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%%% Primary Title: Lie algebras
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%%% Filename: LieAlgebras.tex
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\begin{document}
\subsection{Lie algebras in quantum theories}
Continuous symmetries often have a special \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of underlying continuous \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, called a {\em Lie group}. Briefly, a {\em Lie group} $G$ is generally considered having a (smooth) $C^\infty$ \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} structure, and acts upon itself smoothly. Such a globally smooth structure is surprisingly simple in two ways: it always admits an Abelian \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html}, 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 \htmladdnormallink{computations}{http://planetphysics.us/encyclopedia/LQG2.html} and the initial problem of defining the form and symmetry of the quantum \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html} subject to \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} and initial conditions in the quantum \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} under consideration. However, unlike most \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{abstract algebras}{http://planetphysics.us/encyclopedia/PAdicMeasure.html}, a \htmladdnormallink{Lie Algebra}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html} is not associative, and it is in fact a {\em \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}}. It is also perhaps this feature that makes the Lie algebras somewhat compatible, or consistent, with \htmladdnormallink{quantum logics}{http://planetphysics.us/encyclopedia/LQG2.html} that are also thought to have non-associative, non-distributive and \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} lattice structures.
\subsection{General Lie algebra definition and Examples}
\begin{definition}
A {\em Lie algebra} over a \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} $k$ is a vector space $\mathfrak{g}$ together with a bilinear map $[\ ,\ ] : \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$, called the {\em Lie bracket} and defined by the association $(x,y)\mapsto [x,y]$. The bracket is subject to the following two conditions:
\begin{enumerate}
\item $[x,x] = 0$ for all $x\in\mathfrak{g}$.
\item The {\em Jacobi \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html}}: $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ for all $x,y,z\in\mathfrak{g}$.
\end{enumerate}
\end{definition}
{\bf Examples:}
Any vector space can be made into a Lie algebra simply by setting $[x,y] = 0$ for all \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} $x,y$. Such a Lie algebra is an {\em Abelian} Lie algebra.
If $G$ is a Lie group, then the tangent space at the identity forms a Lie algebra over the real numbers.
$\mathbb{R}^3$ with the \htmladdnormallink{cross product}{http://planetphysics.us/encyclopedia/VectorProduct.html} \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} is a \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} three dimensional (3D) Lie algebra over $\mathbb{R}$.
Consider next the annihilation \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorConcept.html} $a$ and the creation \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorConcept.html} $a\dagger$ in \htmladdnormallink{quantum theory}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}. Then, the Hamiltonian $H$ of a harmonic quantum oscillator, together with the \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} $a$ and $a\dagger$ generate a 4--dimensional (\htmladdnormallink{4D}{http://planetphysics.us/encyclopedia/Curved4DimensionalSpace.html}) Lie algebra with \htmladdnormallink{commutators}{http://planetphysics.us/encyclopedia/Commutator.html}: $[H, a] = −a$, $[H, a\dagger] = a\dagger,$ and $[a, a\dagger] = I$. This Lie algebra is solvable and generates after repeated application of $a\dagger$ all of the eigenvectors of the \htmladdnormallink{quantum harmonic oscillator}{http://planetphysics.us/encyclopedia/LieAlgebraInQuantumTheory.html}.
\end{document}
