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- ===Lie Algebra of a Quantum Harmonic Oscillator=== ...that act as [[../Generator/|generators]] of a [[../TopologicalOrder2/|Lie Algebra]] for this ...8 KB (1,066 words) - 08:36, 11 March 2023
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- A linear (or "classical") Lie algebras is a [[Lie algebra]] whose elements can be represented as [[matrices]] (or [[linear transforma ''See also: [[sl(n)]]'' ([[special linear Lie algebra]]) ...504 bytes (68 words) - 16:05, 21 February 2009
- ...lling form]]) and is pivotal in the structure theory of representations of Lie algebras. *<!--[[wikibooks:Lie algebra]]? --> ...969 bytes (135 words) - 03:38, 1 May 2017
- ...bilinear form is unique up to scaling. A Lie algebra is [[semi-simple Lie algebra|semi-simple]] if and only if its Killing form is [[non-degenerate]]. ...ath> where x,y,z are elements in the algebra and the brackets [] are the [[Lie brackets]] ...1 KB (187 words) - 03:09, 1 May 2017
- ===Lie algebras in quantum theories=== ...inite--Lie algebra that completely specifies locally the properties of the Lie group everywhere. ...4 KB (496 words) - 08:36, 11 March 2023
- ...structs a [[basis (linear algebra)|basis]] for its [[universal envelopping algebra]] <math>U(\mathfrak{g})</math>, called the '''Poincare-Birkhoff-Witt''' ('' *James. E. Humphreys, ''Introduction to Lie algebras and representation theory'', pp.91-93 (detailed) ...1 KB (135 words) - 18:34, 16 November 2014
- ...rving diffeomorphisms of the circle. The representation theory of Virasoro algebra is rich, and has diverse applications in Mathematics and Physics. ''Vir'' is the Lie algebra over the field of complex numbers with the following generators: ...2 KB (251 words) - 23:03, 26 July 2017
- A '''differential graded Lie algebra''' is a graded <math>k</math>-vector space <math>\mathfrak{g}</math> ..., and it inherits a Lie bracket. (In fact, it is an <math>L_\infty</math>-algebra.) ...785 bytes (125 words) - 11:38, 26 March 2016
- ''Hamiltonian algebroids'' are generalizations of the [[../BilinearMap/|Lie algebras]] of canonical transformations. Their [[../Commutator/|commutator]], or Lie bracket, <math>L</math>, is : ...3 KB (465 words) - 15:32, 12 September 2020
- * Lie algebras and their representations. === MARE: A Lie algebra and its representations === ...2 KB (396 words) - 12:05, 20 January 2025
- ...nce, let A and B be square matrices, and let α(s) and β(t) be paths in the Lie group of nonsingular matrices which satisfy ...1 KB (226 words) - 07:28, 22 August 2023
- ...polynomial algebra <math>A[t]</math> is an essential tool to construct an algebra extension <math>B</math> of <math>A</math> in which a given <math>z\in A</m === Remark: Algebra expansion === ...7 KB (1,005 words) - 00:24, 16 June 2024
- ...[../CosmologicalConstant/|field]], then we call <math>A</math> a ''Poisson algebra'' . In this case, we may wish to add the extra requirement ...>A</math> is non-degenerate and is isomorphic as a commutative ring to the algebra of smooth [[../Bijective/|functions]] on a [[../NoncommutativeGeometry4/|ma ...2 KB (332 words) - 15:50, 12 September 2020
- algebra <math>\mathcal{A}</math> as follows. Let <math>\mathcal{A}</math> be the se where the coefficients <math>c_i</math> lie in <math>R</math> and, to every pair of [[../TrivialGroupoid/|objects]] <ma ...2 KB (286 words) - 15:17, 13 October 2023
- ===Lie Algebra of a Quantum Harmonic Oscillator=== ...that act as [[../Generator/|generators]] of a [[../TopologicalOrder2/|Lie Algebra]] for this ...8 KB (1,066 words) - 08:36, 11 March 2023
- [[../QuantumOperatorAlgebra5/|groupoid]] and associated groupoid algebra representations: \mu)<math> in a separable Hilbert space </math>\H<math> induces a *-algebra ...4 KB (624 words) - 12:52, 23 October 2023
- \subsubsection{Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras: Firstly, a specific ''algebra'' consists of a [[../NormInducedByInnerProduct/|vector space]] <math>E</ma ...10 KB (1,513 words) - 15:39, 12 September 2020
- ...bra]] <math>L(G^{\prime})</math> of <math>G^{\prime}</math>. Since the Lie algebra <math>L(G)</math> of <math>G</math> is a subalgebra of <math>L(G^{\prime})< ...3 KB (482 words) - 04:46, 1 July 2015
- ...bra]] <math>L(O^{\prime})</math> of <math>G^{\prime}</math>. Since the Lie algebra <math>L(O)</math> of <math>G</math> is a subalgebra of <math>L(G^{\prime})< ...3 KB (502 words) - 22:29, 11 September 2016
- <math>C_c(\mathsf{G})</math> into a *--algebra once we have defined the involution \subsubsection{Lie groupoids, their dual algebroids and representations ...9 KB (1,354 words) - 03:16, 13 September 2020
- ===C*- and von Neumann algebras: Quantum operator algebra in quantum theories=== ...fication]] is simpler and studied in greater depth than that of general C*-algebra classification theory. ...11 KB (1,607 words) - 05:03, 12 September 2020