/usr/lib/open-axiom/src/algebra/naalgc.spad is in open-axiom-source 1.4.1+svn~2626-2ubuntu2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 | --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev category MONAD Monad
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: *, **
++ Related Constructors: SemiGroup, Monoid, MonadWithUnit
++ Also See:
++ AMS Classifications:
++ Keywords: Monad, binary operation
++ Reference:
++ N. Jacobson: Structure and Representations of Jordan Algebras
++ AMS, Providence, 1968
++ Description:
++ Monad is the class of all multiplicative monads, i.e. sets
++ with a binary operation.
Monad(): Category == SetCategory with
--operations
*: (%,%) -> %
++ a*b is the product of \spad{a} and b in a set with
++ a binary operation.
rightPower: (%,PositiveInteger) -> %
++ rightPower(a,n) returns the \spad{n}-th right power of \spad{a},
++ i.e. \spad{rightPower(a,n) := rightPower(a,n-1) * a} and
++ \spad{rightPower(a,1) := a}.
leftPower: (%,PositiveInteger) -> %
++ leftPower(a,n) returns the \spad{n}-th left power of \spad{a},
++ i.e. \spad{leftPower(a,n) := a * leftPower(a,n-1)} and
++ \spad{leftPower(a,1) := a}.
**: (%,PositiveInteger) -> %
++ a**n returns the \spad{n}-th power of \spad{a},
++ defined by repeated squaring.
add
import RepeatedSquaring(%)
x:% ** n:PositiveInteger == expt(x,n)
rightPower(a,n) ==
one? n => a
res := a
for i in 1..(n-1) repeat res := res * a
res
leftPower(a,n) ==
one? n => a
res := a
for i in 1..(n-1) repeat res := a * res
res
)abbrev category MONADWU MonadWithUnit
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: *, **, 1
++ Related Constructors: SemiGroup, Monoid, Monad
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Keywords: Monad with unit, binary operation
++ Reference:
++ N. Jacobson: Structure and Representations of Jordan Algebras
++ AMS, Providence, 1968
++ Description:
++ MonadWithUnit is the class of multiplicative monads with unit,
++ i.e. sets with a binary operation and a unit element.
++ Axioms
++ leftIdentity("*":(%,%)->%,1) \tab{30} 1*x=x
++ rightIdentity("*":(%,%)->%,1) \tab{30} x*1=x
++ Common Additional Axioms
++ unitsKnown---if "recip" says "failed", that PROVES input wasn't a unit
MonadWithUnit(): Category == Monad with
--constants
1: constant -> %
++ 1 returns the unit element, denoted by 1.
--operations
one?: % -> Boolean
++ one?(a) tests whether \spad{a} is the unit 1.
rightPower: (%,NonNegativeInteger) -> %
++ rightPower(a,n) returns the \spad{n}-th right power of \spad{a},
++ i.e. \spad{rightPower(a,n) := rightPower(a,n-1) * a} and
++ \spad{rightPower(a,0) := 1}.
leftPower: (%,NonNegativeInteger) -> %
++ leftPower(a,n) returns the \spad{n}-th left power of \spad{a},
++ i.e. \spad{leftPower(a,n) := a * leftPower(a,n-1)} and
++ \spad{leftPower(a,0) := 1}.
"**": (%,NonNegativeInteger) -> %
++ \spad{a**n} returns the \spad{n}-th power of \spad{a},
++ defined by repeated squaring.
recip: % -> Union(%,"failed")
++ recip(a) returns an element, which is both a left and a right
++ inverse of \spad{a},
++ or \spad{"failed"} if such an element doesn't exist or cannot
++ be determined (see unitsKnown).
leftRecip: % -> Union(%,"failed")
++ leftRecip(a) returns an element, which is a left inverse of \spad{a},
++ or \spad{"failed"} if such an element doesn't exist or cannot
++ be determined (see unitsKnown).
rightRecip: % -> Union(%,"failed")
++ rightRecip(a) returns an element, which is a right inverse of
++ \spad{a}, or \spad{"failed"} if such an element doesn't exist
++ or cannot be determined (see unitsKnown).
add
import RepeatedSquaring(%)
one? x == x = 1
x:% ** n:NonNegativeInteger ==
zero? n => 1
expt(x,n pretend PositiveInteger)
rightPower(a: %,n: NonNegativeInteger) ==
zero? n => 1
res: % := 1
for i in 1..n repeat res := res * a
res
leftPower(a: %,n: NonNegativeInteger) ==
zero? n => 1
res: % := 1
for i in 1..n repeat res := a * res
res
)abbrev category NARNG NonAssociativeRng
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 03 July 1991
++ Basic Operations: +, *, -, **
++ Related Constructors: Rng, Ring, NonAssociativeRing
++ Also See:
++ AMS Classifications:
++ Keywords: not associative ring
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ NonAssociativeRng is a basic ring-type structure, not necessarily
++ commutative or associative, and not necessarily with unit.
