General Finite-Dimensional Algebras

These algebras are presented in terms of a basis for a free module Mtogether with a set of structure constants defining the multiplication of these basis elements. It is assumed that we have an echelonization algorithm for M so that standard bases may be constructed for submodules.

• Creation of algebras in terms of structure constants
• Direct sum
• Arithmetic including Lie bracket operation
• Identities: associative, commutative, Lie, etc
• Properties of elements: idempotent, unit, zero-divisor, nilpotent
• Trace and minimal polynomial
• Creation of subalgebras, ideals and quotient algebras
• Ideal arithmetic: Sum, product, powers, intersection
• Ideal structure: Jacobson radical, maximal (minimal) left, right, two-sided ideals
• Decomposition: Simplicity, semi-simplicity, composition series