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   1    *----*    MuPAD Pro 4.0.0 -- The Open Computer Algebra System
   2   /|   /|
   3  *----* |    Copyright (c)  1997 - 2006  by SciFace Software
   4  | *--|-*                   All rights reserved.
   5  |/   |/
   6  *----*      Licensed to:   MuPAD Combinat Developer
   7 
   8 
   9  +---+
  10  | T |             MuPAD-Combinat 1.3.3 (development)
  11  +---+---+
  12  | A | K |         an open source MuPAD package for
  13  +---+---+---+
  14      | I | N |    research in Algebraic Combinatorics
  15      +---+---+
  16 
  17 This package provides or extends the following libraries:
  18   combinat, examples, Dom, Cat, output, experimental, IPC, operators
  19 
  20 For quick information on a particular library, please type:
  21   info(library)   or   ?library   (requires MuPAD >= 4.0.0)
  22 
  23 For the full html documentation, please browse through:
  24   http://mupad-combinat.sf.net/ (project web page)
  25   file:/media/AdvDisk/Combinat//index.html (local mirror)
  26 
  27 -- Interface:
  28 packages::Combinat::dotCategories, packages::Combinat::help,
  29 packages::Combinat::viewDot,       packages::Combinat::viewDotTeX,
  30 packages::Combinat::viewTeX                                       
  31 >> TEXTWIDTH:=84:
  32 >> 1+1
  33 
  34                                          2
  35                                                                        Time: 0 msec 
  36 >> export(combinat):
  37 >> partitions::list(5)
  38 
  39      [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
  40                                                                       Time: 68 msec 
  41 >> partitions::count(10)
  42 
  43                                          42
  44                                                                        Time: 4 msec 
  45 >> trees::list(5)
  46 
  47     --   o  ,  o ,  o ,  o  ,  o ,  o ,  o ,   o ,  o ,  o ,  o ,  o ,  o , o --
  48     |  // \\  /|\  /|\  / \   / \  /|\  / \   / \  / \   |    |    |    |   |  |
  49     |           |   |     /\    |  |    | |  /\    |    /|\  / \  / \   |   |  |
  50     |                           |                  |           |  |    / \  |  |
  51     --                                                                      | --
  52                                                                      Time: 392 msec 
  53 >> trees::count(6)
  54 
  55                                          42
  56                                                                        Time: 0 msec 
  57 >> trees::random(50)
