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 sum 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 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~$\Q$.

 \misctitle{Warning} The images of the $\var{lift}_i$ are not guaranteed to form a direct sum.