++ Axioms
++ x*(y+z) = x*y + x*z
++ (x+y)*z = x*z + y*z
++ Common Additional Axioms
++ noZeroDivisors ab = 0 => a=0 or b=0
NonAssociativeRng(): Category == Join(AbelianGroup,Monad) with
associator: (%,%,%) -> %
++ associator(a,b,c) returns \spad{(a*b)*c-a*(b*c)}.
commutator: (%,%) -> %
++ commutator(a,b) returns \spad{a*b-b*a}.
antiCommutator: (%,%) -> %
++ antiCommutator(a,b) returns \spad{a*b+b*a}.
add
associator(x,y,z) == (x*y)*z - x*(y*z)
commutator(x,y) == x*y - y*x
antiCommutator(x,y) == x*y + y*x
)abbrev category NASRING NonAssociativeRing
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +, *, -, **
++ Related Constructors: NonAssociativeRng, Rng, Ring
++ Also See:
++ AMS Classifications:
++ Keywords: non-associative ring with unit
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ A NonAssociativeRing is a non associative rng which has a unit,
++ the multiplication is not necessarily commutative or associative.
NonAssociativeRing(): Category == Join(NonAssociativeRng,MonadWithUnit) with
--operations
characteristic: NonNegativeInteger
++ characteristic() returns the characteristic of the ring.
--we can not make this a constant, since some domains are mutable
coerce: Integer -> %
++ coerce(n) coerces the integer n to an element of the ring.
add
n:Integer
coerce(n) == n * 1$%
)abbrev category NAALG NonAssociativeAlgebra
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +, -, *, **
++ Related Constructors: Algebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ NonAssociativeAlgebra is the category of non associative algebras
++ (modules which are themselves non associative rngs).
++ Axioms
++ r*(a*b) = (r*a)*b = a*(r*b)
NonAssociativeAlgebra(R:CommutativeRing): Category == _
Join(NonAssociativeRng, Module R) with
--operations
plenaryPower : (%,PositiveInteger) -> %
++ plenaryPower(a,n) is recursively defined to be
++ \spad{plenaryPower(a,n-1)*plenaryPower(a,n-1)} for \spad{n>1}
++ and \spad{a} for \spad{n=1}.
add
plenaryPower(a,n) ==
one? n => a
n1 : PositiveInteger := (n-1)::NonNegativeInteger::PositiveInteger
plenaryPower(a,n1) * plenaryPower(a,n1)
)abbrev category FINAALG FiniteRankNonAssociativeAlgebra
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 12 June 1991
++ Basic Operations: +,-,*,**, someBasis
++ Related Constructors: FramedNonAssociativeAlgebra, FramedAlgebra,
++ FiniteRankAssociativeAlgebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra, basis
++ References:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++
++ R. Wisbauer: Bimodule Structure of Algebra
++ Lecture Notes Univ. Duesseldorf 1991
++ Description:
++ A FiniteRankNonAssociativeAlgebra is a non associative algebra over
++ a commutative ring R which is a free \spad{R}-module of finite rank.
FiniteRankNonAssociativeAlgebra(R:CommutativeRing):
Category == NonAssociativeAlgebra R with
someBasis : () -> Vector %
++ someBasis() returns some \spad{R}-module basis.
rank : () -> PositiveInteger
++ rank() returns the rank of the algebra as \spad{R}-module.
conditionsForIdempotents: Vector % -> List Polynomial R
++ conditionsForIdempotents([v1,...,vn]) determines a complete list
++ of polynomial equations for the coefficients of idempotents
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
structuralConstants: Vector % -> Vector Matrix R
++ structuralConstants([v1,v2,...,vm]) calculates the structural
++ constants \spad{[(gammaijk) for k in 1..m]} defined by
++ \spad{vi * vj = gammaij1 * v1 + ... + gammaijm * vm},
++ where \spad{[v1,...,vm]} is an \spad{R}-module basis
++ of a subalgebra.
leftRegularRepresentation: (% , Vector %) -> Matrix R
++ leftRegularRepresentation(a,[v1,...,vn]) returns the matrix of
++ the linear map defined by left multiplication by \spad{a}
++ with respect to the \spad{R}-module basis \spad{[v1,...,vn]}.
rightRegularRepresentation: (% , Vector %) -> Matrix R
++ rightRegularRepresentation(a,[v1,...,vn]) returns the matrix of
++ the linear map defined by right multiplication by \spad{a}
++ with respect to the \spad{R}-module basis \spad{[v1,...,vn]}.
leftTrace: % -> R
++ leftTrace(a) returns the trace of the left regular representation
++ of \spad{a}.
rightTrace: % -> R
++ rightTrace(a) returns the trace of the right regular representation
++ of \spad{a}.
leftNorm: % -> R
++ leftNorm(a) returns the determinant of the left regular representation
++ of \spad{a}.
rightNorm: % -> R
++ rightNorm(a) returns the determinant of the right regular
++ representation of \spad{a}.
coordinates: (%, Vector %) -> Vector R
++ coordinates(a,[v1,...,vn]) returns the coordinates of \spad{a}
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
coordinates: (Vector %, Vector %) -> Matrix R
++ coordinates([a1,...,am],[v1,...,vn]) returns a matrix whose
++ i-th row is formed by the coordinates of \spad{ai}
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
represents: (Vector R, Vector %) -> %
++ represents([a1,...,am],[v1,...,vm]) returns the linear
++ combination \spad{a1*vm + ... + an*vm}.
leftDiscriminant: Vector % -> R
++ leftDiscriminant([v1,...,vn]) returns the determinant of the
++ \spad{n}-by-\spad{n} matrix whose element at the \spad{i}-th row
++ and \spad{j}-th column is given by the left trace of the product
++ \spad{vi*vj}.