  58 
  59                                                 o
  60                                                 |
  61                                             /  / \\
  62                                            /  \|  /\
  63                                 /  /       |\\     |
  64                                  //  \    \
  65                                   |  |    |
  66                                  / \ |   /|\
  67                                     / \    |
  68                                       |
  69                                      /  \
  70                                     /|\ |
  71                                       |/ \
  72                                       | / \
  73                                          /|\
  74                                          |
  75                                                                       Time: 96 msec 
  76 
  77 Sin								      
  78 
  79 >> print(eq) $ eq in binaryTrees::grammar::specification
  80 
  81                             Tree = Union(LeafNil, Node)
  82 
  83              Node = Alias(NodeRaw, x -> dom([op(x, 1), op(op(x, 2))]))
  84 
  85                         NodeRaw = Prod(NodeLabel, ChildList)
  86 
  87                                NodeLabel = Atom(NIL)
  88 
  89                             ChildList = Prod(Tree, Tree)
  90 
  91               Leaf = Alias(LeafLabel, combinat::[email protected]_LIST)
  92 
  93                                LeafLabel = Atom(NIL)
  94 
  95                                 LeafNil = Epsilon(.)
  96 >> r := binaryTrees::grammar::recurrenceRelation():
  97 >>  assume(n>0):
  98 >>  u(n) = factor(op(solve(r, u(n)),1))
  99 
 100                                    2 u(n - 1) (2 n - 1)
 101                             u(n) = --------------------
 102                                            n + 1
 103                                                                      Time: 136 msec 
 104 >> domain Trees
 105 >>  inherits combinat::trees;
 106 >>  category Cat::TreesClass(Ordered = FALSE);
 107 >>  axiom    Ax::normalRep, Ax::canonicalRep;
 108 >>  
 109 >>  info := "A domain for rooted unordered trees";
 110 >>  
 111 >>  grafts :=
 112 >>  proc(x:dom, y:dom) : Type::ListOf(dom)
 113 >>      local i;
 114 >>  begin
 115 >>      [sort(x::append(y))] .           // Base case
 116 >>      (map( (x[i])::grafts(y),         // Recursively
 117 >>            z -> sort(x::subsop(i=z))) //     graft back each result in turn
 118 >>       $ i=1..nops(x))                 // on all childs
 119 >>  end_proc;
 120 >>  end_domain:
 121 >> T := Trees::list(4);
 122 
 123                                -- o,  o ,  o ,  o  --
 124                                |  |   |   / \  /|\  |
 125                                |  |  / \    |       |
 126                                -- |                --
 127                                                                      Time: 100 msec 
 128 >> t1 := T[2]; t2 := subsop(Trees::list(5)[7], 0=0)
 129 
 130                                          o
 131                                          |
 132                                         / \
 133                                                                        Time: 4 msec 
 134 
 135                                          0
 136                                         / \
 137                                         | |
 138                                                                       Time: 52 msec 
 139 >> t1::grafts(t2)
 140 
 141                            --   o  ,  o  ,  o  ,  o   --
 142                            |   / \    |     |     |    |
 143                            |  /\ 0   /|\   / \   / \   |
 144                            |    / \    0     |     |   |
 145                            |    | |   / \    0     0   |
 146                            |          | |   / \   / \  |
 147                            --               | |   | | --
 148                                                                       Time: 36 msec 
