1on list;
2on errcont;
3
4% 1.) Example of ordering of objects such as fer,bos,axp;
5axp(bos(f,0,0))*bos(g,3,1)*fer(k,1,0);
6%fer(k,1,0)*bos(g,3,1)*axp(bos(f,0,0));
7
8% 2.) Example of ordering of fer and fer objects
9fer(f,1,2)*fer(f,1,2);
10% 0
11fer(f,1,2)*fer(g,2,3);
12% -fer(g,2,3)*fer(f,1,2);
13fer(f,1,2)*fer(f,1,3);
14% - fer(f,1,3)*fer(f,1,2);
15fer(f,1,2)*fer(f,2,2);
16% - fer(f,2,2)*fer(f,1,2);
17
18% 3.) Example of ordering of bos and bos objects;
19bos(f,3,0)*bos(g,0,4);
20%bos(g,0,4)*bos(f,3,0);
21bos(f,3,0)*bos(f,0,0);
22%bos(f,3,0)*bos(f,0,0);
23bos(f,3,2)*bos(f,3,5);
24%bos(f,3,5)*bos(f,3,2);
25
26% 4.) ordering of inverse superfunctions;
27% last index in bos objects denotes powers;
28bos(f,0,3)*bos(k,0,2)*bos(zz,0,3,-1)*bos(k,0,2,-1);
29%bos(zz,0,3,-1)*bos(f,0,3);
30bos(c,0,3)*bos(b,0,2)*bos(a,0,3,-1)*bos(b,0,2,-1);
31%bos(c,0,3)*bos(a,0,3,-1);
32
33% 5.) Demostration of inverse rule;
34let inverse;
35bos(f,0,3)**3*bos(k,3,1)**40*bos(f,0,3,-2);
36%bos(k,3,1,40)*bos(f,0,3,1);
37clearrules inverse;
38
39
40% 6.) Demonstration of (susy) derivative operators;
41% Up to now we did not decided on the chirality assumption
42% so let us check first the tradicional algebra os susy derivative;
43let trad;
44
45%first susy derivative
46der(1)*fer(f,1,2)*bos(g,3,1)*axp(bos(h,0,0));
47
48fer(g,2,1)*bos(f,0,2,-2)*axp(fer(k,1,2)*fer(h,2,1))*del(1);
49sub(del=der,ws);
50
51%second susy derivative
52der(2)*fer(g,2,3)*bos(kk,0,3)*axp(bos(f,3,0));
53
54fer(r,2,1)*bos(kk,3,4,-4)*axp(fer(f,1,2)*fer(g,2,1))*del(2);
55sub(del=der,ws);
56
57%usual derivative;
58d(1)*fer(f,1,2)*bos(g,3,1)*axp(bos(h,0,0));
59fer(g,2,1)*bos(f,0,2,-2)*axp(fer(h,1,2)*fer(k,2,1))*d(2);
60sub(d(2)=d(1),ws);
61
62% 7.) the value of action of (susy) derivative;
63
64xxx:=fer(f,1,2)*bos(k,0,2,-2)*axp(fer(h,2,0)*fer(aa,1,3));
65yyy:=fer(g,2,3)*bos(kk,3,1,-3)*axp(bos(f,0,2,-3));
66
67%first susy derivative
68
69pr(1,xxx);
70pr(1,yyy);
71
72%second susy2 derivative;
73pr(2,xxx);
74pr(2,yyy);
75
76% third susy2 derivative;
77
78pr(3,xxx);
79pr(3,yyy);
80
81clearrules trad;
82let chiral;
83pr(3,xxx);
84clearrules chiral;
85let chiral1;
86pr(3,xxx);
87
88clearrules chiral1;
89let trad;
90% usual derivative
91pg(1,xxx);
92pg(3,yyy);
93
94clear xxx,yyy;
95
96% 8.)