++ Note: the same as \spad{determinant(leftTraceMatrix([v1,...,vn]))}.
rightDiscriminant: Vector % -> R
++ rightDiscriminant([v1,...,vn]) returns the determinant of the
++ \spad{n}-by-\spad{n} matrix whose element at the \spad{i}-th row
++ and \spad{j}-th column is given by the right trace of the product
++ \spad{vi*vj}.
++ Note: the same as \spad{determinant(rightTraceMatrix([v1,...,vn]))}.
leftTraceMatrix: Vector % -> Matrix R
++ leftTraceMatrix([v1,...,vn]) is the \spad{n}-by-\spad{n} matrix
++ whose element at the \spad{i}-th row and \spad{j}-th column is given
++ by the left trace of the product \spad{vi*vj}.
rightTraceMatrix: Vector % -> Matrix R
++ rightTraceMatrix([v1,...,vn]) is the \spad{n}-by-\spad{n} matrix
++ whose element at the \spad{i}-th row and \spad{j}-th column is given
++ by the right trace of the product \spad{vi*vj}.
leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
++ leftCharacteristicPolynomial(a) returns the characteristic
++ polynomial of the left regular representation of \spad{a}
++ with respect to any basis.
rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
++ rightCharacteristicPolynomial(a) returns the characteristic
++ polynomial of the right regular representation of \spad{a}
++ with respect to any basis.
--we not necessarily have a unit
--if R has CharacteristicZero then CharacteristicZero
--if R has CharacteristicNonZero then CharacteristicNonZero
commutative?:()-> Boolean
++ commutative?() tests if multiplication in the algebra
++ is commutative.
antiCommutative?:()-> Boolean
++ antiCommutative?() tests if \spad{a*a = 0}
++ for all \spad{a} in the algebra.
++ Note: this implies \spad{a*b + b*a = 0} for all \spad{a} and \spad{b}.
associative?:()-> Boolean
++ associative?() tests if multiplication in algebra
++ is associative.
antiAssociative?:()-> Boolean
++ antiAssociative?() tests if multiplication in algebra
++ is anti-associative, i.e. \spad{(a*b)*c + a*(b*c) = 0}
++ for all \spad{a},b,c in the algebra.
leftAlternative?: ()-> Boolean
++ leftAlternative?() tests if \spad{2*associator(a,a,b) = 0}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
rightAlternative?: ()-> Boolean
++ rightAlternative?() tests if \spad{2*associator(a,b,b) = 0}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
flexible?: ()-> Boolean
++ flexible?() tests if \spad{2*associator(a,b,a) = 0}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
alternative?: ()-> Boolean
++ alternative?() tests if
++ \spad{2*associator(a,a,b) = 0 = 2*associator(a,b,b)}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
powerAssociative?:()-> Boolean
++ powerAssociative?() tests if all subalgebras
++ generated by a single element are associative.
jacobiIdentity?:() -> Boolean
++ jacobiIdentity?() tests if \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0}
++ for all \spad{a},b,c in the algebra. For example, this holds
++ for crossed products of 3-dimensional vectors.
lieAdmissible?: () -> Boolean
++ lieAdmissible?() tests if the algebra defined by the commutators
++ is a Lie algebra, i.e. satisfies the Jacobi identity.
++ The property of anticommutativity follows from definition.
jordanAdmissible?: () -> Boolean
++ jordanAdmissible?() tests if 2 is invertible in the
++ coefficient domain and the multiplication defined by
++ \spad{(1/2)(a*b+b*a)} determines a
++ Jordan algebra, i.e. satisfies the Jordan identity.
++ The property of \spadatt{commutative("*")}
++ follows from by definition.
noncommutativeJordanAlgebra?: () -> Boolean
++ noncommutativeJordanAlgebra?() tests if the algebra
++ is flexible and Jordan admissible.
jordanAlgebra?:() -> Boolean
++ jordanAlgebra?() tests if the algebra is commutative,
++ characteristic is not 2, and \spad{(a*b)*a**2 - a*(b*a**2) = 0}
++ for all \spad{a},b,c in the algebra (Jordan identity).
++ Example:
++ for every associative algebra \spad{(A,+,@)} we can construct a
++ Jordan algebra \spad{(A,+,*)}, where \spad{a*b := (a@b+b@a)/2}.
lieAlgebra?:() -> Boolean
++ lieAlgebra?() tests if the algebra is anticommutative
++ and \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0}
++ for all \spad{a},b,c in the algebra (Jacobi identity).
++ Example:
++ for every associative algebra \spad{(A,+,@)} we can construct a
++ Lie algebra \spad{(A,+,*)}, where \spad{a*b := a@b-b@a}.
if R has IntegralDomain then
-- we not neccessarily have a unit, hence we don't inherit
-- the next 3 functions anc hence copy them from MonadWithUnit:
recip: % -> Union(%,"failed")
++ recip(a) returns an element, which is both a left and a right
++ inverse of \spad{a},
++ or \spad{"failed"} if there is no unit element, if such an
++ element doesn't exist or cannot be determined (see unitsKnown).
leftRecip: % -> Union(%,"failed")
++ leftRecip(a) returns an element, which is a left inverse of \spad{a},
++ or \spad{"failed"} if there is no unit element, if such an
++ element doesn't exist or cannot be determined (see unitsKnown).
rightRecip: % -> Union(%,"failed")
++ rightRecip(a) returns an element, which is a right inverse of
++ \spad{a},
++ or \spad{"failed"} if there is no unit element, if such an
++ element doesn't exist or cannot be determined (see unitsKnown).
associatorDependence:() -> List Vector R
++ associatorDependence() looks for the associator identities, i.e.
