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Lie algebra (9375 bytes)
1: ...]] and differentiable [[manifold|manifolds]]. Lie algebras were introduced to study the concept of [[infini... 5: ... : ''g'' × ''g'' → ''g'', called the ''Lie bracket'', which satisfies the following properti... 27: ...ve as long as '''F''' is not of [[Characteristic (algebra)|characteristic]] 2. 28: ...y the Lie bracket is not in general [[associative algebra|associative]], 32: A '''Lie algebra''' is an object A in the [[category of vector spa... Anyonic Lie algebra (431 bytes) 1: An '''anyonic Lie algebra''' is a U(1) graded [[vector space]] L over '''C'... 9: *[[Lie algebra]] 16: [[Category:Algebra]] Affine Lie algebra (556 bytes) 1: ...s]], an '''affine Lie algebra''' is a [[Kac-Moody algebra]] whose [[generalized Cartan matrix]] is of rank ... 3: ... better understood than that of general Kac-Moody algebras. Also, they appear naturally in [[theoretical p... 7: *Jurgen Fuchs, ''Affine Lie Algebras and Quantum Groups'', (1992) Cambridge Universit... 10: [[Category:Lie algebras]] Semisimple Lie algebra (938 bytes) 1: ... of [[simple Lie algebra]]s, i.e., nontrivial Lie algebras ''g'' whose only [[ideal (ring theory)|ideals]] ... 3: Let ''g'' be a Lie algebra. The following conditions are equivalent: 10: ...le if and only if every [[representation of a Lie algebra|representation]] is completely reducible, that is... 17: [[Category:Lie algebras]] Algebra representation of a Lie superalgebra (1502 bytes) 1: ...or space]], A is a [[representation of a Lie superalgebra|vector space rep]] of L and in addition, the elem... 11: Now, for the case of a '''representation of a Lie algebra''', we simply drop all the gradings and the (-1) ... 13: ...iative [[superalgebra]]? We have a [[Poisson superalgebra]]! 15: Let's have more fun. A Lie (super)algebra is an algebra and it also happens to be an [[adjoint representa... 17: ...ial case of an [[algebra representation of a Hopf algebra]]. Hyper generalized orthogonal Lie algebra (4138 bytes) 11: .... [[Nikolai Ivanovich Lobachevsky|Lobachevskian]] algebra is <math>\mathfrak{so}</math> ('''n''', 1) (not [... 13: ...('''n''',1,1). In general it represents the [[Lie algebra]] of the [[semidirect product]] of traslations in... 14: ...mathfrak{so}</math>('''n''','''m''') as its [[Lie algebra]]. 18: ... is '''the''' [[Lie_algebra#Classification_of_Lie_Algebras|very conmmutative]] part of '''Galilean group'''... 20: ... (in some pages even planckian!) factors. The Lie algebra <math>\mathfrak{g}</math> is [[linear span|spanne... Derivative algebra (abstract algebra) (1213 bytes) 1: ...act algebra]], a '''derivative algebra''' is an [[algebraic structure]] of the signature 9: is a [[Boolean algebra]] and <sup>D</sup> is a [[unary operator]], the '... 15: ...]] for the [[modal logic]] ''wK4'' that [[Boolean algebra]]s play for ordinary [[propositional logic]]. 19: ...nd modality via topology'', Annals of Pure and Applied Logic, 127 (2004) 155-170 21: * McKinsey, J.C.C. and Tarski, A., ''The Algebra of Topology'', Annals of Mathematics, 45 (1944) 1... Algebra representation of a Hopf algebra (1034 bytes) 1: ...ra]] is. It turns out [[Hopf algebra]]s also have algebra reps with an additional structure over and above ... 3: ...] <math>\rho:H\otimes A\rightarrow A</math> is an algebra representation of H if, in addition to being a (v... 5: ...ebra representation of a Lie superalgebra]] and [[Algebra representation of a group]] are all special cases... 