nLab
algebra

Algebra is the manipulation of symbols without (necessarily) regard for their meaning, especially in a way that can be formalized in cartesian logic?. It is often seen as dual to geometry.

The word ‘algebra’ is often also used for an algebraic structure:

• often by default an associative unital algebra;
• more generally a monoid object;
• more generally in a different way, a nonassociative algebra;
• an algebra over an operad, of a monad, a PROP, etc;
• an algebra for an endofunctor;
• a model of any algebraic theory or anything studied in universal algebra;
• higher categorical analogues, many object/family versions of algebras, for example algebroids, and pseudoalgebra?s (or 2-algebra?s) over pseudomonads (or 2-monads).

Various fields of mathematics or mathematical concepts can be manipulated in an algebraic or symbolic way, and such approaches or formalized subfields have names like categorical algebra?, homological algebra, homotopical algebra and so on. Methods of combinatorics which involve much algebra, and manipulations with formal power series? in particular, are called algebraic combinatorics?.

The nlab has a number of entries on particular algebraic structures (monoid, semigroup, group, ring, noetherian ring, quasigroup, associative algebra, Lie algebra, coalgebra, dg-algebra, bialgebra, graded algebra, Hopf algebra, coring, quasitriangular bialgebra, lattice, rig, $\Omega$-group, field, perfect field, free field, vertex operator algebra etc.), their structural features, parts, “envelopes” or localizations (ideal, center, centralizer, normal subgroup, normal closure, normalizer, holomorph, Ore set, Ore localization, enveloping algebra, universal enveloping algebra…) and on algebraic structures internal to other categories (topological group, Lie group, Lie groupoid, algebraic group, formal group etc).

For many algebraic strutures a notion of action is defined; they embody “symmetry algebras” of some other algebraic objects. The possibilities for realizing a given algebra in such a way are systematically studied in a field called representation theory.

Further references

• See Wikipedia