++ finds a basis of the solutions of the linear combinations of the
++ six permutations of \spad{associator(a,b,c)} which yield 0,
++ for all \spad{a},b,c in the algebra.
++ The order of the permutations is \spad{123 231 312 132 321 213}.
leftMinimalPolynomial : % -> SparseUnivariatePolynomial R
++ leftMinimalPolynomial(a) returns the polynomial determined by the
++ smallest non-trivial linear combination of left powers of \spad{a}.
++ Note: the polynomial never has a constant term as in general
++ the algebra has no unit.
rightMinimalPolynomial : % -> SparseUnivariatePolynomial R
++ rightMinimalPolynomial(a) returns the polynomial determined by the
++ smallest non-trivial linear
++ combination of right powers of \spad{a}.
++ Note: the polynomial never has a constant term as in general
++ the algebra has no unit.
leftUnits:() -> Union(Record(particular: %, basis: List %), "failed")
++ leftUnits() returns the affine space of all left units of the
++ algebra, or \spad{"failed"} if there is none.
rightUnits:() -> Union(Record(particular: %, basis: List %), "failed")
++ rightUnits() returns the affine space of all right units of the
++ algebra, or \spad{"failed"} if there is none.
leftUnit:() -> Union(%, "failed")
++ leftUnit() returns a left unit of the algebra
++ (not necessarily unique), or \spad{"failed"} if there is none.
rightUnit:() -> Union(%, "failed")
++ rightUnit() returns a right unit of the algebra
++ (not necessarily unique), or \spad{"failed"} if there is none.
unit:() -> Union(%, "failed")
++ unit() returns a unit of the algebra (necessarily unique),
++ or \spad{"failed"} if there is none.
-- we not necessarily have a unit, hence we can't say anything
-- about characteristic
-- if R has CharacteristicZero then CharacteristicZero
-- if R has CharacteristicNonZero then CharacteristicNonZero
unitsKnown
++ unitsKnown means that \spadfun{recip} truly yields reciprocal
++ or \spad{"failed"} if not a unit,
++ similarly for \spadfun{leftRecip} and
++ \spadfun{rightRecip}. The reason is that we use left, respectively
++ right, minimal polynomials to decide this question.
add
--n := rank()
--b := someBasis()
--gamma : Vector Matrix R := structuralConstants b
-- here is a problem: there seems to be a problem having local
-- variables in the capsule of a category, furthermore
-- see the commented code of conditionsForIdempotents, where
-- we call structuralConstants, which also doesn't work
-- at runtime, i.e. is not properly inherited, hence for
-- the moment we put the code for
-- conditionsForIdempotents, structuralConstants, unit, leftUnit,
-- rightUnit into the domain constructor ALGSC
V ==> Vector
M ==> Matrix
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
SUP ==> SparseUnivariatePolynomial
NNI ==> NonNegativeInteger
-- next 2 functions: use a general characteristicPolynomial
leftCharacteristicPolynomial a ==
n := rank()$%
ma : Matrix R := leftRegularRepresentation(a,someBasis()$%)
mb : Matrix SUP R := zero(n,n)
for i in 1..n repeat
for j in 1..n repeat
mb(i,j):=
i=j => monomial(ma(i,j),0)$SUP(R) - monomial(1,1)$SUP(R)
monomial(ma(i,j),1)$SUP(R)
determinant mb
rightCharacteristicPolynomial a ==
n := rank()$%
ma : Matrix R := rightRegularRepresentation(a,someBasis()$%)
mb : Matrix SUP R := zero(n,n)
for i in 1..n repeat
for j in 1..n repeat
mb(i,j):=
i=j => monomial(ma(i,j),0)$SUP(R) - monomial(1,1)$SUP(R)
monomial(ma(i,j),1)$SUP(R)
determinant mb
leftTrace a ==
t : R := 0
ma : Matrix R := leftRegularRepresentation(a,someBasis()$%)
for i in 1..rank()$% repeat
t := t + elt(ma,i,i)
t
rightTrace a ==
t : R := 0
ma : Matrix R := rightRegularRepresentation(a,someBasis()$%)
for i in 1..rank()$% repeat
t := t + elt(ma,i,i)
t
leftNorm a == determinant leftRegularRepresentation(a,someBasis()$%)
rightNorm a == determinant rightRegularRepresentation(a,someBasis()$%)
antiAssociative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? ( (b.i*b.j)*b.k + b.i*(b.j*b.k) ) =>
messagePrint("algebra is not anti-associative")$OutputForm
return false
messagePrint("algebra is anti-associative")$OutputForm
true
jordanAdmissible?() ==
b := someBasis()
n := rank()
recip(2 * 1$R) case "failed" =>
messagePrint("this algebra is not Jordan admissible, as 2 is not invertible in the ground ring")$OutputForm
false
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for l in 1..n repeat
not zero? ( _
antiCommutator(antiCommutator(b.i,b.j),antiCommutator(b.l,b.k)) + _
antiCommutator(antiCommutator(b.l,b.j),antiCommutator(b.k,b.i)) + _
