (-1)\) and \(\d (-2)\) respectively and moreover two diﬀerent types of neutral integration operators \(\d (-3)\) and \(\d (-4)\). In the ﬁrst two cases they act according to the formula

\begin{equation*} \d (-1)\,\bos (f,0,0) = \sum _{i=1}^{\infty } (-1)^{i}\,\bos (f,0,i-1)\,\d (-1)^{i} \end{equation*}

for the right integration and

\begin{equation*} \bos (f,0,0)\,\d (-2) = \sum _{i=1}^{\infty } \d (-2)^{i}\,\bos (f,0,i-1) \end{equation*}

for the left integration.

Before using these operators the precision of the integration must be speciﬁed by an assignment of the form ww := number, which sets the actual upper limit to be used on the sums above instead of inﬁnity. If required this precision can be changed by reassignment. Both operators are deﬁned by their action and by the properties

\begin{gather*} \d (1)\,\d (-1)=\d (-1)\,\d (1)=\d (2)\,\d (-1)=\d (2)\,\d (-1)=1, \\ \der (1)\,\d (-1)=\d (-1)\,\der (1), \\ \d (-1)\,\del (1)=\del (1)\,\d (-1), \end{gather*}

and analogously for \(\d (-2)\) and \(\der (2), \del (2)\).

The neutral operator does not show any action on an expression but has several properties. More precisely

\begin{gather*} \d (1)\,\d (-3)=\d (-3)\,\d (1)=\d (2)\,\d (-3)=\d (-3)\,\d (2)=1, \\ \der (k)\,\d (-3)=\d (-3)\,\der (k), \\ \d (-3)\,\del (k)=\del (k)\,\d (-3), \end{gather*}

while for \(\d (-4)\)

\begin{gather*} \d (1)\,\d (-4)=\d (-4)\,\d (1)=\d (2)\,\d (-4)=\d (-4)\,\d (2)=1, \\ \der (k)\,\d (-4)=\d (-4)\,\der (k), \end{gather*}

where \(k=1,2\).

From the last two formulas we see that the \(\d (-3)\) operator is transparent under \(\del \) operators while the \(\d (-4)\) operator stops the \(\del \) action.

Similarly to \(\d (-3)\) or \(\d (-4)\) it is also possible to use the neutral diﬀerentiation operator denoted by \(\d (3)\). It has the properties

\begin{gather*} \d (3)\,\d (-4)=\d (-4)\,\d (3)=\d (3)\,\d (-3)=\d (-3)\,\d (3)=1, \\ \der (k)\,\d (3)=\d (3)\,\der (k), \\ \d (3)\,\del (k)=\del (k)\,\d (3), \end{gather*}

where \(k=1,2\).

We can have also “accelerated” integration operators denoted by \(\mathbf {dr}(-n)\) where \(n\) is a natural number. The action of these operators is exactly the same as \(\d (-1)^n\) but instead of using the integration formulas \(n\) times in the case of \(\d (-1)^n\), \(\mathbf {dr}(-n)\) uses the following formula only once:

\begin{equation*} \mathbf {dr}(-n)\,\bos (f,0,0) = \sum ^\mathit {ww}_{s=0}(-1)^{s} \begin {pmatrix} n+s-1 \\ n-1 \end {pmatrix} \bos (f,0,s)\,\mathbf {dr}(-n-s). \end{equation*}

Similarly to the \(\d (-1)\) case, we have to declare also the “precision” of integration if we would like to use the accelerated integration operators. The command let cutoff and assignment of the form cut := number allow us to annihilate the higher-order terms in the \(\mathbf {dr}\) integration procedure. Moreover, the command let drr automatically changes the usual integration \(\d (-1)\) into accelerated integration \(\mathbf {dr}\). The command let nodrr changes \(\mathbf {dr}\) integration into \(\d (-1)\).

