Complete reducibility in some algebraic systems

Complete reducibility in some algebraic systems

Creators Kashiwagi Yoshimi

Creators Kikumasa Isao

In some algebraic systems we find algebraic objects which are direct sums of indecomposable or simple or irreducible sub-objects. Conditions for such complete reducibility are different, but results are quite similar. Examples are the Krull-Remak-Schmidt's Theorem about the decomposition of groups with chain conditions into indecomposable subgroups and the decomposition of semi-simple Lie algebras over a field of characteristic 0 into simple ideals. We will see this phenomenon in error-correcting codes, non-associative algebras (especially Lie algebras), and modules over a commutative ring, which are completely different algebraic systems. Proofs except for error-correcting codes are traditional in this paper, but the viewpoint seems to be now.

[ISSN]0387-4087

[NCID]AN00244148