# split_index

Split the range of an index into two subsets

Replace a sum by a sum-of-sums, abstractly. Concretely, replaces all
index contractions of a given type by a sum of two terms, each with
indices of a different type. Useful for Kaluza-Klein reductions and
the like. An example makes this more clear:{M,N,P,Q,R}::Indices(full).
{m,n,p,q,r}::Indices(space1).
{a,b,c,d,e}::Indices(space2).

ex:=A_{M p} B_{M p};
split_index(_, $M,m,a$);

\(\displaystyle{}A_{M p} B_{M p}\)

\(\displaystyle{}A_{m p} B_{m p}+A_{a p} B_{a p}\)

ex:=A_{M p} B_{M p};
split_index(_, $M,m,4$);

\(\displaystyle{}A_{M p} B_{M p}\)

\(\displaystyle{}A_{m p} B_{m p}+A_{4 p} B_{4 p}\)

Note that the two index types into which the original indices should be
split can be either symbolic (as in the first case above) or numeric
(as in the second case).