# Changeset cfc9c0 in git

Ignore:
Timestamp:
Apr 28, 2005, 6:32:06 PM (18 years ago)
Branches:
(u'jengelh-datetime', 'ceac47cbc86fe4a15902392bdbb9bd2ae0ea02c6')(u'spielwiese', 'a800fe4b3e9d37a38c5a10cc0ae9dfa0c15a4ee6')
Children:
fefe03c337c2d780101b98e7858eb2538ed41587
Parents:
fdebd3cb4759f0739bae5afd99a64ab39ae4efb4
Message:
*levandov: corrections, related to the documentation

git-svn-id: file:///usr/local/Singular/svn/trunk@7932 2c84dea3-7e68-4137-9b89-c4e89433aadc
Location:
Singular/LIB
Files:
3 edited

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 rfdebd3 /////////////////////////////////////////////////////////////////////////////// version="$Id: ncdecomp.lib,v 1.7 2005-03-25 18:38:46 levandov Exp$"; version="$Id: ncdecomp.lib,v 1.8 2005-04-28 16:32:06 levandov Exp$"; category="Noncommutative"; info=" /////////////////////////////////////////////////////////////////////////////// proc CentralQuot(module I, ideal G) "USAGE:  CentralQuot(M, T), for a module M and an ideal T, RETURN:  module of the central quotient I:G, "USAGE:  CentralQuot(M, G), for a module M and an ideal G, PURPOSE: compute the central quotient I:G RETURN:  module NOTE:    the output module is not necessarily a Groebner basis, SEE ALSO: CentralSaturation, CenCharDec proc CentralSaturation(module M, ideal T) "USAGE:  CentralSaturation(M, T), for a module M and an ideal T, RETURN:  module of the central saturation of M by T (also denoted by M:T^{\infty}), PURPOSE: compute the central saturation of M by T (also denoted by M:T^{\infty}) RETURN:  module NOTE:    the output module is not necessarily a Groebner basis, SEE ALSO: CentralQuot, CenCharDec proc CenCharDec(module I, def #) "USAGE:  CenCharDec(I, C);  I a module, C an ideal/list of generators of the center; PURPOSE: compute a finite central character decomposition (or point out that there is no finite one), PURPOSE: compute a finite central character decomposition (or determine that there is no finite one), RETURN:  a list L, where each entry consists of three records: @*       L[*][1] ('ideal' type), the central character as the maximal ideal in the center, @*       L[*][2] ('module' type), the Groebner basis of the weight module, corresponding to the character, @*       L[*][3] ('int' type) is the K-dimension of the weight module (-1 is returned for an infinite dimension); NOTE:     some modules have no finite decomposition (in such case one gets warning message) @*       L[*][3] ('int' type) is the K-dimension of the weight module (-1 is returned in the case of infinite dimension); NOTE:     actual decomposition is a sum of L[i][2]; @*        some modules have no finite decomposition (in such case one gets warning message) SEE ALSO: CentralQuot, CentralSaturation EXAMPLE: example CenCharDec; shows examples "USAGE:  IntersectWithSub(M,Z),  M an ideal, Z an ideal/list of pairwise commutative elements PURPOSE: computes an intersection of M with the subalgebra, generated by Z RETURN:  ideal (of two--sided generators, not a Groebner basis!) RETURN:  ideal (of two-sided generators, not a Groebner basis!) NOTE:    usually one puts generators of the center into Z EXAMPLE: example IntersectWithSub; shows an example
 rfdebd3 /////////////////////////////////////////////////////////////////////////////// version="$Id: nctools.lib,v 1.10 2005-02-23 18:10:46 levandov Exp$"; version="$Id: nctools.lib,v 1.11 2005-04-28 16:32:06 levandov Exp$"; category="Noncommutative"; info=" RootOfUnity(n);         the minimal polynomial for the n-th primitive root of unity, Weyl([p]);              creates Weyl algebra structure in a basering (two different realizations), CreateWeyl(n, [p]);     returns n-th Weyl algebra in (x(i),D(i)) presentation; in char p, if an integer p is given, Heisenberg(N, [p,d]);   returns N-th  Heisenberg algebra in (x(i),y(i),h) realization, CreateWeyl(n, [p]);     returns n-th Weyl algebra in (x(i),D(i)) presentation, Heisenberg(N, [p,d]);   returns n-th  Heisenberg algebra in (x(i),y(i),h) realization, Exterior();             returns qring, the exterior algebra of a basering, Fin_dim_algebra(matrix M, list #); creates finite dimensional algebra structure from the basering and the multiplication matrix M, fin_dim_algebra(matrix