antiCommutator(antiCommutator(b.k,b.j),antiCommutator(b.i,b.l)) _
) =>
messagePrint("this algebra is not Jordan admissible")$OutputForm
return false
messagePrint("this algebra is Jordan admissible")$OutputForm
true
lieAdmissible?() ==
n := rank()
b := someBasis()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (commutator(commutator(b.i,b.j),b.k) _
+ commutator(commutator(b.j,b.k),b.i) _
+ commutator(commutator(b.k,b.i),b.j)) =>
messagePrint("this algebra is not Lie admissible")$OutputForm
return false
messagePrint("this algebra is Lie admissible")$OutputForm
true
-- conditionsForIdempotents b ==
-- n := rank()
-- gamma : Vector Matrix R := structuralConstants b
-- listOfNumbers : List String := [string(q)$String for q in 1..n]
-- symbolsForCoef : Vector Symbol :=
-- [concat("%", concat("x", i))::Symbol for i in listOfNumbers]
-- conditions : List Polynomial R := []
-- for k in 1..n repeat
-- xk := symbolsForCoef.k
-- p : Polynomial R := monomial( - 1$Polynomial(R), [xk], [1] )
-- for i in 1..n repeat
-- for j in 1..n repeat
-- xi := symbolsForCoef.i
-- xj := symbolsForCoef.j
-- p := p + monomial(_
-- elt((gamma.k),i,j) :: Polynomial(R), [xi,xj], [1,1])
-- conditions := cons(p,conditions)
-- conditions
structuralConstants b ==
--n := rank()
-- be careful with the possibility that b is not a basis
m : NonNegativeInteger := (maxIndex b) :: NonNegativeInteger
sC : Vector Matrix R := [new(m,m,0$R) for k in 1..m]
for i in 1..m repeat
for j in 1..m repeat
covec : Vector R := coordinates(b.i * b.j, b)
for k in 1..m repeat
setelt( sC.k, i, j, covec.k )
sC
if R has IntegralDomain then
leftRecip x ==
zero? x => "failed"
lu := leftUnit()
lu case "failed" => "failed"
b := someBasis()
xx : % := (lu :: %)
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
listOfPowers : List % := [xx]
while rank(cond) = k repeat
k := k+1
xx := xx*x
listOfPowers := cons(xx,listOfPowers)
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
invC := recip vectorOfCoef.1
invC case "failed" => "failed"
invCR : R := - (invC :: R)
reduce(_+,[(invCR*vectorOfCoef.i)*power for i in _
2..maxIndex vectorOfCoef for power in reverse listOfPowers])
rightRecip x ==
zero? x => "failed"
ru := rightUnit()
ru case "failed" => "failed"
b := someBasis()
xx : % := (ru :: %)
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
listOfPowers : List % := [xx]
while rank(cond) = k repeat
k := k+1
xx := x*xx
listOfPowers := cons(xx,listOfPowers)
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
invC := recip vectorOfCoef.1
invC case "failed" => "failed"
invCR : R := - (invC :: R)
reduce(_+,[(invCR*vectorOfCoef.i)*power for i in _
2..maxIndex vectorOfCoef for power in reverse listOfPowers])
recip x ==
lrx := leftRecip x
lrx case "failed" => "failed"
rrx := rightRecip x
rrx case "failed" => "failed"
(lrx :: %) ~= (rrx :: %) => "failed"
lrx :: %
leftMinimalPolynomial x ==
zero? x => monomial(1$R,1)$(SparseUnivariatePolynomial R)
b := someBasis()
xx : % := x
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
while rank(cond) = k repeat
k := k+1
xx := x*xx
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
res : SparseUnivariatePolynomial R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial R)
res
rightMinimalPolynomial x ==
zero? x => monomial(1$R,1)$(SparseUnivariatePolynomial R)
b := someBasis()
xx : % := x
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
while rank(cond) = k repeat
k := k+1
xx := xx*x
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
res : SparseUnivariatePolynomial R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial R)
res
associatorDependence() ==
n := rank()
b := someBasis()
cond : Matrix(R) := new(n**4,6,0$R)$Matrix(R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
a123 : Vector R := coordinates(associator(b.i,b.j,b.k),b)
a231 : Vector R := coordinates(associator(b.j,b.k,b.i),b)
a312 : Vector R := coordinates(associator(b.k,b.i,b.j),b)
a132 : Vector R := coordinates(associator(b.i,b.k,b.j),b)
a321 : Vector R := coordinates(associator(b.k,b.j,b.i),b)
a213 : Vector R := coordinates(associator(b.j,b.i,b.k),b)
for r in 1..n repeat
z:= z+1
setelt(cond,z,1,elt(a123,r))
setelt(cond,z,2,elt(a231,r))
setelt(cond,z,3,elt(a312,r))
setelt(cond,z,4,elt(a132,r))
setelt(cond,z,5,elt(a321,r))
setelt(cond,z,6,elt(a213,r))
nullSpace(cond)
jacobiIdentity?() ==
n := rank()
b := someBasis()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? ((b.i*b.j)*b.k + (b.j*b.k)*b.i + (b.k*b.i)*b.j) =>
messagePrint("Jacobi identity does not hold")$OutputForm
return false