20.61.10 Useful Commands

Combinations

We encounter, in many practical applications, the problem of constructing diﬀerent possible combinations of superfunction and super-pseudo-diﬀerential elements with given conformal dimensions. We provide three diﬀerent procedures in order to realize this requirement:

\begin{gather*} \mathbf {w\_comb}(\mathit {list},n,m,x), \\ \mathbf {fcomb}(\mathit {list},n,m,x), \\ \mathbf {pse\_ele}(n,\mathit {list},m). \end{gather*}

All these commands are based on the gradations trick, to associate with superfunctions and superderivatives the scaling parameter conformal dimension. We consider here \(k/2\) and \(k\) (\(k\) a positive integer) gradation only.

The command \(\mathbf {w\_comb}\) gives the most general form of superfunction combinations of given gradation. It is a four-argument procedure in which:

1.: the ﬁrst argument is a list in which each element is a three-element list in which the ﬁrst element is the name of the superfunction from which we would like to construct our combinations, the second denotes its gradation, and the last can take two values: f or b to indicate that the superfunction is respectively superfermionic or superbosonic;
2.: the second argument is a number, the desired gradation;
3.: the third argument is an arbitrary non-numerical value which enumerates the free parameters in our combinations;
4.: the fourth argument takes one of two values: f or b to indicate that whole combinations should be respectively fermionic or bosonic.

Examples:

\begin{eqnarray*} \mathbf {w\_comb}(\{\{ f,1,b \},\{g,1,b\}\},2,z,b) &\Rightarrow & z1\,\bos (f,3,0) + z2\,\bos (f,0,1) + \mbox {}\\ && z3\,\bos (f,0,0)^2 \\ \mathbf {w\_comb}(\{\{ f,1,b\}\},3/2,g,f) &\Rightarrow & g1\,\fer (f,1,0) + g2\,\fer (f,2,0) \end{eqnarray*}

The command \(\mathbf {fcomb}\), similarly to \(\mathbf {w\_comb}\), gives the general form of an arbitrary combination of superfunctions modulo divergence terms. It is a four-argument command with the same meaning of arguments as for \(\mathbf {w\_comb}\). This command ﬁrst calls \(\mathbf {w\_comb}\), then eliminates in the canonical way SuSy derivatives by integration by parts of \(\mathbf {w\_comb}\). By canonical we understand that (SuSy) derivatives are removed ﬁrst from the superfunction which is ﬁrst in the list of superfunctions in the \(\mathbf {fcomb}\) command, next from the second, etc.

In order to illustrate the canonical manner of elimination of (SuSy) derivatives let us consider some expression which is constructed from \(f, g\) and \(h\) superfunctions and their (SuSy) derivatives. This expression is ﬁrst split into three sub-expressions called the f-expression, g-expression and h-expression. The f-expression contains only combinations of \(f\) with \(f\) or \(g\) or (and) \(h\), while the g-expression contains only combinations of \(g\) with \(g\) or \(h\) and the h-expression contains only combinations of \(h\) with \(h\). The command \(\mathbf {fcomb}\) removes ﬁrst (SuSy) derivatives from \(f\) in f-expression, then from \(g\) in g-expression, and ﬁnally from \(h\) in h-expression. Consider this example:

\begin{equation*} \fer (f,1,0)\,\fer (g,2,0) + \bos (g,0,0)\,\bos (g,3,0). \end{equation*}

Let us now assume that we have \(f,g\) order; then the f-expression is \(\fer (f,1,0)\,\fer (g,2,0)\), while the g-expression is \(\bos (g,0,1)\,\bos (g,3,0)\). Now canonical elimination gives

\begin{equation*} - \bos (f,0,0)\,\bos (g,3,0) + 2\,\bos (g,0,0)\,\bos (g,3,1), \end{equation*}

while assuming \(g,f\) order gives

\begin{equation*} - \bos (f,3,0)\,\bos (g,0,0) + 2\,\bos (g,0,0)\,\bos (g,3,1). \end{equation*}

Example:

\begin{multline*} \mathbf {fcomb}(\{\{u,1\}\},4,h) \Rightarrow \\ h(1)\,\fer (u,2,0)\,\fer (u,1,0)\,\bos (u,0,0) + h(2)\,\bos (u,3,0)\,\bos (u,0,0)^2 + \mbox {}\\ h(3)\,\bos (u,0,2)\,\bos (u,0,0) + h(4)\,\bos (u,0,0)^4 \end{multline*}