 149 >> domain preLieAlgebra(R = Dom::ExpressionField(): Cat::Ring)
 150 >>  inherits Dom::FreeModule(R, Trees);
 151 >>  category Cat::AlgebraWithBasis(R); // Non Associative...
 152 >>  
 153 >>  v := dom::term(Trees([NIL]));
 154 >>  
 155 >>  mult2Basis :=
 156 >>  proc(t1 : Trees,
 157 >>       t2 : Trees)
 158 >>      local t;
 159 >>  begin
 160 >>      dom::plus(dom::term(t) $ t in Trees::grafts(t1,t2));
 161 >>  end;
 162 >>  end_domain:
 163 >> pr := preLieAlgebra():
 164 >> d := pr::v
 165 
 166                                         B(o)
 167                                                                        Time: 4 msec 
 168 >> d := d * pr::v
 169 
 170                                         B o
 171                                          (|)
 172                                                                        Time: 8 msec 
 173 >> d := d * pr::v
 174 
 175                                   B  o   + B/ o \
 176                                    (/ \)    | | |
 177                                             \ | /
 178                                                                        Time: 4 msec 
 179 >> d := d * pr::v
 180 
 181                       3 B/  o  \ + B/  o  \ + B/ o \ + B  o
 182                          | / \ |    |  |  |    | | |    (/|\)
 183                          \   | /    \ / \ /    | | |
 184                                                \ | /
 185                                                                       Time: 12 msec 
 186 >> d := d * pr::v;
 187 
 188 B   o    + 6 B/  o  \ + 4 B/  o  \ + 3 B/  o  \ + B/  o  \ + B/ o \ + 4 B/  o   \ +
 189  (// \\)      | /|\ |      | / \ |      |  |  |    |  |  |    | | |      | / \  |
 190               \   | /      |   | |      | / \ |    |  |  |    | | |      \   /\ /
 191                            \   | /      \   | /    \ / \ /    | | |
 192                                                               \ | /
 193 
 194    B/  o  \ + 3 B/  o  \
 195     |  |  |      | / \ |
 196     \ /|\ /      \ | | /
 197                                                                       Time: 40 msec 
 198 >> _plus(coeff(d))
 199 
 200                                          24
 201                                                                        Time: 0 msec 
 202 >> d := d * pr::v;
 203 
 204 15 B/  o  \ + 10 B/  o   \ + 10 B/  o  \ + 5 B/  o   \ + B/   o   \ + 6 B/  o  \ +
 205     | /|\ |       | / \  |       | / \ |      | / \  |    |   |   |      |  |  |
 206     \  || /       \ | /\ /       | | | |      \  /|\ /    \ // \\ /      | /|\ |
 207                                  \   | /                                 \   | /
 208 
 209    10 B/  o   \ + 15 B/  o   \ + 5 B/  o  \ + 4 B/  o  \ + 3 B/  o  \ + B/  o  \ +
 210        | /|\  |       | / \  |      | / \ |      |  |  |      |  |  |    |  |  |
 211        \   /\ /       |   /\ |      |   | |      | / \ |      |  |  |    |  |  |
 212                       \    | /      |   | |      |   | |      | / \ |    |  |  |
 213                                     \   | /      \   | /      \   | /    \ / \ /
 214 
 215    B/ o \ + 5 B/  o   \ + 4 B/  o   \ + B/  o  \ + 3 B/  o  \ + 10 B/  o  \ +
 216     | | |      | / \  |      |  |   |    |  |  |      |  |  |       | /|\ |
 217     | | |      |   |  |      | / \  |    |  |  |      | / \ |       |   | |
 218     | | |      \  / \ /      \   /\ /    \ /|\ /      \ | | /       \   | /
 219     | | |
 220     \ | /
 221 
 222    10 B/   o   \ + B   o
 223        | // \\ |    (//|\\)
 224        \     | /
 225                                                                       Time: 88 msec 
 226 >> _plus(coeff(d))
 227 
 228                                         120
 229                                                                        Time: 0 msec 
 230 >> d := d * pr::v;
 231 
 232 B    o     + 15 B/   o   \ + 45 B/   o   \ + 20 B/   o    \ + 20 B/   o   \ +
 233  (/// \\\)       | //|\\ |       | // \\ |       | // \\  |       | // \\ |