M, list #); creates finite dimensional algebra structure from the basering and the multiplication matrix M, AUXILIARY PROCEDURES: Newton(f);              Newton diagram of a polynomial f, Newton(f);              Newton diagram of a polynomial, NCRelations(r);         recovers the non-commutative relations of a G-algebra, IsCentral(p,[v]);       check for the commutativity of polynomial p with the G-algebra, IsCentral(p,[v]);       check for the commutativity of a polynomial in the G-algebra, Is_NC();                checks whether basering is noncommutative, UpOneMatrix(N);         returns NxN matrix with 1's in the whole upper triagle, /////////////////////////////////////////////////////////////////////////////// // This procedure computes a weights vector for a G-Algebra // r must be a G-algebra // This procedure computes a weights vector for a G-algebra r proc Gweights(def r) "USAGE:   Gweights(r); r a ring or a square matrix RETURN:   a weight vector for the G-algebra: r itself, if it is of the type ring, PURPOSE: compute the weight vector for the G-algebra: either r itself, if it is of the type ring, or for a G-algebra, defined by the square matrix r RETURN:   intvec NOTE:    with Gweights you obtain a vector, which must be used to redefine the G-Algebra. If the input is a matrix and the output is the zero vector then there is not a G-algebra structure associated to these relations with respect to the given variables. Another possibility is to use wRing or weightedRing to obtain directly the G-Algebra. EXAMPLE: example Gweights; shows examples proc weightedRing(def r) "USAGE:   weightedRing(r); r a ring RETURN:  ring with the relations of r with order changed to comply with the ordering condition for G-algebras NOTE:    you have to activate this ring with the "setring" command PURPOSE:  equip a ring with such weights,that the relations of a new ring comply with the ordering condition for G-algebras RETURN:  ring NOTE:    you have to activate this ring with the \"setring\" command EXAMPLE: example weightedRing; shows examples SEE ALSO: wRing, Gweights proc Newton(poly f) "USAGE:   Newton(f); f a poly RETURN:  intmat, representing the Newton diagram of f PURPOSE: compute the Newton diagram of f RETURN:  intmat NOTE:    each row is the exponent of a monomial of f EXAMPLE: example Newton; shows examples proc NCRelations(def r) "USAGE:   NCRelations(r); r a ring RETURN:  a list with two elements, both elements are of type matrix and represent the matrices C,D defining the non-commutative relations of the G-algebra r PURPOSE: recover the noncommutative relations via matrices C and D from a noncommutative ring RETURN:  list L with two elements, both elements are of type matrix: L[1] = matrix of coefficients C, L[2] = matrix of polynomials D EXAMPLE: example NCRelations; shows examples "{ /////////////////////////////////////////////////////////////////////////////// proc Fin_dim_algebra(matrix M, list #) "USAGE:   Fin_dim_algebra(M,[r]); M a matrix, r an optional ring. RETURN:  nothing. Creates finite dimensional algebra structure in a ring r (if it is given) or in a basering (by default) from the matrix M. Exports the ideal called Quot for further qring definition. NOTE: matrix M will be read according to relations x_j*x_i = M[i,j]; Quot is not given in its two-sided Groebner basis. EXAMPLE: Fin_dim_algebra; shows examples proc fin_dim_algebra(matrix M, list #) "USAGE:   fin_dim_algebra(M,[r]); M a matrix, r an optional ring. PURPOSE: creates a finite dimensional algebra structure in a ring r (if it is given) or in a basering (by default) from the matrix M. Exports the ideal called Quot for further qring definition. RETURN:  nothing NOTE: matrix M will be read according to relations Xj*Xi = M[i,j]; Quot is not given in its two-sided Groebner basis. EXAMPLE: example fin_dim_algebra; shows examples " { matrix S[3][3]; S[2,3]=a*x(1); S[3,2]=-b*x(1); Fin_dim_algebra(S); fin_dim_algebra(S); Quot = twostd(Quot); qring Qr = Quot; proc IsCentral(poly p, list #) "USAGE:   IsCentral(p,[v]); p poly, v an integer (with v!=0 procedure will be verbose) RETURN:  integer (1 if p commutes with all variables, 0 otherwise) "USAGE:   IsCentral(p,[v]); p poly, v an integer (with v!