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7.5.7.0. SuperCommutative
Procedure from library nctools.lib (see nctools_lib).
- Usage:
- SuperCommutative([b,[e, [Q, [flag]]]]);
- Return:
- qring
- Purpose:
- create the super-commutative algebra (as a GR-algebra) 'over' a basering,
- Note:
- activate this qring with the "setring" command.
- Note:
- if b==e then the resulting ring is commutative unles 'flag' is given and non-zero.
- Theory:
- given a basering, this procedure introduces the anticommutative relations x(j)x(i)=-x(i)x(j) for all e>=j>i>=b,
moreover, creates a factor algebra modulo the two-sided ideal, generated by x(b)^2, ..., x(e)^2[ + Q]
Example:
| LIB "nctools.lib";
ring R = 0,(x(1..4)),dp; // global!
def ER = SuperCommutative(); // the same as Exterior (b = 1, e = N)
setring ER; ER;
==> // characteristic : 0
==> // number of vars : 4
==> // block 1 : ordering dp
==> // : names x(1) x(2) x(3) x(4)
==> // block 2 : ordering C
==> // noncommutative relations:
==> // x(2)x(1)=-x(1)*x(2)
==> // x(3)x(1)=-x(1)*x(3)
==> // x(4)x(1)=-x(1)*x(4)
==> // x(3)x(2)=-x(2)*x(3)
==> // x(4)x(2)=-x(2)*x(4)
==> // x(4)x(3)=-x(3)*x(4)
==> // quotient ring from ideal
==> _[1]=x(4)^2
==> _[2]=x(3)^2
==> _[3]=x(2)^2
==> _[4]=x(1)^2
"Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "].";
==> Alternating variables: [ 1 , 4 ].
kill R; kill ER;
ring R = 0,(x(1..4)),(lp(1), dp(3)); // global!
def ER = SuperCommutative(2); // b = 2, e = N
setring ER; ER;
==> // characteristic : 0
==> // number of vars : 4
==> // block 1 : ordering lp
==> // : names x(1)
==> // block 2 : ordering dp
==> // : names x(2) x(3) x(4)
==> // block 3 : ordering C
==> // noncommutative relations:
==> // x(3)x(2)=-x(2)*x(3)
==> // x(4)x(2)=-x(2)*x(4)
==> // x(4)x(3)=-x(3)*x(4)
==> // quotient ring from ideal
==> _[1]=x(4)^2
==> _[2]=x(3)^2
==> _[3]=x(2)^2
"Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "].";
==> Alternating variables: [ 2 , 4 ].
kill R; kill ER;
ring R = 0,(x(1..6)),(ls(2), dp(2), lp(2)); // local!
def ER = SuperCommutative(3,4); // b = 3, e = 4
setring ER; ER;
==> // characteristic : 0
==> // number of vars : 6
==> // block 1 : ordering ls
==> // : names x(1) x(2)
==> // block 2 : ordering dp
==> // : names x(3) x(4)
==> // block 3 : ordering lp
==> // : names x(5) x(6)
==> // block 4 : ordering C
==> // noncommutative relations:
==> // x(4)x(3)=-x(3)*x(4)
==> // quotient ring from ideal
==> _[1]=x(3)^2
==> _[2]=x(4)^2
"Alternating variables: [", AltVarStart(), ",", AltVarEnd(), "].";
==> Alternating variables: [ 3 , 4 ].
kill R; kill ER;
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