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