7: [[Category:Abstract algebra]] Lie (12421 bytes) 1: ...falsehoods. For other meanings of this word see [[Lie (disambiguation)]].'' 3: ...lie, and especially a person who habitually tells lies, is a '''liar'''. 7: ...ilosophy|philosopers]] are divided over whether a lie is ever allowable but are generally opposed - [[P... 11: ...e moderated by the importance of the matter being lied about. 13: ...based on choosing some very large but comfortable lie which is hard to challenge for social status or o... Lie Zi (682 bytes) 1: '''Lie Zi''' or '''Lieh Tzu''' is a famous legendary [[Taoist]] sage men... 3: The Taoist compilation text known as the ''Lie Zi'' is attributed to this name and was inscribed... 6: * ''The Book of Lieh-tzǔ: A Classic of Tao'' translated by A.C. ... 12: [[fr:Lie Zi]] Lie subgroup (1042 bytes) 1: ...''h'' of ''H'' is a [[Lie subalgebra]] of the Lie algebra ''g'' of ''G''. 3: .... one that winds around in ''G''. Then there is a Lie group [[homomorphism]] φ : '''R''' → ''G... 5: ...s of the Lie algebra ''g'' of ''G'' correspond to Lie subgroups ''H'' of ''G''. There is no criterion s... 7: [[Category:Lie groups]] Lie bracket (108 bytes) 1: A '''lie bracket''' can refer to: 3: *[[Lie algebra]] 4: *[[Lie derivative]] 8: [[fr:Crochet de Lie]] Lie algebroid (586 bytes) 1: ...ule]] that has both a [[Lie bracket]] and a [[Lie algebra]] morphism, known as an [[anchor map]], given as 7: ...ie groupoid]]. It is a generalization of a [[Lie algebra]] for [[groupoid]]s. 9: [[Category:Lie algebras]] Lie groupoid (291 bytes) 1: ...tiable functions]]. It is a generalization of a [[Lie group]]. Croppies Lie Down (1346 bytes) 1: '''''Croppies Lie Down''''' is an [[Ulster]] [[Protestantism|Protes... 14: :And make all the traitors and croppies lie down. 15: :Down, down, croppies lie down. 21: .../Croppies_lie_down.html Full lyrics to ''Croppies Lie Down''] Lie group (8772 bytes) 1: ...ical structures. They were introduced by [[Sophus Lie]] in [[1870]] in order to study symmetries of dif... 3: ...more complete list of examples see the [[table of Lie groups]]. 5: == Types of Lie groups == 7: One classifies Lie groups regarding their algebraic properties ([[simple group|simple]], [[semisimp... 11: ... whose inverse is also a homomorphism. Isomorphic Lie groups do not need to be distinguished for all pr... Lie superalgebra (3148 bytes) 1: ...these theories, the ''even'' elements of the superalgebra correspond to [[boson]]s and ''odd'' elements to ... 3: ...] which is the graded version of a ordinary [[Lie algebra]]. The product map is written as <math>[\cdot,\cd... 5: [[Image:Liealgebra.png]] 7: ... product [·, ·], called the '''Lie superbracket''' or '''supercommutator''', satisfi... 16: ... the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. ... Élie Halévy (3694 bytes) 1: ...ce|French]] [[philosopher]] and [[historian]] '''Élie Halévy''' ([[September 6]], [[1870]] - [[August 2... 3: ...ather was the playwright [[Ludovic Halévy]], and Élie grew up surrounded by musicians, scholars, and po... Élie Ducommun (1346 bytes) 1: '''Élie Ducommun''' ([[February 19]], [[1833]]-[[December... 5: ...//www.nobel-winners.com/Peace/elie_ducommun.html Elie Ducommun Bio] 7: [[Category:1833 births|Ducommun, Élie]] 8: [[Category:1906 deaths|Ducommun, Élie]] 9: [[Category:Nobel Peace Prize winners|Ducommun, Élie]] Lie (album) (1312 bytes) 1: '''''Lie''''' is an album by [[Charles Manson]]. Recorded...
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