nLab
algebra

This page is about algebra as a theory. If you are looking for the term algebra as an object see associative algebra or algebra over an operad or the like. See below for more.


geometry \leftarrow Isbell duality \rightarrow algebra


Contents

Idea

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. While modern algebra has ties and applications nearly everywhere in mathematics, traditionally closest ties are with the number theory and algebraic 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 pseudoalgebras (or 2-algebras) 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 nnlab 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, Ω\Omega-group, field, perfect field, skewfield, free field, vector space, vertex operator algebra, crossed module, chain complex, hypermonoid, hyperring, hyperfield etc.), entries on 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, dg-algebra etc).

There are also few pages on various invariants of algebraic objects or operations on algebraic expressions, e.g. on resultants of polynomials, determinant of a matrix, quasideterminant of a matrix with noncommutative entries.

For many algebraic structures a notion of action is defined; they embody “symmetry algebras” of some other algebraic objects. An action is expressed via a representation of one object as a subobject of a full object of another; or as a combination of the object which acts and which is acted upon (e.g. action groupoid). Objects with action are modules of the appropriate kind (possibly dualized: comodule, contramodule; multiple, e.g. bimodule; or homotopized like A A_\infty-modules). The possibilities for realizing a given algebra via symmetries of another object are systematically studied in a field called representation theory.

See also

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

Further references

Revised on September 19, 2012 20:04:43 by Urs Schreiber (131.174.188.151)