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Continuous symmetries often have a special type of underlying continuous group, called a Lie group. Briefly, a Lie group  is generally considered having a (smooth)  manifold structure, and acts upon itself smoothly. Such a globally smooth structure is surprisingly simple in two ways: it always admits an Abelian fundamental group, and seemingly also related to this global property, it admits an associated,
unique–as well as finite–Lie algebra that completely specifies locally the properties of the Lie group everywhere. There is a finite Lie algebra of quantum commutators and their unique (continuous) Lie groups. Thus, Lie algebras can greatly simplify quantum computations and the initial problem of defining the form and symmetry of the quantum hamiltonian subject to boundary and initial conditions in the quantum system under consideration. However, unlike most regular abstract algebras, a Lie algebra is not associative, and it is in fact a vector space. It is also perhaps this feature that makes the Lie algebras somewhat compatible, or consistent, with quantum logics that are also thought to have non-associative, non-distributive and non-commutative lattice structures.
Definition 0.1 A Lie algebra over a field  is a vector space
 together with a bilinear map
![$[\ ,\ ] : \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$](http://images.physicslibrary.org/cache/objects/711/l2h/img5.png "$[\ ,\ ] : \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$") , called the Lie bracket and defined by the association
![$(x,y)\mapsto [x,y]$](http://images.physicslibrary.org/cache/objects/711/l2h/img6.png "$(x,y)\mapsto [x,y]$") . The bracket is subject to the following two conditions:
![$[x,x] = 0$](http://images.physicslibrary.org/cache/objects/711/l2h/img7.png "$[x,x] = 0$") for all
 .- The Jacobi identity:
![$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$](http://images.physicslibrary.org/cache/objects/711/l2h/img9.png "$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$") for all
 .
Examples:
Any vector space can be made into a Lie algebra simply by setting ![$[x,y] = 0$](http://images.physicslibrary.org/cache/objects/711/l2h/img11.png "$[x,y] = 0$") for all vectors  . Such a Lie algebra is an Abelian Lie algebra.
If  is a Lie group, then the tangent space at the identity forms a Lie algebra over the real numbers.
 with the cross product operation is a non-Abelian three dimensional (3D) Lie algebra over
 .
Consider next the annihilation operator  and the creation operator  in quantum theory. Then, the Hamiltonian  of a harmonic quantum oscillator, together with the operators  and  generate a 4–dimensional (4D) Lie algebra with commutators:
![$[H, a] = \^aˆ’a$](http://images.physicslibrary.org/cache/objects/711/l2h/img21.png "$[H, a] = \^aˆ’a$") ,
![$[H, a\dagger] = a\dagger,$](http://images.physicslibrary.org/cache/objects/711/l2h/img22.png "$[H, a\dagger] = a\dagger,$") and
![$[a, a\dagger] = I$](http://images.physicslibrary.org/cache/objects/711/l2h/img23.png "$[a, a\dagger] = I$") . This Lie algebra is solvable and generates after repeated application of  all of the eigenvectors of the quantum harmonic oscillator.
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"Lie algebras" is owned by bci1.
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See Also: homotopy addition lemma and corollary, quantum operator concept, quantum harmonic oscillator and Lie algebra, index of algebraic topology, commutator algebra
| Also defines: |
harmonic quantum oscillator, finite Lie algebra of quantum commutators, Lie group, tangent space, globally smooth structure, Abelian Lie algebra, Lie algebra, bilinear map, non-associative structures |
| Keywords: |
harmonic quantum oscillator, finite Lie algebra of quantum commutators, Lie group, tangent space, globally smooth structure, Abelian Lie algebra |
Cross-references: quantum harmonic oscillator, commutators, 4D, operators, quantum theory, non-Abelian, operation, cross product, vectors, identity, field, non-commutative, quantum logics, vector space, abstract algebras, regular, system, boundary, hamiltonian, computations, fundamental group, manifold, group, type
There are 42 references to this object.
This is version 11 of Lie algebras, born on 2009-05-01, modified 2009-05-01.
Object id is 711, canonical name is LieAlgebras.
Accessed 2210 times total.
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