messagePrint("Jacobi identity holds")$OutputForm
true
lieAlgebra?() ==
not antiCommutative?() =>
messagePrint("this is not a Lie algebra")$OutputForm
false
not jacobiIdentity?() =>
messagePrint("this is not a Lie algebra")$OutputForm
false
messagePrint("this is a Lie algebra")$OutputForm
true
jordanAlgebra?() ==
b := someBasis()
n := rank()
recip(2 * 1$R) case "failed" =>
messagePrint("this is not a Jordan algebra, as 2 is not invertible in the ground ring")$OutputForm
false
not commutative?() =>
messagePrint("this is not a Jordan algebra")$OutputForm
false
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for l in 1..n repeat
not zero? (associator(b.i,b.j,b.l*b.k)+_
associator(b.l,b.j,b.k*b.i)+associator(b.k,b.j,b.i*b.l)) =>
messagePrint("not a Jordan algebra")$OutputForm
return false
messagePrint("this is a Jordan algebra")$OutputForm
true
noncommutativeJordanAlgebra?() ==
b := someBasis()
n := rank()
recip(2 * 1$R) case "failed" =>
messagePrint("this is not a noncommutative Jordan algebra, as 2 is not invertible in the ground ring")$OutputForm
false
not flexible?()$% =>
messagePrint("this is not a noncommutative Jordan algebra, as it is not flexible")$OutputForm
false
not jordanAdmissible?()$% =>
messagePrint("this is not a noncommutative Jordan algebra, as it is not Jordan admissible")$OutputForm
false
messagePrint("this is a noncommutative Jordan algebra")$OutputForm
true
antiCommutative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in i..n repeat
not zero? (i=j => b.i*b.i; b.i*b.j + b.j*b.i) =>
messagePrint("algebra is not anti-commutative")$OutputForm
return false
messagePrint("algebra is anti-commutative")$OutputForm
true
commutative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in i+1..n repeat
not zero? commutator(b.i,b.j) =>
messagePrint("algebra is not commutative")$OutputForm
return false
messagePrint("algebra is commutative")$OutputForm
true
associative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? associator(b.i,b.j,b.k) =>
messagePrint("algebra is not associative")$OutputForm
return false
messagePrint("algebra is associative")$OutputForm
true
leftAlternative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.j,b.i,b.k)) =>
messagePrint("algebra is not left alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,a,b) = 0")$OutputForm
true
rightAlternative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.i,b.k,b.j)) =>
messagePrint("algebra is not right alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,b) = 0")$OutputForm
true
flexible?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.k,b.j,b.i)) =>
messagePrint("algebra is not flexible")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,a) = 0")$OutputForm
true
alternative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.j,b.i,b.k)) =>
messagePrint("algebra is not alternative")$OutputForm
return false
not zero? (associator(b.i,b.j,b.k) + associator(b.i,b.k,b.j)) =>
messagePrint("algebra is not alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0")$OutputForm
true
leftDiscriminant v == determinant leftTraceMatrix v
rightDiscriminant v == determinant rightTraceMatrix v
coordinates(v:Vector %, b:Vector %) ==
m := new(#v, #b, 0)$Matrix(R)
for i in minIndex v .. maxIndex v for j in minRowIndex m .. repeat
setRow!(m, j, coordinates(qelt(v, i), b))
m
represents(v, b) ==
m := minIndex v - 1
reduce(_+,[v(i+m) * b(i+m) for i in 1..maxIndex b])
leftTraceMatrix v ==
matrix [[leftTrace(v.i*v.j) for j in minIndex v..maxIndex v]$List(R)
for i in minIndex v .. maxIndex v]$List(List R)
rightTraceMatrix v ==
matrix [[rightTrace(v.i*v.j) for j in minIndex v..maxIndex v]$List(R)
for i in minIndex v .. maxIndex v]$List(List R)
leftRegularRepresentation(x, b) ==
m := minIndex b - 1
matrix
[parts coordinates(x*b(i+m),b) for i in 1..rank()]$List(List R)
rightRegularRepresentation(x, b) ==
m := minIndex b - 1
matrix
[parts coordinates(b(i+m)*x,b) for i in 1..rank()]$List(List R)
)abbrev category FRNAALG FramedNonAssociativeAlgebra
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +,-,*,**,basis
++ Related Constructors: FiniteRankNonAssociativeAlgebra, FramedAlgebra,
++ FiniteRankAssociativeAlgebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra, basis
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ FramedNonAssociativeAlgebra(R) is a
++ \spadtype{FiniteRankNonAssociativeAlgebra} (i.e. a non associative
++ algebra over R which is a free \spad{R}-module of finite rank)
++ over a commutative ring R together with a fixed \spad{R}-module basis.