 234                  \     | /       \    || /       \     /\ /       |     | |
 235                                                                   \     | /
 236 
 237    60 B/  o  \ + 45 B/  o   \ + 15 B/  o   \ + 15 B/  o  \ + 18 B/  o   \ +
 238        | /|\ |       | /|\  |       | /|\  |       | /|\ |       | / \  |
 239        |  || |       |   /\ |       |   |  |       |   | |       |   /\ |
 240        \   | /       \    | /       \  / \ /       |   | |       \   || /
 241                                                    \   | /
 242 
 243    15 B/  o  \ + 10 B/  o   \ + 10 B/  o  \ + 6 B/  o   \ + 5 B/  o   \ +
 244        |  |  |       |  |   |       |  |  |      | / \  |      |  |   |
 245        | /|\ |       | / \  |       | / \ |      |   |  |      | / \  |
 246        \  || /       \ | /\ /       | | | |      \  /|\ /      \  /|\ /
 247                                     \   | /
 248 
 249    B/   o   \ + 6 B/  o  \ + 24 B/  o    \ + 10 B/  o   \ + 15 B/  o   \ +
 250     |   |   |      |  |  |       | / \   |       |  |   |       |  |   |
 251     |   |   |      |  |  |       |   /\  |       | /|\  |       | / \  |
 252     \ // \\ /      | /|\ |       \    /\ /       \   /\ /       |   /\ |
 253                    \   | /                                      \    | /
 254 
 255    15 B/  o   \ + 18 B/  o   \ + 6 B/  o  \ + 5 B/  o  \ + 4 B/  o  \ +
 256        | / \  |       | / \  |      | / \ |      |  |  |      |  |  |
 257        | | |  |       |   |  |      |   | |      | / \ |      |  |  |
 258        \  / \ /       |  / \ |      |   | |      |   | |      | / \ |
 259                       \    | /      |   | |      |   | |      |   | |
 260                                     \   | /      \   | /      \   | /
 261 
 262    3 B/  o  \ + B/  o  \ + B/ o \ + 6 B/  o   \ + 5 B/  o   \ + 4 B/  o   \ +
 263       |  |  |    |  |  |    | | |      | / \  |      |  |   |      |  |   |
 264       |  |  |    |  |  |    | | |      |   |  |      | / \  |      |  |   |
 265       |  |  |    |  |  |    | | |      |   |  |      |   |  |      | / \  |
 266       | / \ |    |  |  |    | | |      \  / \ /      \  / \ /      \   /\ /
 267       \   | /    \ / \ /    | | |
 268                             \ | /
 269 
 270    B/  o  \ + 3 B/  o  \ + 24 B/  o   \ + 10 B/  o  \ + 15 B/  o  \ +
 271     |  |  |      |  |  |       | / \  |       |  |  |       | / \ |
 272     |  |  |      |  |  |       |   /\ |       | /|\ |       | | | |
 273     |  |  |      | / \ |       |    | |       |   | |       |   | |
 274     \ /|\ /      \ | | /       \    | /       \   | /       \   | /
 275 
 276    45 B/  o   \ + 36 B/  o   \ + 60 B/  o   \ + 15 B/  o    \ + 10 B/   o   \ +
 277        | / \  |       | / \  |       | /|\  |       | /| \  |       |   |   |
 278        | | /\ |       |  /|\ |       \  |/\ /       \   /|\ /       | // \\ |
 279        \    | /       \    | /                                      \     | /
 280 
 281    6 B/  o    \ + B/   o   \ + 15 B/  o   \ + 20 B/  o   \ + 10 B/  o  \ +
 282       | / \   |    |   |   |       | / \  |       | / \  |       | / \ |
 283       \  //\\ /    \ //|\\ /       \ |/|\ /       | | /\ |       | | | |
 284                                                   \ |    /       \ | | /
 285 
 286    10 B/   o   \ + 15 B/  o  \
 287        |  / \  |       | /|\ |
 288        \ /\ /\ /       \ ||| /
 289                                                                      Time: 188 msec 
 290 >> _plus(coeff(d))
 291 
 292                                         720
 293                                                                        Time: 0 msec 
 294 >> // Identité Prélie :
 295 >>  bool(     (x*y)*z  -  x*(y*z)   =   (x*z)*y  -  x*(z*y)     )
 296 >>  $ x in pr::basis(3) $ y in pr::basis(3) $ z in pr::basis(3);
 297 
 298                    TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE
 299                                                                      Time: 120 msec 
 300 >> read("experimental/2005-09-08-David.mu"):
