algebra in nLab
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
- group, normal subgroup
- action, Cayley's theorem
- centralizer, normalizer
- abelian group, cyclic group
- group extension, Galois extension
- algebraic group, formal group
- Lie group, quantum group
Ring theory
Module theory
Gebras
Higher algebra
Algebraic theories
Algebras and modules
Higher algebras
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symmetric monoidal (∞,1)-category of spectra
Model category presentations
Geometry on formal duals of algebras
Theorems
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, near-ring, Ω\Omega-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 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
| A\phantom{A}geometryA\phantom{A} | A\phantom{A}categoryA\phantom{A} | A\phantom{A}dual categoryA\phantom{A} | A\phantom{A}algebraA\phantom{A} |
|---|---|---|---|
| A\phantom{A}topologyA\phantom{A} | A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A} | A\phantom{A}↪Gelfand-KolmogorovAlg ℝ op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A} | A\phantom{A}commutative algebraA\phantom{A} |
| A\phantom{A}topologyA\phantom{A} | A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A} | A\phantom{A}≃Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A} | A\phantom{A}comm. C-star-algebraA\phantom{A} |
| A\phantom{A}noncomm. topologyA\phantom{A} | A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A} | A\phantom{A}≔Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A} | A\phantom{A}general C-star-algebraA\phantom{A} |
| A\phantom{A}algebraic geometryA\phantom{A} | A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A} | A\phantom{A}≃almost by def.TopAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} A\phantom{A} | AA\phantom{A} \phantom{A} A\phantom{A}commutative ringA\phantom{A} |
| A\phantom{A}noncomm. algebraicA\phantom{A} A\phantom{A}geometryA\phantom{A} | A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A} | A\phantom{A}≔Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A} | A\phantom{A}fin. gen. A\phantom{A}associative algebraA\phantom{A}A\phantom{A} |
| A\phantom{A}differential geometryA\phantom{A} | A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A} | A\phantom{A}↪Pursell's theoremTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A} | A\phantom{A}commutative algebraA\phantom{A} |
| A\phantom{A}supergeometryA\phantom{A} | A\phantom{A}SuperSpaces Cart ℝ n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A} | A\phantom{A}↪Pursell's theorem Alg ℤ 2AAAA op ↦ C ∞(ℝ n)⊗∧ •ℝ q\array{ \overset{\phantom{\text{Pursell's theorem}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A} | A\phantom{A}supercommutativeA\phantom{A} A\phantom{A}superalgebraA\phantom{A} |
| A\phantom{A}formal higherA\phantom{A} A\phantom{A}supergeometryA\phantom{A} A\phantom{A}(super Lie theory)A\phantom{A} | ASuperL ∞Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A} | A↪ALada-MarklA sdgcAlg op ↦ CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A} | A\phantom{A}differential graded-commutativeA\phantom{A} A\phantom{A}superalgebra A\phantom{A} (“FDAs”) |
in physics:
| A\phantom{A}algebraA\phantom{A} | A\phantom{A}geometryA\phantom{A} |
|---|---|
| A\phantom{A}Poisson algebraA\phantom{A} | A\phantom{A}Poisson manifoldA\phantom{A} |
| A\phantom{A}deformation quantizationA\phantom{A} | A\phantom{A}geometric quantizationA\phantom{A} |
| A\phantom{A}algebra of observables | A\phantom{A}space of statesA\phantom{A} |
| A\phantom{A}Heisenberg picture | A\phantom{A}Schrödinger pictureA\phantom{A} |
| A\phantom{A}AQFTA\phantom{A} | A\phantom{A}FQFTA\phantom{A} |
| A\phantom{A}higher algebraA\phantom{A} | A\phantom{A}higher geometryA\phantom{A} |
| A\phantom{A}Poisson n-algebraA\phantom{A} | A\phantom{A}n-plectic manifoldA\phantom{A} |
| A\phantom{A}En-algebrasA\phantom{A} | A\phantom{A}higher symplectic geometryA\phantom{A} |
| A\phantom{A}BD-BV quantizationA\phantom{A} | A\phantom{A}higher geometric quantizationA\phantom{A} |
| A\phantom{A}factorization algebra of observablesA\phantom{A} | A\phantom{A}extended quantum field theoryA\phantom{A} |
| A\phantom{A}factorization homologyA\phantom{A} | A\phantom{A}cobordism representationA\phantom{A} |
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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Serge Lang: Algebra, 3rd ed. Springer (2002) [doi:10.1007/978-1-4613-0041-0]
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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:
- Wikipedia, Algebra
Last revised on December 31, 2024 at 15:42:09. See the history of this page for a list of all contributions to it.