Lie algebras sit inside a chain of algebraic structures: sets, binary operations, semigroups, monoids, groups, rings, modules, vector spaces, bilinear maps, ideals, and generators.

The value of tracing that chain is orientation. Advanced structures are easier to use when their dependencies and operations are visible.

Mathematics noteOriginally posted 2020-09-01; expanded here around the structure map.

Article focus: Lie algebras, Lie groups, topology, algebraic structures, modules, rings, groups, and vector spaces.

The Structure Map

The original note followed Lie algebras through modules, algebraic structures, division rings, rings, semirings, ringoids, subgroups, abelian groups, groups, monoids, semigroups, magmas, sets, binary operations, bilinear mappings, bilinear operators, ideals, generators, matroids, and vector spaces.

The useful question was whether there is a periodic-table-style map from magmas to vector spaces through groups and rings.

Why It Helps

A Lie group can be read as a group with manifold structure. A Lie algebra gives a related linearized structure. Seeing the dependency chain helps connect algebraic spaces with topology, functional analysis, tensor calculus, and variational calculus.