 301 //////////////////////////////////////////////////////////////////////
 302 Loading worksheet: Twisted Kac algebras
 303 Cf.  p. 715 of '2-cocycles and twisting of Kac algebras'
 304 
 305 Version: $Id: 2005-09-08-David.mu 7495 2008-04-30 20:01:44Z nthiery $
 306 To update to the latest version, go to the MuPAD-Combinat directory and type:
 307        svn update
 308 Content:
 309   G := DihedralGroup(4)
 310   SkewTensorProduct(A, B)	-- Skew tensor product of A and B (A being the dual of B)
 311   coidealDual([ p ])	-- Basis of the dual of the left coideal generated by p
 312 
 313   TwistedDihedralGroupAlgebra:
 314     KD4 := TwistedDihedralGroupAlgebra(4):
 315     KD4 := KD(4):			-- shortcut
 316     KD4::G = KD4::group			-- KD4 expressed on group elements
 317     KD4::G([3,1])			-- a^3 b
 318     KD4::M = KD4::matrix		-- KD4 expressed as block diagonal matrices
 319     KD4::G::tensorSquare		-- the tensor product KD4::G # KD4::G
 320     KD4::M::tensorSquare		--
 321 
 322     KD4::coeffRing			-- the coefficient field
 323     KD4::coeffRing::primitiveUnitRoot(4)-- the complex value I
 324 
 325     KD4::M(x), KD4::G(x)		-- conversions between bases
 326     KD4::e(1), KD4::e(2,2,1)		-- matrix units
 327     KD4::p(2,2,j), KD4::r(2,2,j)	-- some projectors of the j-th block
 328     KD4::p1, KD4::p2, KD4::q1, KD4::q1	-- some projectors
 329     KD4::G::Omega			-- Omega in the group basis
 330     KD4::M::tensorSquare( KD4::G::Omega )-- Omega in the matrix basis
 331     KD4::M::coproductAsMatrix(e(1))	-- the coproduct of e(1) as a matrix
 332 
 333     KD4::automorphismReverseOddBlocks   -- some (potential) automorphisms
 334     KD4::automorphismTransposeEvenBlocks--
 335     KD4::automorphismTransposeOddBlocks -- (not an automorphism for KD4!)
 336 
 337     // To get shorter notations:
 338     export(KD4, Alias, e, p1, p2, q1, q2):
 339     alias(view = KD4::M::coproductAsMatrix):
 340     // Then you can do:
 341     e(2,2,1) ...
 342     view(e(1))
 343 
 344   TwistedQuaternionGroupAlgebra(N)
 345     KQ4 := TwistedDihedralGroupAlgebra(4):
 346     KQ4 := KD(4):		-- shortcut
 347     Same usage as for KD(N)
 348 
 349   algebraClosure([a,b,c])
 350   coidealClosure([a,b,c])
 351   coidealAndAlgebraClosure([a,b,c])
 352   echelonForm([a,b,c], Reduced)
 353   Isomorphism KD(2N) <-> KQ(2N)
 354   The most natural isomorphism, in the G basis:
 355    KQ4::G(KD4::G([1,0])):              -- The image of a of KD4 in KQ4
 356    KD4::G(KQ4::G([0,1])):              -- The image of b of KQ4 in KD4
 357 
 358   The 8 possible isomorphisms in the M basis:
 359    phi := isomorphismKDMKQM(4, 3, TRUE)-- isomorphism KD(4)::M -> KQ(4)::M
 360    KD4::M::isHopfAlgebraMorphism(f);
 361    inv := KD4::M::inverseOfModuleMorphism(phi);
 362    KQ4::M::isHopfAlgebraMorphism(inv);
 363 
 364 
 365 A sample computation:
 366   M := KQ(4):
 367   Fbasis := coidealAndAlgebraClosure([M::e(1) + M::e(2)]):
 368   F := Dom::SubFreeModule(Fbasis, [Cat::FiniteDimensionalHopfAlgebraWithBasis(M::coeffRing)]):
 369   Fdual := Dom::DualOfFreeModule(F):
 370   G := Fdual::intrisicGroup():
 371   G::list();   // C'est le groupe dihedral D4
 372 
 373 //////////////////////////////////////////////////////////////////////
 374 >> KD3 := KD(3):
 375 >> KD3::categories
 376 
 377 [Cat::HopfAlgebraWithSeveralBases(Q(II, epsilon)),
 378 
 379    TwistedDihedralOrQuaternionGroupAlgebra(3),
 380 
 381    Cat::AlgebraWithSeveralBases(Q(II, epsilon)), Cat::Algebra(Q(II, epsilon)),
 382 
 383    Cat::ModuleWithSeveralBases(Q(II, epsilon)), Cat::Ring,
 384 
 385    Cat::Module(Q(II, epsilon)), Cat::DomainWithSeveralRepresentations, Cat::Rng,
 386 
 387    Cat::SemiRing, Cat::LeftModule(KD(3, Q(II, epsilon))),
 388 
 389    Cat::LeftModule(Q(II, epsilon)), Cat::RightModule(Q(II, epsilon)),
 390 
 391    Cat::UseOverloading, Cat::FacadeDomain, Cat::SemiRng, Cat::Monoid,
 392 
 393    Cat::AbelianGroup, Cat::SemiGroup, Cat::CancellationAbelianMonoid,
 394 
 395    Cat::AbelianMonoid, Cat::AbelianSemiGroup, Cat::Object, Cat::BaseCategory]
 396                                                                        Time: 4 msec 
 397 >> [aa,bb] := KD3::group::algebraGenerators::list()