Finally, the command \(\mathbf {pse\_ele}\) gives the general form of an element of the pseudo-SuSy derivative algebra [3]. Such an element can be written down symbolically as

\begin{equation*} (\bos + \fer \,\der (1)+\fer \,\der (2)+\bos \,\der (1)\,\der (2))\,\d (1)^n \end{equation*}

for the traditional and chiral representations, or

\begin{equation*} (\bos + \fer \,\der (1)+\fer \,\der (2)+\bos \,\der (3))\,\d (1)^n \end{equation*}

for the chiral1 representation, where \(\bos \) and \(\fer \) denote arbitrary superfunctions. The mentioned command allows us to obtain such an element of the given gradation which is constructed from some set of superfunctions of given gradation. This command takes three arguments:

\begin{equation*} \mathbf {pse\_ele}(\mathit {wx},\mathit {wy},\mathit {wz}). \end{equation*}

The ﬁrst argument denotes the gradation of the SuSy-pseudo-element, and the second denotes the names and gradations of the superfunctions from which we would like to construct our element. This second argument \(wy\) is in the form of a list exactly the same as in the \(\mathbf {w\_comb}\) command. The last argument denotes the names which enumerate the free parameters in our combination.

Parts of the pseudo-SuSy-diﬀerential elements

In order to obtain the components of the (pseudo)-SuSy element we have three diﬀerent commands:

\begin{gather*} \mathbf {s\_part}(\mathit {expression},m), \\ \mathbf {d\_part}(\mathit {expression},n), \\ \mathbf {sd\_part}(\mathit {expression},m,n), \end{gather*}

where \(m,n=0,1,2,3,\ldots \).

The \(\mathbf {s\_part}\) command gives the coeﬃcient standing in the \(m\)-th SuSy derivative. However, notice that for \(m=3\) we should consider the coeﬃcients standing in the \(\der (1)\,\der (2)\) operator for the traditional or chiral representations while for the chiral1 representation the terms standing in the \(\der (3)\) operator. The \(\mathbf {d\_part}\) command give the coeﬃcients standing in the same power of \(\d (1)\), while \(\mathbf {sd\_part}\) gives the term standing in the \(m\)-th SuSy derivative and \(n\)-th power of the usual derivative.

Example: Given the REDUCE input

ala := bos(g,0,0) + fer(f,3,0)*der(1) +
  (fer(h,2,0)*der(2) + bos(r,0,0)*der(1)*der(2))*d(1);

we have

\begin{eqnarray*} \mathbf {s\_part}(\mathit {ala},3) &\Rightarrow & \fer (f,3,0) \\ \mathbf {d\_part}(\mathit {ala},1) &\Rightarrow & \fer (h,2,0)\,\der (2) + \bos (r,0,0)\,\der (1)\,\der (2) \\ \mathbf {sd\_part}(\mathit {ala},0,0) &\Rightarrow & \bos (g,0,0) \end{eqnarray*}

Adjoint

The adjoint \(\mathit {PP}^*\) of some SuSy operator \(\mathit {PP}\) is deﬁned in standard form by

\begin{equation*} \langle \alpha , \mathit {PP}\,\beta \rangle = \langle \beta , \mathit {PP}^*\,\alpha \rangle \end{equation*}

where \(\alpha \) and \(\beta \) are test superboson functions and the scalar product is deﬁned by

\begin{equation*} \langle \alpha , \beta \rangle = \int \alpha \beta \,d\theta _{1}\,d\theta _{2}, \end{equation*}

where we use the Berezin integral deﬁnition [1]

\begin{align*} \int \theta _{i}\,d\theta _{j} &= \delta _{i,j}, \\ \int d\theta _{i} &= 0. \end{align*}

For this operation we have the command

\begin{equation*} \mathbf {cp}(\mathit {expression}). \end{equation*}

Examples:

\begin{eqnarray*} \mathbf {cp}(\der (1)) &\Rightarrow & -\der (1), \\ \mathbf {cp}(\del (1)*\fer (r,1,0)*\der (1)) &\Rightarrow & \fer (r,1,1) + \fer (r,1,0)\,\d (1) - \mbox {}\\ && \del (1)\,\bos (r,0,1), \end{eqnarray*}

The last example illustrates that it is possible to deﬁne \(\mathbf {cp}(\del (1)\,\fer (r,1,0)\,\der (1))\) in the diﬀerent but equivalent manner as \(\fer (r,1,0)\,\d (1) - \bos (r,0,1)\,\der (1)\).

