Function: algsimpledec
Section: algebras
C-Name: algsimpledec
Prototype: GD0,L,
Help: algsimpledec(al,{flag=0}): decomposition into simple algebras of the
 semisimple algebra al.
Doc: \var{al} being the output of \tet{algtableinit} representing a semisimple
 algebra, returns a \typ{VEC} $[\var{al}_1,\var{al}_2,\dots,\var{al}_n]$ such
 that~\var{al} is isomorphic to the direct product of the simple algebras
 $\var{al}_i$. When $\var{flag}=1$, each component is instead 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
 on the $i$-th factor and a section of it.
 The images of the $\var{lift}_i$ form a direct sum, so that the images
 of~$1\in\var{al}_i$ under~$\var{lift}_i$ are central primitive idempotents
 of~\var{al}. 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~$\Q$.