FramedNonAssociativeAlgebra(R:CommutativeRing): Category == _
Join(FiniteRankNonAssociativeAlgebra(R),Eltable(Integer,R)) with
--operations
basis: () -> Vector %
++ basis() returns the fixed \spad{R}-module basis.
coordinates: % -> Vector R
++ coordinates(a) returns the coordinates of \spad{a}
++ with respect to the
++ fixed \spad{R}-module basis.
coordinates: Vector % -> Matrix R
++ coordinates([a1,...,am]) returns a matrix whose i-th row
++ is formed by the coordinates of \spad{ai} with respect to the
++ fixed \spad{R}-module basis.
structuralConstants:() -> Vector Matrix R
++ structuralConstants() calculates the structural constants
++ \spad{[(gammaijk) for k in 1..rank()]} defined by
++ \spad{vi * vj = gammaij1 * v1 + ... + gammaijn * vn},
++ where \spad{v1},...,\spad{vn} is the fixed \spad{R}-module basis.
conditionsForIdempotents: () -> List Polynomial R
++ conditionsForIdempotents() determines a complete list
++ of polynomial equations for the coefficients of idempotents
++ with respect to the fixed \spad{R}-module basis.
represents: Vector R -> %
++ represents([a1,...,an]) returns \spad{a1*v1 + ... + an*vn},
++ where \spad{v1}, ..., \spad{vn} are the elements of the
++ fixed \spad{R}-module basis.
convert: % -> Vector R
++ convert(a) returns the coordinates of \spad{a} with respect to the
++ fixed \spad{R}-module basis.
convert: Vector R -> %
++ convert([a1,...,an]) returns \spad{a1*v1 + ... + an*vn},
++ where \spad{v1}, ..., \spad{vn} are the elements of the
++ fixed \spad{R}-module basis.
leftDiscriminant : () -> R
++ leftDiscriminant() returns the
++ determinant of the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by the left trace of the product \spad{vi*vj}, where
++ \spad{v1},...,\spad{vn} are the
++ elements of the fixed \spad{R}-module basis.
++ Note: the same as \spad{determinant(leftTraceMatrix())}.
rightDiscriminant : () -> R
++ rightDiscriminant() returns the determinant of the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by the right trace of the product \spad{vi*vj}, where
++ \spad{v1},...,\spad{vn} are the elements of
++ the fixed \spad{R}-module basis.
++ Note: the same as \spad{determinant(rightTraceMatrix())}.
leftTraceMatrix : () -> Matrix R
++ leftTraceMatrix() is the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by left trace of the product \spad{vi*vj},
++ where \spad{v1},...,\spad{vn} are the
++ elements of the fixed \spad{R}-module
++ basis.
rightTraceMatrix : () -> Matrix R
++ rightTraceMatrix() is the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by the right trace of the product \spad{vi*vj}, where
++ \spad{v1},...,\spad{vn} are the elements
++ of the fixed \spad{R}-module basis.
leftRegularRepresentation : % -> Matrix R
++ leftRegularRepresentation(a) returns the matrix of the linear
++ map defined by left multiplication by \spad{a} with respect
++ to the fixed \spad{R}-module basis.
rightRegularRepresentation : % -> Matrix R
++ rightRegularRepresentation(a) returns the matrix of the linear
++ map defined by right multiplication by \spad{a} with respect
++ to the fixed \spad{R}-module basis.
if R has Field then
leftRankPolynomial : () -> SparseUnivariatePolynomial Polynomial R
++ leftRankPolynomial() calculates the left minimal polynomial
++ of the generic element in the algebra,
++ defined by the same structural
++ constants over the polynomial ring in symbolic coefficients with
++ respect to the fixed basis.
rightRankPolynomial : () -> SparseUnivariatePolynomial Polynomial R
++ rightRankPolynomial() calculates the right minimal polynomial
++ of the generic element in the algebra,
++ defined by the same structural
++ constants over the polynomial ring in symbolic coefficients with
++ respect to the fixed basis.
apply: (Matrix R, %) -> %
++ apply(m,a) defines a left operation of n by n matrices
++ where n is the rank of the algebra in terms of matrix-vector
++ multiplication, this is a substitute for a left module structure.
++ Error: if shape of matrix doesn't fit.
--attributes
--attributes
--separable <=> discriminant() ~= 0
add
V ==> Vector
M ==> Matrix
P ==> Polynomial
F ==> Fraction
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
CVMP ==> CoerceVectorMatrixPackage(R)
--GA ==> GenericNonAssociativeAlgebra(R,rank()$%,_
-- [random()$Character :: String :: Symbol for i in 1..rank()$%], _
-- structuralConstants()$%)
--y : GA := generic()
if R has Field then
leftRankPolynomial() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
listOfNumbers : List String := [string(q)$String for q in 1..n]
symbolsForCoef : Vector Symbol :=
[concat("%", concat("x", i))::Symbol for i in listOfNumbers]
xx : M P R := new(1,n,0)
mo : P R
x : M P R := new(1,n,0)
for i in 1..n repeat
mo := monomial(1, [symbolsForCoef.i], [1])$(P R)
qsetelt!(x,1,i,mo)
y : M P R := copy x
k : PositiveInteger := 1
cond : M P R := copy x
-- multiplication in the generic algebra means using
-- the structural matrices as bilinear forms.