 398 
 399                                     [B(a), B(b)]
 400                                                                        Time: 4 msec 
 401 >> bb^2
 402 
 403                                         B(1)
 404                                                                       Time: 12 msec 
 405 >> aa^2, aa^6, bb*aa
 406 
 407                                    2            5
 408                                 B(a ), B(1), B(a  b)
 409                                                                        Time: 4 msec 
 410 >> (1 - aa^3)*(bb + aa^3) + 1/2*bb*aa^3
 411 
 412                                            3            3
 413                        -1 B(1) + B(b) + B(a ) + -1/2 B(a  b)
 414                                                                       Time: 24 msec 
 415 >> KD3::M(aa + 2*bb)
 416 
 417           +-                                                           -+
 418           |  3,  0,  0, 0,    0,         0,           0,          0     |
 419           |                                                             |
 420           |  0, -1,  0, 0,    0,         0,           0,          0     |
 421           |                                                             |
 422           |  0,  0, -3, 0,    0,         0,           0,          0     |
 423           |                                                             |
 424           |  0,  0,  0, 1,    0,         0,           0,          0     |
 425           |                                                             |
 426           |  0,  0,  0, 0, epsilon,      2,           0,          0     |
 427           |                                                             |
 428           |  0,  0,  0, 0,    2,    1 - epsilon,      0,          0     |
 429           |                                                             |
 430           |  0,  0,  0, 0,    0,         0,      epsilon - 1,     2     |
 431           |                                                             |
 432           |  0,  0,  0, 0,    0,         0,           2,      -epsilon  |
 433           +-                                                           -+
 434                                                                       Time: 48 msec 
 435 >> coproduct(aa^3), coproduct(bb)
 436 
 437                                 3       3
 438                              B(a ) # B(a ), B(b) # B(b)
 439                                                                      Time: 496 msec 
 440 >> coproduct(aa)
 441 
 442                                                         2                   2
 443 7/16 B(a) # B(a) + 1/16 B(a) # B(a b) + -1/16 B(a) # B(a ) + 1/16 B(a) # B(a  b) +
 444 
 445                   4                   4                     5
 446    3/16 B(a) # B(a ) + 1/16 B(a) # B(a  b) + 3/16 B(a) # B(a ) +
 447 
 448                   5
 449    1/16 B(a) # B(a  b) + 1/16 B(a b) # B(a) + 1/16 B(a b) # B(a b) +
 450 
 451                     2                     2      /   II        \             4
 452    1/16 B(a b) # B(a ) + 1/16 B(a b) # B(a  b) + | - -- - 1/16 | B(a b) # B(a ) +
 453                                                  \    8        /
 454 
 455    / II        \             4      / II        \             5
 456    | -- - 1/16 | B(a b) # B(a  b) + | -- - 1/16 | B(a b) # B(a ) +
 457    \  8        /                    \  8        /
 458 
 459    /   II        \             5               2                   2
 460    | - -- - 1/16 | B(a b) # B(a  b) + -1/16 B(a ) # B(a) + 1/16 B(a ) # B(a b) +
 461    \    8        /
 462 
 463             2       2            2       2               2       4
 464    -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a  b) + -1/16 B(a ) # B(a ) +
 465 
 466            2       4               2       5            2       5
 467    1/16 B(a ) # B(a  b) + -1/16 B(a ) # B(a ) + 1/16 B(a ) # B(a  b) +
 468 
 469            2                     2                       2         2
 470    1/16 B(a  b) # B(a) + 1/16 B(a  b) # B(a b) + 1/16 B(a  b) # B(a ) +
 471 
 472            2         2      /   II        \    2         4
