Function: algsimpledec
Section: algebras
C-Name: algsimpledec
Prototype: GD0,L,
Help: algsimpledec(al,{maps=0}): [J,dec] where J is the Jacobson radical of al
 and dec is the decomposition into simple algebras of the semisimple algebra
 al/J.
Doc: \var{al} being the output of \tet{algtableinit}, returns a \typ{VEC}
 $[J,[\var{al}_{1},\dots,\var{al}_{n}]]$ where $J$ is a basis of the
 Jacobson radical of \var{al} and~$\var{al}/J$ is isomorphic to the direct
 product of the simple algebras~$\var{al}_{i}$. When $\var{maps}=1$,
 each~$\var{al}_{i}$ is replaced with a \typ{VEC}
 $[\var{al}_{i},\var{proj}_{i},\var{lift}_{i}]$ where $\var{proj}_{i}$
 and~$\var{lift}_{i}$
 are matrices respectively representing the projection map~$\var{al} \to
 \var{al}_{i}$ and a section of it. Modulo~$J$, the images of the
 $\var{lift}_{i}$
 form a direct sum in~$\var{al}/J$, so that the images of~$1\in\var{al}_{i}$
 under~$\var{lift}_{i}$ are central primitive idempotents of~$\var{al}/J$. The
 factors are sorted by increasing dimension, then increasing dimension of the
 center. This ensures that the ordering of the isomorphism classes of the
 factors is deterministic over finite fields, but not necessarily over~$\Q$.