From a practical point of view, we do not deﬁne conjugation for the \(\d (-1)\) and \(\d (-2)\) operators, because then we should deﬁne the precision of the action of the operators \(\d (-1)\) and \(\d (-2)\), and even then we would obtain very complicated formulas. However, if somebody decides to apply this conjugation to \(\d (-1)\) or \(\d (-2)\), it is recommended ﬁrst to change by hand these operators into \(\d (-3)\), next to compute \(\mathbf {cp}\) and change \(\d (-3)\) back into \(\d (-1)\) or \(\d (-2)\) together with the declaration of the precision.

Projection

In many cases, especially in the SuSy approach to soliton theory, we have to obtain the projection onto the invariant subspace (with respect to the commutator) of the pseudo-SuSy-diﬀerential algebra. There are three diﬀerent subspaces [4] and hence we have the two-argument command

\begin{equation*} \mathbf {rzut}(\mathit {expression},n) \end{equation*}

in which \(n=0,1,2\).

Example: Given the REDUCE input

ewa := bos(f,0,0) + bos(f3,0,0)*der(1)*der(2) +
   bos(g,0,0)*d(1) + bos(g3,0,0)*d(1)*der(1)*der(2) +
   fer(f1,1,0)*der(1) + fer(f2,2,0)*der(2) +
   fer(g1,1,0)*d(1)*der(1) + fer(g2,2,0)*d(1)*der(2);

we have

\begin{equation*} \mathbf {rzut}(\mathit {ewa},0) \Rightarrow \mathit {ewa}, \end{equation*}

\begin{equation*} \mathbf {rzut}(\mathit {ewa},1) \Rightarrow \mathit {ewa}-\bos (f,0,0), \end{equation*}

\begin{multline*} \mathbf {rzut}(\mathit {ewa},2) \Rightarrow \bos (f3,0,0)\,\der (1)\,\der (2) + \mbox {} \\ \big ( \fer (g1,1,0)\,\der (1) + \fer (g2,2,0)\,\der (2) + \bos (g3,0,0)\,\der (1)\,\der (2) \big )\,\d (1). \end{multline*}

Analogue of coeff

Motivated by practical applications, we constructed for our supersymmetric functions three commands, which allow us to obtain a list of the same combinations of some superfunctions and (SuSy) derivatives from some given operator-valued expression. Each command takes one argument and returns a list. We use the following REDUCE input to illustrate each command:

magda := fer(f,1,0)*fer(f,2,0)*a1 + der(1);

The ﬁrst command is

\begin{equation*} \mathbf {lyst}(\mathit {expression}). \end{equation*}

For example

\begin{equation*} \mathbf {lyst}(\mathit {magda}) \Rightarrow \{\fer (f,1,0)\,\fer (f,2,0)\,a1, \der (1)\}. \end{equation*}

The second command is

\begin{equation*} \mathbf {lyst1}(\mathit {expression}) \end{equation*}

with the output in the form of a list in which each element is constructed from the coeﬃcients and (SuSy) operators of the corresponding element in the lyst list. For example

\begin{equation*} \mathbf {lyst1}(\mathit {magda}) \Rightarrow \{a1,\der (1)\}. \end{equation*}

The third command is

\begin{equation*} \mathbf {lyst2}(\mathit {expression}) \end{equation*}

with the output in the form of a list in which each element is constructed from coeﬃcients in the given expression. For example

\begin{equation*} \mathbf {lyst2}(\mathit {magda}) \Rightarrow \{a1,1\}. \end{equation*}

Simpliﬁcation

If we encounter during the process of computation an expression such as

\begin{equation*} \fer (f,1,0)\,\d (-3)\,\fer (f,2,0)\,\d (1), \end{equation*}

it is not reduced further. To facilitate simpliﬁcation, we can replace \(\d (1)\) with \(\d (2)\), or vice versa. In order to do this replacement we have the command

\begin{equation*} \mathbf {chan}(\mathit {expression}) \end{equation*}

Example:

\begin{multline*} \mathbf {chan}(\fer (f,1,0)*\d (-3)*\fer (f,2,0)*\d (1)) \Rightarrow \\ -\fer (f,2,0)\,\fer (f,1,0) - \fer (f,1,0)\,\d (-3)\,\fer (f,2,1). \end{multline*}

Notice that as a result we remove the \(\d (1)\) operator.