 473    1/16 B(a  b) # B(a  b) + | - -- - 1/16 | B(a  b) # B(a ) +
 474                             \    8        /
 475 
 476    /   II        \    2         4      / II        \    2         5
 477    | - -- - 1/16 | B(a  b) # B(a  b) + | -- - 1/16 | B(a  b) # B(a ) +
 478    \    8        /                     \  8        /
 479 
 480    / II        \    2         5              4
 481    | -- - 1/16 | B(a  b) # B(a  b) + 3/16 B(a ) # B(a) +
 482    \  8        /
 483 
 484    / II        \    4                      4       2
 485    | -- - 1/16 | B(a ) # B(a b) + -1/16 B(a ) # B(a ) +
 486    \  8        /
 487 
 488    / II        \    4       2              4       4
 489    | -- - 1/16 | B(a ) # B(a  b) + 3/16 B(a ) # B(a ) +
 490    \  8        /
 491 
 492    /   II        \    4       4               4       5
 493    | - -- - 1/16 | B(a ) # B(a  b) + -1/16 B(a ) # B(a ) +
 494    \    8        /
 495 
 496    /   II        \    4       5              4
 497    | - -- - 1/16 | B(a ) # B(a  b) + 1/16 B(a  b) # B(a) +
 498    \    8        /
 499 
 500    /   II        \    4                       4         2
 501    | - -- - 1/16 | B(a  b) # B(a b) + 1/16 B(a  b) # B(a ) +
 502    \    8        /
 503 
 504    / II        \    4         2      / II        \    4         4
 505    | -- - 1/16 | B(a  b) # B(a  b) + | -- - 1/16 | B(a  b) # B(a ) +
 506    \  8        /                     \  8        /
 507 
 508            4         4      /   II        \    4         5
 509    1/16 B(a  b) # B(a  b) + | - -- - 1/16 | B(a  b) # B(a ) +
 510                             \    8        /
 511 
 512            4         5              5           /   II        \    5
 513    1/16 B(a  b) # B(a  b) + 3/16 B(a ) # B(a) + | - -- - 1/16 | B(a ) # B(a b) +
 514                                                 \    8        /
 515 
 516             5       2    /   II        \    5       2               5       4
 517    -1/16 B(a ) # B(a ) + | - -- - 1/16 | B(a ) # B(a  b) + -1/16 B(a ) # B(a ) +
 518                          \    8        /
 519 
 520    / II        \    5       4              5       5
 521    | -- - 1/16 | B(a ) # B(a  b) + 3/16 B(a ) # B(a ) +
 522    \  8        /
 523 
 524    / II        \    5       5              5
 525    | -- - 1/16 | B(a ) # B(a  b) + 1/16 B(a  b) # B(a) +
 526    \  8        /
 527 
 528    / II        \    5                       5         2
 529    | -- - 1/16 | B(a  b) # B(a b) + 1/16 B(a  b) # B(a ) +
 530    \  8        /
 531 
 532    /   II        \    5         2      / II        \    5         4
 533    | - -- - 1/16 | B(a  b) # B(a  b) + | -- - 1/16 | B(a  b) # B(a ) +
 534    \    8        /                     \  8        /
 535 
 536            5         4      /   II        \    5         5            5         5
 537    1/16 B(a  b) # B(a  b) + | - -- - 1/16 | B(a  b) # B(a ) + 1/16 B(a  b) # B(a  b)
 538                             \    8        /
 539                                                                      Time: 260 msec 
 540 >> K := KD3::G:   // un simple raccourci
 541 >>  checkAntipode := K::mu @ ( K::id # K::antipode ) @ K::coproduct:
 542 >> checkAntipode(x) $ x in K::basis::list()
 543 
 544        B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1), B(1)
 545                                                                     Time: 2020 msec 
 546 >> e := KD3::e:
 547 >> K2basis := coidealAndAlgebraClosure([ e(1)+e(2) ])
 548 
 549 -- +-                        -+  +-                        -+
 550 |  |  1, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 551 |  |                          |  |                          |
 552 |  |  0, 1, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 553 |  |                          |  |                          |
 554 |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 1, 0, 0, 0, 0, 0  |
 555 |  |                          |  |                          |
 556 |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 1, 0, 0, 0, 0  |
 557 |  |                          |, |                          |,
 558 |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 559 |  |                          |  |                          |