97% And now let us change traditional algebra on the chiral algebra;
98clearrules trad;
99let chiral;
100% And now we compute the same derivative but in the chiral
101% representation;
102
103%first susy derivative
104der(1)*fer(f,1,2)*bos(g,3,1)*axp(bos(h,0,0));
105
106fer(g,2,1)*bos(f,0,2,-2)*axp(fer(k,1,2)*fer(h,2,1))*del(1);
107sub(del=der,ws);
108
109%second susy derivative
110der(2)*fer(g,2,3)*bos(kk,0,3)*axp(bos(f,3,0));
111
112fer(r,2,1)*bos(kk,3,4,-4)*axp(fer(f,1,2)*fer(g,2,1))*del(2);
113sub(del=der,ws);
114;
115
116% 9.) the value of action of (susy) derivative;
117
118xxx:=fer(f,1,2)*bos(k,0,2,-2)*axp(fer(h,2,0)*fer(aa,1,3));
119yyy:=fer(g,2,3)*bos(kk,3,1,-3)*axp(bos(f,0,2,-3));
120
121%first susy derivative
122
123pr(1,xxx);
124pr(1,yyy);
125
126%second susy2 derivative;
127pr(2,xxx);
128pr(2,yyy);
129
130clear xxx,yyy;
131% We return back to the traditional algebra;
132
133clearrules chiral;
134let trad;
135
136% 10.) The components of super-objects;
137
138xxx:=fer(f,2,3)*bos(g,3,2,2);
139
140% all components;
141fpart(xxx);
142
143%bosonic sector;
144bpart(xxx);
145
146%the given component
147bf_part(xxx,0);
148
149%the given component in the bosonic sector;
150
151b_part(xxx,0);
152b_part(xxx,1);
153
154
155
156clear zzz;
157clearrules trad;
158let chiral;
159zzz:=bos(f,3,1,-1)*bos(g,0,1,2);
160b_part(zzz,0);
161b_part(zzz,3);
162clearrules chiral;
163let chiral1;
164b_part(zzz,0);
165b_part(zzz,3);
166clearrules chiral1;
167let trad;
168
169%11 matrix represenattion of operators;
170lax:=der(1)*der(2)+bos(u,0,0);
171
172macierz(lax,b,b);
173macierz(lax,f,b);
174macierz(lax,b,f);
175macierz(lax,f,f);
176
177% 12.) Demonstration of chirality properties;
178clearrules trad;
179let chiral;
180b_chiral:={f0};
181b_antychiral:={f1};
182f_chiral:={f2};
183f_antychiral:={f3};
184for k:=0:3 do  write fer(f0,k,0);
185for k:=0:3 do  write fer(f1,k,0);
186for k:=0:3 do  write fer(f2,k,0);
187for k:=0:3 do  write fer(f3,k,0);
188for k:=0:3 do  write bos(f1,k,0);
189for k:=0:3 do  write bos(f2,k,0);
190for k:=0:3 do  write bos(f2,k,0);
191for k:=0:3 do  write bos(f3,k,0);
192
193% 13.) Integrations;
194
195d(-1)*xxx;
196
197%we have to declare ww;
198ww:=2;
199
200d(-1)*xxx;
201xxx*d(-2);
202d(-3)*xxx;
203
204ww:=4;
205d(-1)**5:=0;d(-2)**5:=0;
206
207d(-1)*yyy;
208yyy*d(-2);
209
210clear d(-1)**5,d(-2)**5;
211
212
213on list;
214
215% 14.) The accelerations of integrations;
216
217clear ww;
218ww:=3;
219let drr;
220let cutoff;
221cut:=4;
222d(-1)*xxx;
223d(-1)**2*yyy;
224clear ww,cut;
225ww:=4;
226cut:=5;
227d(-1)**3*yyy;
228d(-1)*xxx;
229
230clearrules cutoff;clearrules drr;
231clear cut,ww;
232
233% it is possible to use directly accelerated integrations oprators dr;
234ww:=4;
235dr(-2)*fer(f,1,2)*bos(kk,0,2);
236
237on time;
238showtime;
239
240dr(-3)*bos(g,3,1)*bos(ff,3,2);
241
242showtime;
243%if you try usual integration
244
245d(-1)**3*bos(g,3,1)*bos(ff,3,2);
246
247showtime;
248
249% then the time - diffrences is evident. In this example d(-1)
250% integration is 10 times slower then  dr integrations.
