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7.3.16 nc_algebra

Syntax:

nc_algebra( matrix_expression C, matrix_expression D )
nc_algebra( number_expression n, matrix_expression D )
nc_algebra( matrix_expression C, poly_expression p )
nc_algebra( number_expression n, poly_expression p )
Type:
ring
Purpose:
Executed in the basering r, say, in k variables $ x_1, \ldots, x_k ,\;$nc_algebra creates and returns the non-commutative extension of r subject to relations $ \{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}, 1 \leq i <j \leq k \}, $ where $c_{ij}$ and $d_{ij}$ must be put into two strictly upper triangular matrices C with entries $c_{ij}$ from the ground field of r and D with (commutative) polynomial entries $d_{ij}$ from r. See all the details in G-algebras.
If $\forall i<j$, $c_{ij} = n$,one can input the number n instead of matrix C.
If $\forall i<j$, $d_{ij} = p$,one can input the polynomial p instead of matrix D.

Note: The returned ring should be activated afterwards, using the command setring.

Remark:
At present, PLURAL does not check the non-degeneracy conditions (see G-algebras) while setting an algebra.

Example:
 
LIB "nctools.lib";
// ------- first example: C, D are matrices --------
ring r1 = (0,Q),(x,y,z),Dp;
minpoly = rootofUnity(6);
matrix C[3][3];
matrix D[3][3];
C[1,2]=Q2;   C[1,3]=1/Q2;  C[2,3]=Q2;
D[1,2]=-Q*z; D[1,3]=1/Q*y; D[2,3]=-Q*x;
def S=nc_algebra(C,D);
// this algebra is a quantum deformation U'_q(so_3),
// where Q is a 6th root of unity
setring S;S;
==> // coefficients: QQ[Q]/(Q2-Q+1)
==> // number of vars : 3
==> //        block   1 : ordering Dp
==> //                  : names    x y z
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    yx=(Q-1)*xy+(-Q)*z
==> //    zx=(-Q)*xz+(-Q+1)*y
==> //    zy=(Q-1)*yz+(-Q)*x
kill r1,S;
// ----- second example: number n=1, D is a matrix
ring r2=0,(Xa,Xb,Xc,Ya,Yb,Yc,Ha,Hb),dp;
matrix d[8][8];
d[1,2]=-Xc; d[1,4]=-Ha; d[1,6]=Yb;  d[1,7]=2*Xa;
d[1,8]=-Xa; d[2,5]=-Hb; d[2,6]=-Ya; d[2,7]=-Xb;
d[2,8]=2*Xb; d[3,4]=Xb; d[3,5]=-Xa; d[3,6]=-Ha-Hb;
d[3,7]=Xc;   d[3,8]=Xc; d[4,5]=Yc; d[4,7]=-2*Ya;
d[4,8]=Ya;  d[5,7]=Yb; d[5,8]=-2*Yb;
d[6,7]=-Yc; d[6,8]=-Yc;
def S=nc_algebra(1,d);  // this algebra is U(sl_3)
setring S;S;
==> // coefficients: QQ
==> // number of vars : 8
==> //        block   1 : ordering dp
==> //                  : names    Xa Xb Xc Ya Yb Yc Ha Hb
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    XbXa=Xa*Xb-Xc
==> //    YaXa=Xa*Ya-Ha
==> //    YcXa=Xa*Yc+Yb
==> //    HaXa=Xa*Ha+2*Xa
==> //    HbXa=Xa*Hb-Xa
==> //    YbXb=Xb*Yb-Hb
==> //    YcXb=Xb*Yc-Ya
==> //    HaXb=Xb*Ha-Xb
==> //    HbXb=Xb*Hb+2*Xb
==> //    YaXc=Xc*Ya+Xb
==> //    YbXc=Xc*Yb-Xa
==> //    YcXc=Xc*Yc-Ha-Hb
==> //    HaXc=Xc*Ha+Xc
==> //    HbXc=Xc*Hb+Xc
==> //    YbYa=Ya*Yb+Yc
==> //    HaYa=Ya*Ha-2*Ya
==> //    HbYa=Ya*Hb+Ya
==> //    HaYb=Yb*Ha+Yb
==> //    HbYb=Yb*Hb-2*Yb
==> //    HaYc=Yc*Ha-Yc
==> //    HbYc=Yc*Hb-Yc
kill r2,S;
// ---- third example: C is a matrix, p=0 is a poly
ring r3=0,(a,b,c,d),lp;
matrix c[4][4];
c[1,2]=1; c[1,3]=3; c[1,4]=-2;
c[2,3]=-1; c[2,4]=-3; c[3,4]=1;
def S=nc_algebra(c,0); // it is a quasi--commutative algebra
setring S;S;
==> // coefficients: QQ
==> // number of vars : 4
==> //        block   1 : ordering lp
==> //                  : names    a b c d
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    ca=3ac
==> //    da=-2ad
==> //    cb=-bc
==> //    db=-3bd
kill r3,S;
// -- fourth example : number n = -1, poly p = 3w
ring r4=0,(u,v,w),dp;
def S=nc_algebra(-1,3w);
setring S;S;
==> // coefficients: QQ
==> // number of vars : 3
==> //        block   1 : ordering dp
==> //                  : names    u v w
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    vu=-uv+3w
==> //    wu=-uw+3w
==> //    wv=-vw+3w
kill r4,S;
See also ncalg_lib; nctools_lib; qmatrix_lib.