 560 |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 561 |  |                          |  |                          |
 562 |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 563 |  |                          |  |                          |
 564 |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 565 -- +-                        -+  +-                        -+
 566 
 567    +-                         -+  +-                        -+
 568    |  0, 0, 0, 0, 0,  0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 569    |                           |  |                          |
 570    |  0, 0, 0, 0, 0,  0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 571    |                           |  |                          |
 572    |  0, 0, 0, 0, 0,  0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 573    |                           |  |                          |
 574    |  0, 0, 0, 0, 0,  0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 575    |                           |, |                          |,
 576    |  0, 0, 0, 0, 0, -1, 0, 0  |  |  0, 0, 0, 0, 1, 0, 0, 0  |
 577    |                           |  |                          |
 578    |  0, 0, 0, 0, 1,  0, 0, 0  |  |  0, 0, 0, 0, 0, 1, 0, 0  |
 579    |                           |  |                          |
 580    |  0, 0, 0, 0, 0,  0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 581    |                           |  |                          |
 582    |  0, 0, 0, 0, 0,  0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |
 583    +-                         -+  +-                        -+
 584 
 585    +-                        -+  +-                        -+ --
 586    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 587    |                          |  |                          |  |
 588    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 589    |                          |  |                          |  |
 590    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 591    |                          |  |                          |  |
 592    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 593    |                          |, |                          |  |
 594    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 595    |                          |  |                          |  |
 596    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 597    |                          |  |                          |  |
 598    |  0, 0, 0, 0, 0, 0, 1, 0  |  |  0, 0, 0, 0, 0, 0, 0, 0  |  |
 599    |                          |  |                          |  |
 600    |  0, 0, 0, 0, 0, 0, 0, 0  |  |  0, 0, 0, 0, 0, 0, 0, 1  |  |
 601    +-                        -+  +-                        -+ --
 602                                                                    Time: 13601 msec 
 603 >> K2 := Dom::SubFreeModule(K2basis,
 604 >>      [Cat::FiniteDimensionalHopfAlgebraWithBasis(KD3::coeffRing)]):
 605 >> K2::isCommutative(), K2::isCocommutative()
 606 
 607                                     TRUE, FALSE
 608                                                                       Time: 56 msec 
 609 >> K2dual := K2::Dual():
 610 >> K2dual::groupLikeElements()
 611 
 612  _          _          _          _              _           _
 613 [B([3, 3]), B([1, 1]), B([8, 8]), B([7, 7]), -II B([6, 5]) + B([5, 5]),
 614 
 615       _           _
 616    II B([6, 5]) + B([5, 5])]
 617                                                                     Time: 1236 msec 
 618 >> G := K2dual::intrinsicGroup():
 619 >>  G::list()
 620 
 621                      [[], [1], [1, 1], [2], [1, 2], [1, 1, 2]]
 622                                                                        Time: 0 msec 
 623 >> K2dual::isSemiSimple()
 624 
 625                                         TRUE
 626                                                                       Time: 56 msec 
 627 >> K2dual::simpleModulesDimensions()
 628 
 629                                      [2, 1, 1]
 630                                                                      Time: 704 msec 

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