Function: algradical
Section: algebras
C-Name: algradical
Prototype: G
Help: algradical(al): Jacobson radical of the algebra al.
Doc: \var{al} being a table algebra output by \tet{algtableinit}, returns a
basis of the Jacobson radical of the algebra \var{al} over its prime field
($\Q$ or $\F_p$).
Here is an example with $A = \Q[x]/(x^2)$, with the basis~$(1,x)$:
\bprog
? mt = [matid(2),[0,0;1,0]];
? A = algtableinit(mt);
? algradical(A) \\ = (x)
%3 =
[0]
[1]
@eprog
Another one with $2\times 2$ upper triangular matrices over $\Q$, with basis
$I_2$, $a = \kbd{[0,1;0,0]}$ and $b = \kbd{[0,0;0,1]}$, such that $a^2 =
0$, $ab = a$, $ba = 0$, $b^2 = b$:
\bprog
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
? A = algtableinit(mt);
? algradical(A) \\ = (a)
%6 =
[0]
[1]
[0]
@eprog