=0, procedure will be verbose) PURPOSE: check whether p is central in a basering (that is, commutes with everything) RETURN:  integer (1, if p commutes with all variables, 0 otherwise) EXAMPLE: example IsCentral; shows examples "{ proc UpOneMatrix(int N) "USAGE:   UpOneMatrix(N); N an integer, the number of columns RETURN:  intmat, NxN matrix with 1's in the whole upper triagle PURPOSE: compute an NxN matrix with 1's in the whole upper triagle RETURN:  intmat NOTE: helpful for setting noncommutative algebras with complicated coefficient matrices /////////////////////////////////////////////////////////////////////////////// proc ndc(list #) "USAGE:   ndc([v]); v an optional integer. If v!=0 procedure will be verbose RETURN:  ideal of non-degeneracy conditions of the basering "USAGE:   ndc([v]); v an optional integer (if v!=0, procedure will be verbose) PURPOSE: compute the non-degeneracy conditions of the basering, RETURN:  ideal EXAMPLE: example ndc; shows examples " proc RootOfUnity(int n) "USAGE:   RootOfUnity(n); n an integer RETURN:  number, the n-th primitive root of unity (for use as minpoly) PURPOSE: compute the minimal polynomial for the n-th primitive root of unity RETURN:  number NOTE: works only in field extensions by one element EXAMPLE: example RootOfUnity; shows examples proc Weyl(list #) "USAGE:   Weyl([p]); p an optional integer. RETURN:  nothing. Creates Weyl algebra structure in a basering. By default PURPOSE: create a Weyl algebra structure on a basering. By default mimics (x(1..N),d(1..N)) realization. If p is given and is not zero, uses (x(1),d(1),x(2),d(2),... ) realization. RETURN:  nothing SEE ALSO: CreateWeyl EXAMPLE: example Weyl; shows examples " proc Heisenberg(int N, list #) "USAGE:   Heisenberg(N, [p,d]); N an integer (setting 2*N+1 variables), p an optional integer (field characteristic), d an optional integer (power of h in the commutator) RETURN:  N-th  Heisenberg algebra in x(i),y(i),h realization NOTE: you have to activate this ring with the "setring" command PURPOSE: create a N-th  Heisenberg algebra in x(i),y(i),h realization RETURN: nothing NOTE: you have to activate this ring with the \"setring\" command EXAMPLE: example Heisenberg; shows examples " "EXAMPLE:";echo=2; def a = Heisenberg(2); setring a; a; setring a;   a; def H3 = Heisenberg(3, 7, 2); setring H3;  H3; } proc Exterior(list #) "USAGE:   Exterior(); RETURN:  qring, the exterior algebra of a basering NOTE: you have to activate this qring with the "setring" command PURPOSE:  create the exterior algebra of a basering, RETURN:  qring NOTE: you have to activate this qring with the \"setring\" command EXAMPLE: example Exterior; shows examples " proc CreateWeyl(int n, list #) "USAGE:   CreateWeyl(n,[p]); n an integer, n>0; p an optional integer (field characteristic) RETURN:  a ring, describing n-th Weyl algebra NOTE:    You have to activate this ring with the "setring" command. PURPOSE: create a n-th Weyl algebra RETURN:  ring NOTE:    you have to activate this ring with the \"setring\" command. The presentation of n-th Weyl algebra is classical: D(i)x(i)=x(i)D(i)+1 proc Is_NC() "USAGE:   Is_NC(); RETURN:   1, if basering is noncommutative; 0 otherwise. PURPOSE: check whether a basering is commutative or not RETURN:   int (1, if basering is noncommutative; 0 otherwise) EXAMPLE: example Is_NC; shows examples "{
 rfdebd3 version="$Id: qmatrix.lib,v 1.8 2005-02-23 18:10:46 levandov Exp$"; version="$Id: qmatrix.lib,v 1.9 2005-04-28 16:32:06 levandov Exp$"; category="Noncommutative"; info=" proc LengthSymElement(intvec v) "USAGE:   LengthSymElement(v); v an intvec representing an element of S(n) RETURN:  int, the length of v PURPOSE: determine the length of v RETURN:  int NOTE:    if v doesn't represent an element of S(n), the output may have no sense SEE ALSO: SymGroup, LengthSym proc LengthSym(intmat M) "USAGE:   LengthSym(M); M an intmat representing a subset of S(n) (each row must be an element of S(n)) RETURN:  intvec, the i-th element is the length of the i-th row of M PURPOSE: determine a vector, which i-th element is the length of the i-th row of M RETURN:  intvec NOTE:    If M is not a subset of S(n), the output may not have meaning SEE ALSO: SymGroup, LengthSymElement