O(2) Invariance

In many cases in supersymmetric theories we deal with the O(2) invariance of SuSy indices. This invariance follows from the physical assumption of nonprivileging the “fermionic” coordinates in the superspace. In order to check whether our formula possesses such invariance we can use

\begin{equation*} \mathbf {odwa}(\mathit {expression}) \end{equation*}

This procedure replaces in the given expression \(\der (1)\) with \(-\der (2)\) and \(\der (2)\) with \(\der (1)\). Next, it changes, in the same manner, the values of the action of these operators on the superfunctions.

Macierz

We can deﬁne the supercomponent form for the \(\mathbf {pse\_ele}\) objects similarly to the representation of the superfunctions. Usually we can consider such an object as the matrix which acts on the components of the superfunction. It is realized in our program using the command

\begin{equation*} \mathbf {macierz}(\mathit {expression},x,y), \end{equation*}

where \(\mathit {expression}\) is the formula under consideration. The argument \(x\) can take two values, b or f, depending on whether we would like to consider the bosonic (b) part or fermionic (f) part of the expression. The last argument denotes the option in which we act on the bosonic or fermionic superfunction. It takes two values: f for fermionic test superfunction or b for bosonic. More explicitly, we obtain

\begin{equation*} \mathbf {macierz}(\der (1)*\der (2),b,f) \Rightarrow \begin {pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \d (1) & 0 \\ 0 & -\d (1) & 0 & 0 \\ -\d (1)^2 & 0 & 0 & 0 \end {pmatrix}, \end{equation*}

\begin{equation*} \mathbf {macierz}(\der (1)*\der (2),f,b) \Rightarrow \begin {pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \d (1) \\ -\d (1) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {pmatrix}. \end{equation*}

20.61.11 Functional Gradients

In the SuSy soliton approach we very frequently encounter the problem of computing the gradient of the given functional. The usual deﬁnition of the gradient [2] is adopted in the supersymmetry also:

\begin{equation*} H'[v] = \langle \operatorname {grad} H, v \rangle = \frac {\partial }{\partial \epsilon } H(u+\epsilon v) \mid _{\epsilon =0}, \end{equation*}

where \(H\) denotes some functional which depends on \(u\), \(v\) denotes a vector along which we compute the gradient, and \(\langle \cdot ,\cdot \rangle \) denotes the relevant scalar product.

We implemented all that in our package for the traditional algebra only. In order to compute the gradient with respect to some superfunction use

\begin{equation*} \mathbf {gra}(\mathit {expression},f), \end{equation*}

where \(\mathit {expression}\) is the given density of the functional and \(f\) denotes the ﬁrst argument in the superfunction operator (name of the superfunction).

Example:

\begin{equation*} \mathbf {gra}(\bos (f,3,0)*\fer (f,1,0),f) \Rightarrow -2\,\fer (f,2,1) \end{equation*}

For practical use we provide two additional commands:

\begin{gather*} \mathbf {dyw}(\mathit {expression},f), \\ \mathbf {war}(\mathit {expression},f). \end{gather*}

The ﬁrst computes the variation of \(\mathit {expression}\) with respect to superfunction \(f\); the second removes (via integration by parts) SuSy derivatives from various functions and ﬁnally produces a list of factorized \(\fer \) and \(\bos \) superfunctions. When the given expression is a full (SuSy) derivative, the result of the \(\mathbf {dyw}\) command is 0 and hence this command is very useful in veriﬁcations of (SuSy) divergences of expressions.

When the result of applications of the \(\mathbf {dyw}\) command is not zero then we would like to have the system of equations on the coeﬃcients standing in the same factorized \(\fer \) and \(\bos \) superfunction. We can quickly obtain such a list using the command \(\mathbf {war}(\mathit {expression},f)\) with the same meaning for the arguments as in the \(\mathbf {dyw}\) command.