251
252off time;
253
254let cutoff;
255cut:=5;
256dr(-2)*fer(f,1,2)*bos(aa,0,1);
257dr(-3)*bos(g,3,1)*bos(bb,0,3);
258clear ww,cut;
259ww:=6;
260cut:=7;
261dr(-3)*fer(k,2,3)*bos(h,0,2);
262dr(-4)*bos(h,0,3)*bos(k,0,2);
263
264clear ww,cut;
265clearrules cutoff;
266
267% 15.) The combinations
268
269%the combinations of dim 7 constructed from fields of
270% the 2 ,3 dimensions, free  parameters are numerated by "a";
271
272w_comb({{f,2,b},{g,3,b}},7,a,b);
273w_comb({{f,2,f},{g,3,f}},4,s,f);
274
275% and now compute the last example but withouth the (susy)divergence
276%terms;
277
278fcomb({{f,2,b},{g,3,b}},5,c,b);
279fcomb({{f,1,f}},4,r,f);
280
281
282% 16.) The element of pseudo - susy -differential algebra;
283
284pse_ele(2,{{f,2,b}},c);
285pse_ele(3,{{f,2,b}},c);
286pse_ele(4,{{f,2,b}},c);
287pse_ele(3,{{f,1,b},{g,2,b}},r);
288
289% The components of the elements of pseudo - susy - differential algebra;
290
291xxx:=pse_ele(2,{{f,1,b},{g,2,b}},r);
292
293for k:=0:3 do write s_part(xxx,k);
294
295for k:=0:2 do write d_part(xxx,k);
296
297for k:=0:2 do for l:=0:3 do write sd_part(xxx,l,k);
298
299clear xxx;
300
301% 17.) Projection onto invariant subspace;
302
303xxx:=
304w_comb({{f,1,b}},2,a,b)*d(1)+
305w_comb({{f,1,b}},3,b,b)*der(1)*der(2)+
306w_comb({{f,1,b}},5/2,c,b)*der(1)+
307w_comb({{f,1,b}},3,ee,b)*d(1)^2+
308w_comb({{f,1,b}},7/2,fe,b)*d(1)*der(2)+
309w_comb({{f,1,b}},3,g,b)*der(1)*der(2)*d(1);
310
311for k:=0:2 do write rzut(xxx,k);
312
313clear xxx;
314
315% 18.) Test for the adjoint operators;
316
317cp(der(1));
318cp(der(1)*der(2));
319clearrules trad;
320let chiral1;
321cp(der(3));
322cp(der(1)*d(1));
323clearrules chiral1;
324let trad;
325cp(d(1));
326cp(d(2));
327as:=fer(f,1,0)*d(-3)*fer(g,2,0)+fer(h,1,2)*d(-3)*fer(kk,2,1);
328cp(as);
329cp(as*as);
330
331as:=fer(f,1,0);
332cp(as);
333cp(ws);
334
335clear as;
336
337as:=bos(f,0,0);
338as1:=as*der(1);
339cp(as1);
340cp(ws);
341cp(as1)+der(1)*as;
342
343as2:=as*der(1)*der(2);
344cp(as2);
345cp(ws);
346cp(as2) - der(1)*der(2)*as;
347
348
349clear as;
350as:=mat((fer(f,1,0)*der(1),bos(g,0,0)*d(-3)*bos(h,0,0)),
351(fer(h,2,1),fer(h,1,2)*d(-3)*fer(k,2,3)));
352cp(as);
353clear as;
354
355
356% 19.) Analog of coeff
357
358xxx:=pse_ele(2,{{f,1,b}},a);
359yyy:=lyst(xxx);
360zzz:=lyst1(xxx);
361yyy:=lyst2(xxx);
362clear xxx,yyy,zzz;
363
364% 20.) Simplifications;
365
366% we would like to compute third generalizations of the SUSY KdV
367% equation
368% example from Z.Popowicz Phys.Lett.A.174 (1993) p.87
369
370lax:=d(1)+d(-3)*der(1)*der(2)*bos(u,0,0);
371lb2:=lax^2;
372la2:=chan(lb2);
373lb3:=lax*la2;
374la3:=chan(lb3);
375lax3:=rzut(la3,1);
376comm:=lax*lax3 - lax3*lax;
377com:=chan(comm);
378result:=sub(der=del,com);
379%the equation is
380equ:=sub(del(1)=1,del(2)=1,d(-3)=1,result);
381
382clear lax,lb2,la2,lb3,la3,lax3,comm,com,result;
383
384% we now compute the same but starting from
385% different realizations of susy algebra
386%
387clearrules trad;
388let chiral1;
389lax:=d(1)+d(-3)*del(3)*bos(u,0,0);
390la2:=chan(lax^2);
391la3:=rzut(chan(lax*la2),0);
392com:=chan(lax*la3-la3*lax);
393equ_chiral1:=sub(d(-3)=1,del(3)=1,com);
394clear lax,lb2,la2,lb3,la3,lax3,lax,comm,com,result;
395clearrules chiral1;
396let trad;
397
398% 21.) Conservation laws;
399% we would like to check the conservations laws for our third
400%generalization of susy kdv equation;
401%
402
403ham:=fcomb({{u,1,b}},3,a,b);
404
405conserv:=dot_ham({{u,equ}},ham);
406% we check now on susy-divergence behaviour;
407%
408az:=war(conserv,u);
409solve(az);
410clear equ,ha,conserv,az;
411
412% 22.) The residue of Lax operator
413% we would like to find conservation laws for Lax susy KdV
