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3.15 ncalgebra

Syntax:

ncalgebra( matrix_expression C, matrix_expression D )
ncalgebra( number_expression n, matrix_expression D )
ncalgebra( matrix_expression C, poly_expression p )
ncalgebra( number_expression n, poly_expression p )
Type: ring
Purpose:
Executed in the basering r, say, in k variables $ x_1, \ldots, x_k ,\;$creates 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 $d_{ij}$ should be organized into two strictly upper triangular matrices C with entries $ c_{ij}$ from the ground field of r and D with polynomial entries $d_{ij}$ from r.

Remark:
At present, we do not perform checks of leading monomial condition and non-degeneracy conditions (see G-algebras), while setting an algebra.

Example:
 
ring r1=(0,Q),(x,y,z),Dp;
minpoly=Q^4+Q^2+1;
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; 
ncalgebra(C,D);
// it is quantum deformation U'_q(so_3),
// with q=Q^2 specialized at the 3rd root of unity
r1;
==> //   characteristic : 0
==> //   1 parameter    : Q 
==> //   minpoly        : (Q4+Q2+1)
==> //   number of vars : 3
==> //        block   1 : ordering Dp
==> //                  : names    x y z 
==> //        block   2 : ordering C
==> //   noncommutative relations:
==> //    yx=(Q2)*xy+(-Q)*z
==> //    zx=(-Q2-1)*xz+(-Q3-Q)*y
==> //    zy=(Q2)*yz+(-Q)*x
kill r1;
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;
ncalgebra(1,d);
// it is U(sl_3)
r2;
==> //   characteristic : 0
==> //   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;
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;
ncalgebra(c,0);
// it is some quasi--commutative algebra
r3;
==> //   characteristic : 0
==> //   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;
ring r4=0,(t,u,v,w),dp;
ncalgebra(-1,0);
// it is anticommutative algebra
r4;
==> //   characteristic : 0
==> //   number of vars : 4
==> //        block   1 : ordering dp
==> //                  : names    t u v w 
==> //        block   2 : ordering C
==> //   noncommutative relations:
==> //    ut=-tu
==> //    vt=-tv
==> //    wt=-tw
==> //    vu=-uv
==> //    wu=-uw
==> //    wv=-vw
kill r4;


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