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Contents
This page is about the algebra as a subfield of mathematics and variations on the general notion of an 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.
Context
Algebra
- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law
Group theory
Ring theory
Module theory
Higher algebra
geometry Isbell duality 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:
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 lab 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, near-ring, -group, field, perfect field, skewfield, free field, vector space, vertex operator algebra, crossed module, chain complex, hypermonoid, hyperring, hyperfield, truss, brace 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.) as well as on higher/homotopical versions like L-infinity algebra, A-infinity 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 -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
geometry | category | dual category | algebra |
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topology | | | commutative algebra |
topology | | | comm. C-star-algebra |
noncomm. topology | | | general C-star-algebra |
algebraic geometry | | | commutative ring |
noncomm. algebraic geometry | | | fin. gen. associative algebra |
differential geometry | | | commutative algebra |
supergeometry | | | supercommutative superalgebra |
formal higher supergeometry (super Lie theory) | | | differential graded-commutative superalgebra (“FDAs”) |
in physics:
References
Introductory textbooks:
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Carl Faith, Algebra: Rings, Modules, Categories I, Grundlehren der mathematischen Wissenschaften 190, Springer (1973) [doi:10.1007/978-3-642-80634-6]
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Anthony Knapp, Basic Algebra, Springer (2006) [doi:10.1007/978-0-8176-4529-8, pdf]
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Paolo Aluffi, Algebra: Chapter 0, Graduate Studies in Mathematics 104, AMS (2009) [ISBN:978-1-4704-1168-8]
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Joseph A. Gallian, Contemporary Abstract Algebra, Chapman and Hall/CRC (2020) [doi:10.1201/9781003142331, webpage, pdf]
See also:
and see the references at ring, module, etc.