Examples: Given the REDUCE input

xxx := fer(f,1,0)*fer(f,2,0) + x*bos(f,3,0)^2;

we obtain

\begin{gather*} \mathbf {dyw}(\mathit {xxx},f) \Rightarrow \{ -2\,\bos (f,3,0)\,\bos (f,0,0), -2x\,\bos (f,0,2)\,\bos (f,0,0) \}, \\ \mathbf {war}(\mathit {xxx},f) \Rightarrow \{ -2, -2x \}. \end{gather*}

20.61.12 Conservation Laws

In many cases we would like to know whether a given expression is a conservation law for some Hamiltonian equation. We can quickly check it using

\begin{equation*} \mathbf {dot\_ham}(\mathit {equation},\mathit {expression}) \end{equation*}

where \(\mathit {equation}\) is a list of two-element lists in which the ﬁrst element denotes the function while the second denotes its ﬂow. The second argument should be understood as the density of some conserved current. For example, for the SuSy version of the Nonlinear Schrödinger Equation [7] we could use the following REDUCE input:

ew := {{q, -bos(q,0,2) + bos(q,0,0)^3*bos(r,0,0)^2
         2*bos(q,0,0)*pr(3,bos(q,0,0)*bos(r,0,0))},
       {r, bos(r,0,2) - bos(q,0,0)^2*bos(r,0,0)^3 +
         2*bos(r,0,0)*pr(3,bos(q,0,0)*bos(r,0,0))}};
ham := bos(q,0,1)*bos(r,0,0) +
         x*bos(q,0,0)^2*bos(r,0,0)^2;
yyy := dot_ham(ew,ham);

The result of the previous computation is a complicated expression that is not zero. We would like to interpret it as a full (SuSy) divergence and we can quickly verify it by using the command \(\mathbf {war}\). We can solve the resulting list of equations using known techniques. For example, in our previous case we obtain

\begin{gather*} \mathbf {war}(yyy,q) \Rightarrow \{-4x,-8x,-4x\}, \\ \mathbf {war}(yyy,r) \Rightarrow \{4x,8x,4x\}, \end{gather*}

and we conclude that \(\mathit {ham}\) is a constant of motion if \(x=0\).

It is also possible to apply the command \(\mathbf {dot\_ham}\) to the pseudo-SuSy-diﬀerential element. This is very useful in the SuSy approach to the Lax operator in which we would like to check the validity of the formula

\begin{equation*} \partial _{t}L := [ L,A ], \end{equation*}

where \(A\) is some (SuSy) operator.

20.61.13 Jacobi Identity

The Jacobi identity for some SuSy Hamiltonian operators is veriﬁed using the relation

\begin{equation*} \langle \alpha , P'_{P(\beta )} \gamma \rangle + \text {all cyclic permutations of } \alpha ,\beta ,\gamma = 0, \end{equation*}

where \(P'\) denotes the directional derivative along the \(P(\beta )\) vector and \(\langle \cdot ,\cdot \rangle \) denotes the scalar product. The directional derivative is deﬁned in the standard manner as [44]:

\begin{equation*} F^{'}(u)[v] = \frac {\partial }{\partial \epsilon } F(u+\epsilon v)\mid _{\epsilon =0}, \end{equation*}

where \(F\) is some functional depending on \(u\), and \(v\) is a directional vector.

In this package we have several commands that allow us to verify the Jacobi identity. We have the possibility to compute, independently of verifying the Jacobi identity, the directional derivative for the given Hamiltonian operator along the given vector using

\begin{equation*} \mathbf {n\_gat}(\mathit {pp}, \mathit {wim}), \end{equation*}

where \(\mathit {pp}\) is a scalar or matrix Hamiltonian operator and \(\mathit {wim}\) denotes the components of a vector along which we compute the derivative. It has the form of a list in which each element has the representation

\begin{equation*} \bos (f) \Rightarrow \mathit {expression}. \end{equation*}

The expression \(\bos (f)\) above denotes the shift of the \(\bos (f,0,0)\) superfunction according to the deﬁnition of the directional derivative.

