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Lie algebra

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an alternating bilinear map g × g → g {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}} , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra…

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Definition

Basic examples

Basic constructions

Examples

Representations

Structure theory and classification

Relation to Lie groups

Real form and complexification

Lie algebra with additional structures

Lie ring

Definition using category-theoretic notation

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Lie algebra

Nodes225
Edges224
Triples294
Avg. degree1.99
Density0.008889
Components1

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Lie algebra

Top relations

related to Sources · 82
Lie algebra → Academic Press, American Mathematical Soc, An Elementary Introduction, Annals, Anthony, Application, Atomic Spectra, Automorphisms, Bourbaki, Brian, Cambridge University Press, CBO9780511526008, Chapters, Collected Papers, Connor, Daniel, Dover, Erdmann, Eugene, Examples
related to Classification · 18
Lie algebra → An, Bn, Cartan, Cn, Dn, E6, E7, E8, F4, G2, Here, Killing, Lie, Moreover, Namely, The, The Levi, This
related to External links · 16
Lie algebra → An Elementary Introduction, Archived, Course, Douglas, EMS Press, Encyclopedia, Introduction, Kac, Lie, Lie Algebras, Mathematics, McKenzie, MIT, Physicists, Retrieved, Victor
related to Relation to Lie groups · 14
Lie algebra → Although Lie, Baker, Campbell, Conversely, Each Lie, For, Hausdorff, Lie, Lie's, SO, SU, The, This, This Lie
related to Universal enveloping algebra · 12
Lie algebra → Birkhoff, F-vector, Here, In, It, Let, Lie, Poincaré, The, To, Witt, XY-YX
related to Goals of representation theory · 9
Lie algebra → Although Ado's, For, Indeed, Lie, One, Rather, The, Weyl, Weyl's
related to The Lie algebra of matrices · 9
Lie algebra → For, It, Lie, On, The, The Jacobi, This, With, XY-YX
is a · 8
Lie algebra → algebra over a field for which the multiplication operation, Lie algebra, Lie subalgebra g, space b n, space of all linear maps from a vector space to itself, space u n, symplectic Lie algebra s p, vector space g
related to Infinite dimensions · 8
Lie algebra → It, Lie, Moody, The, The Kac, The Lie, The Moyal, The Virasoro
related to Lie ring · 8
Lie algebra → Jacobi, Lazard, Lie, Namely, R-bilinear, R-module, The, This

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lie algebra displaystyle mathfrak algebras group space mathbb bracket field groups vector identity linear example every matrices simple semisimple gl

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Lie algebrais aalgebra over a field for which the multiplication operation0.90text
Lie algebrais aspace of all linear maps from a vector space to itself0.90text
Lie algebrais avector space g0.90text
Lie algebrais asymplectic Lie algebra s p0.90text
Lie algebrais aLie subalgebra g0.90text
Lie algebrais aspace b n0.90text
Lie algebrais aspace u n0.90text
Lie algebrais aLie algebra0.90text
ginstance ofIt is customary to denote a Lie algebra by a lower-case fraktur letter0.80text
hinstance ofIt is customary to denote a Lie algebra by a lower-case fraktur letter0.80text
binstance ofIt is customary to denote a Lie algebra by a lower-case fraktur letter0.80text
ninstance ofIt is customary to denote a Lie algebra by a lower-case fraktur letter0.80text

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