414% equation considered in the previous example
415%
416lax:=d(1)-d(-3)*del(1)*der(2)*bos(u,0,0);
417lb2:=lax^2;
418la2:=chan(lb2);
419lb4:=la2^2;
420kxk^3:=0;
421la4:=chan(lb4);
422lc4:=sub(kxk=1,qq=-3,sub(d(-3)=kxk*d(qq),la4));
423lb5:=lax*lc4;
424lc5:=s_part(lb5,3);
425la5:=lc5-sub(d(-3)=0,lc5);
426ld5:=chan(la5);
427konserv:=sub(d(-3)=1,d_part(ld5,-1));
428clear lax,lb2,la2,lb4,kxk,la4,lc4,lb5,lc5,la5,konserv;
429
430%22.) The N=2 SuSy Boussinesq equation
431% example from Z.Popowicz Phys.LettB.319 (1993) 478-484
432
433clearrules trad;
434let chiral;
435
436lax:=del(1)*(d(1)^2+bos(j,0,0)*d(1)+bos(tt,0,0))*der(2);
437la2:=del(1)*(d(1)+2*bos(j,0,0)/3)*der(2);
438com:=sub(del(1)=1,der(2)=1,lax*la2-la2*lax);
439operator boss;
440boss(j,t):=d_part(com,1);
441boss(tt,t):=d_part(com,0);
442
443% let us shift bos(tt,0,0) to
444
445bos(tt,0,0):=bos(tx,0,0)/2+bos(j,0,0)**2/6 + bos(j,0,1)/2;
446bos(tt,0,1):=pg(1,bos(tt,0,0));
447bos(tt,0,2):=pg(1,bos(tt,0,1));
448fer(tt,1,0):=pr(1,bos(tt,0,0));
449fer(tt,2,0):=pr(2,bos(tt,0,0));
450
451% then the equations of motion are;
452
453bos(j,t):=boss(j,t);
454bos(tx,t):=2*(boss(tt,t) - boss(j,t)*bos(j,0,0)/3-
455            pg(1,boss(j,t))/2);
456
457clear lax,la2;
458clearrules chiral;
459let trad;
460
461%23.) the Jacobi identity;
462% we will find the N=2 susy extension of the Virasoro algebra.
463% First we found the most general form of the susy-pseudo-differential
464% element of the dimension two.
465
466vira:=pse_ele(2,{{f,1,b}},a);
467
468% This vira should be antisymmetrical so we found
469
470ewa:=vira+cp(vira);
471
472%we first solve ewa in order to found free coefficients;
473
474load_package groebner;
475adam:=groesolve(sub(der(1)=1,der(2)=1,d(1)=1,lyst1(ewa)));
476
477% we define now the most general antisymmetrical susy-pseudo-symmetrical
478% element of conformal dimension two.
479
480vira:=sub(adam,vira);
481
482% we make additional assumption that our Poisson tensor vira should be O(2)
483% invariant under the change of susy derivatives;
484
485dad:=odwa(vira)-vira;
486factor der;
487wyr1:=sub(der(1)=1,der(2)=1,lyst1(dad));
488remfac der;
489dad:=groesolve(wyr1);
490vira:=sub(dad,vira);
491% we check wheather it is really O(2) invariant;
492vira-odwa(vira);
493% O.K
494%so
495%now we check the Jacobi identity
496
497jjacob:=fjacob(vira,f);
498
499% we now check jjacob on the susy-divergence behaviour w.r. to the test
500% superfunction !#a;
501
502az:=war(jjacob,!#a);
503as:=groesolve(az);
504array ew(3);
505
506for k:=1:2 do ew(k):=part(as,k);
507
508% as we see we have two different solutions
509% first give us classical realizations of the Virasoro algebra
510% (without the center term)  which is
511
512sub(ew(1),vira);
513
514
515% the second solution give us desired susy generalizations of
516% Virasoro algebra
517
518sub(ew(2),vira);
519
520% the coefficient "a" could be absorbed by redefinations of
521% bos(f,0,0)
522% we check that previous result satisfies the antisymmetric requirements
523
524ws + cp(ws);
525
526clearrules trad;
527let chiral1 ;
528
529% We check that for chiral1 realization the following operator
530vira:=der(3)*d(1)+bos(j,0,1)+bos(j,0,0)*d(1)+
531        fer(j,1,0)*der(2)+fer(j,2,0)*der(1);
532% satisfy the Jacobi identity;
533jjacob:=fjacob(vira,j);
534
535az:=war(jjacob,!#a);
536
537%24 superintegration
538clearrules chiral1;
539let trad;
540
541
542as:=s_int(0,bos(f,3,0)^2-bos(f,0,1)^2,{f});
543as1:=sub(d(-3)=0,ws);
544as2:=sub(d(-3)=1,as-as1);
545as3:=s_int(1,as2,{f});
546as4:=sub(del(-1)=0,ws);
547as4:=sub(del(-1)=1,as3-as4);
548as5:=s_int(2,as4,{f});
549
550end;
551