In order to compute the Jacobi identity we use the command

\begin{equation*} \mathbf {fjacob}(\mathit {pp}, \mathit {wim}) \end{equation*}

with the same meaning for \(\mathit {pp}\) and \(\mathit {wim}\) as in the \(\mathbf {n\_gat}\) command.

Notice that the ordering of the components in the \(\mathit {wim}\) list is important and is connected with the interpretation of the components of the Hamiltonian operator \(\mathit {pp}\) as a set of Poisson brackets constructed just from elements of the \(\mathit {wim}\) list. For example, in our scheme, the ﬁrst component of \(\mathit {wim}\) is always connected with the element from which we create the Poisson bracket and which corresponds to the ﬁrst element on the diagonal of \(\mathit {pp}\), the second element of \(\mathit {wim}\) with the second element on the diagonal of \(\mathit {pp}\), etc.

As the result of the application of the \(\mathbf {fjacob}\) command to some Hamiltonian operator we obtain a complicated formula, not necessarily equal to zero but which should be expressed as a (SuSy) divergence. However, we can quickly verify it using the same method as for the \(\mathbf {dot\_ham}\) command, which was described in the previous subsection.

Usually, after the application of the \(\mathbf {fjacob}\) command to some matrix Hamiltonian operator we obtain a huge expression, which is too complicated to analyze even when we would like to check its (SuSy) divergence. In this case we could extract from the \(\mathbf {fjacob}\) expression terms containing given components of vector test functions ﬁxed by us. We can use the command

\begin{equation*} \mathbf {jacob}(\mathit {pp}, \mathit {wim}, \mathit {mm}) \end{equation*}

where \(\mathit {pp}\) and \(\mathit {wim}\) have the same meaning as for the \(\mathbf {fjacob}\) command while \(\mathit {mm}\) is a three-element list denoting the components: \(\{\alpha ,\beta ,\gamma \}\).

This command is not prepared to compute in full the Jacobi identity, which contains the integration operators. We do not implement here the symbolic integration of superfunctions in order to simplify the ﬁnal results.

20.61.14 Objects, Commands and Let Rules

Objects

\begin{equation*} \begin {array}{lllll} \bos (f,n,m) & \bos (f,n,m,k) & \fer (f,n,m) & \axp (f) & \fun (f,n) \\ \fun (f,n,m) & \gras (f,n) & \mathbf {axx}(f) & \d (1) & \d (2) \\ \d (3) & \d (-1) & \d (-2) & \d (-3) & \d (-4) \\ \mathbf {dr}(-n) & \der (1) & \der (2) & \del (1) & \del (2) \end {array} \end{equation*}

Commands

fpart(expression)	bpart(expression)	bf_part(expression,\(n\))
b_part(expression,\(n\))	pr(\(n\),expression)	pg(\(n\),expression)
w_comb({{\(f\),\(n\),\(x\)},…},\(m\),\(z\),\(y\))	fcomb({{\(f\),\(n\),\(x\)},…},\(m\),\(z\),\(y\))	pse_ele(\(n\),{{\(f\),\(n\)},…},\(z\))
s_part(expression,\(n\))	d_part(expression,\(n\))	sd_(expression,\(n\),\(m\))
cp(expression)	rzut(expression,\(n\))	lyst(expression)
lyst1(expression)	lyst2(expression)	chan(expression)
odwa(expression)	gra(expression,\(f\))	dyw(expression,\(f\))
war(expression,\(f\))	dot_ham(equations,expression)	n_gat(operator,list)
fjacob(operator,list)	jacob(operator,list,{\(\alpha ,\beta ,\gamma \)})	macierz(expression,\(x\),\(y\))
s_int(numbers,expression,list)

Let Rules

trad chiral chiral1 inverse drr nodrr

Acknowledgement

The author would like to thank to Dr W. Neun for valuable remarks.

Bibliography

Please see the original version of this document, which is available formatted as https://reduce-algebra.sourceforge.io/extra-docs/susy2.pdf and as the LaTeX source ﬁle susy2.tex (in the REDUCE packages/susy2 directory).

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