-- left multiplication by x, we prepare for that:
genGamma : V M P R := coerceP$CVMP gamma
x := reduce(horizConcat,[x*genGamma(i) for i in 1..#genGamma])
while rank(cond) = k repeat
k := k+1
for i in 1..n repeat
setelt(xx,[1],[i],x*transpose y)
y := copy xx
cond := horizConcat(cond, xx)
vectorOfCoef : Vector P R := (nullSpace(cond)$Matrix(P R)).first
res : SparseUnivariatePolynomial P R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial P R)
res
rightRankPolynomial() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
listOfNumbers : List String := [string(q)$String for q in 1..n]
symbolsForCoef : Vector Symbol :=
[concat("%", concat("x", i))::Symbol for i in listOfNumbers]
xx : M P R := new(1,n,0)
mo : P R
x : M P R := new(1,n,0)
for i in 1..n repeat
mo := monomial(1, [symbolsForCoef.i], [1])$(P R)
qsetelt!(x,1,i,mo)
y : M P R := copy x
k : PositiveInteger := 1
cond : M P R := copy x
-- multiplication in the generic algebra means using
-- the structural matrices as bilinear forms.
-- left multiplication by x, we prepare for that:
genGamma : V M P R := coerceP$CVMP gamma
x := reduce(horizConcat,[genGamma(i)*transpose x for i in 1..#genGamma])
while rank(cond) = k repeat
k := k+1
for i in 1..n repeat
setelt(xx,[1],[i],y * transpose x)
y := copy xx
cond := horizConcat(cond, xx)
vectorOfCoef : Vector P R := (nullSpace(cond)$Matrix(P R)).first
res : SparseUnivariatePolynomial P R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial P R)
res
leftUnitsInternal : () -> REC
leftUnitsInternal() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
cond : Matrix(R) := new(n**2,n,0$R)$Matrix(R)
rhs : Vector(R) := new(n**2,0$R)$Vector(R)
z : Integer := 0
addOn : R := 0
for k in 1..n repeat
for i in 1..n repeat
z := z+1 -- index for the rows
addOn :=
k=i => 1
0
setelt(rhs,z,addOn)$Vector(R)
for j in 1..n repeat -- index for the columns
setelt(cond,z,j,elt(gamma.k,j,i))$Matrix(R)
solve(cond,rhs)$LSMP
leftUnit() ==
res : REC := leftUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no left unit")$OutputForm
"failed"
represents (res.particular :: V R)
leftUnits() ==
res : REC := leftUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no left unit")$OutputForm
"failed"
[represents(res.particular :: V R)$%, _
map(represents, res.basis)$ListFunctions2(Vector R, %) ]
rightUnitsInternal : () -> REC
rightUnitsInternal() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
condo : Matrix(R) := new(n**2,n,0$R)$Matrix(R)
rhs : Vector(R) := new(n**2,0$R)$Vector(R)
z : Integer := 0
addOn : R := 0
for k in 1..n repeat
for i in 1..n repeat
z := z+1 -- index for the rows
addOn :=
k=i => 1
0
setelt(rhs,z,addOn)$Vector(R)
for j in 1..n repeat -- index for the columns
setelt(condo,z,j,elt(gamma.k,i,j))$Matrix(R)
solve(condo,rhs)$LSMP
rightUnit() ==
res : REC := rightUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no right unit")$OutputForm
"failed"
represents (res.particular :: V R)
rightUnits() ==
res : REC := rightUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no right unit")$OutputForm
"failed"
[represents(res.particular :: V R)$%, _
map(represents, res.basis)$ListFunctions2(Vector R, %) ]
unit() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
cond : Matrix(R) := new(2*n**2,n,0$R)$Matrix(R)
rhs : Vector(R) := new(2*n**2,0$R)$Vector(R)
z : Integer := 0
u : Integer := n*n
addOn : R := 0
for k in 1..n repeat
for i in 1..n repeat
z := z+1 -- index for the rows
addOn :=
k=i => 1
0
setelt(rhs,z,addOn)$Vector(R)
setelt(rhs,u,addOn)$Vector(R)
for j in 1..n repeat -- index for the columns
setelt(cond,z,j,elt(gamma.k,j,i))$Matrix(R)
setelt(cond,u,j,elt(gamma.k,i,j))$Matrix(R)
res : REC := solve(cond,rhs)$LSMP
res.particular case "failed" =>
messagePrint("this algebra has no unit")$OutputForm
"failed"
represents (res.particular :: V R)
apply(m:Matrix(R),a:%) ==
v : Vector R := coordinates(a)
v := m *$Matrix(R) v
convert v
structuralConstants() == structuralConstants basis()
conditionsForIdempotents() == conditionsForIdempotents basis()
convert(x:%):Vector(R) == coordinates(x, basis())
convert(v:Vector R):% == represents(v, basis())
leftTraceMatrix() == leftTraceMatrix basis()
rightTraceMatrix() == rightTraceMatrix basis()
leftDiscriminant() == leftDiscriminant basis()
rightDiscriminant() == rightDiscriminant basis()
leftRegularRepresentation x == leftRegularRepresentation(x, basis())
rightRegularRepresentation x == rightRegularRepresentation(x, basis())
coordinates(x: %) == coordinates(x, basis())
represents(v:Vector R):%== represents(v, basis())
coordinates(v:Vector %) ==
m := new(#v, rank(), 0)$Matrix(R)
for i in minIndex v .. maxIndex v for j in minRowIndex m .. repeat
setRow!(m, j, coordinates qelt(v, i))
m
|