Each [Lie group](Lie%20groups.md) $\mbox{G}$ has an associated _Lie Algebra._ The Lie algebra of $\mbox{G}$ is a [non-associative Algebra](Non-associative%20algebras.md), denoted by $\mathfrak{g}$, where $\mathfrak{g}$ is a [vector space](Algebras.md#^198aff) over a field $\mathbb{F}.$ For elements of this vector space there exists a [Lie bracket,](Lie%20bracket.md) which we write as $[.,.]:(X,Y)\in\mathfrak{g}\times\mathfrak{g}\rightarrow[X,Y]\in\mathfrak{g}.$ ^73b451 Here, $[.,.]$ is a [Lie bracket](Lie%20bracket.md) with inputs $X$ and $Y,$ and an output, $[X,Y]$ where $X,Y,[X,Y]\in\mathfrak{g}.$ This map has the following properties $\forall X,Y,Z \in \mathfrak{g}$:   In this definition we don't consider anything relating to whether there is a [matrix representation](Lie%20algebra%20representation.md) of $\mathfrak{g},$ thus this is often referred to as the definition for an _Abstract Lie Algebra._ %%Relate the Jacobi identity to the Jacobi identity. As a starting point see Woit's text.%% # Abelian Lie algebras  ([... see more](Abelian%20Lie%20algebras.md)) # Lie brackets of Lie algebras Here we list examples of Lie Brackets corresponding to [Lie algebras.](Lie%20algebras.md)   %%There should be a way to categorize Lie algebras by their particular bilinear operators. Could you discuss this?%% # Lie algebras of matirx groups %%Consider if the entry for matrix Lie algebras need to be spun off. To do this generalize whatever rules there are for Lie Groups in general.%% ## Properties of Lie algebras of [[Matrix Lie group]]s ## Property 1 Let $\mbox{G}$ be a [[Matrix Lie group]]. The Lie Algebra of $\mbox{G}$ denoted by $\mathfrak{g}$ is the set of all matrices $X$ such that the [matrix exponential](Matrix%20exponentials.md), $e^{tX}$ is in $\mbox{G}$ for all real numbers $t$. ## Property 2 $X$ is in $\mathfrak{g}$ if and only if the entire [one-parameter groups](One-parameter%20groups.md) generated by $X$ is in $\mbox{G}$. Here $X$ is also the [[Lie algebra representation]] of $\mathfrak{g}$. ## Property 3 $e^{tX}$ being in $\mbox{G}$ does not guarantee $X$ is in $\mathfrak{g}$. However, every finite dimensional _real_ Lie Algebra is the Lie algebra of some [[Matrix Lie group]]. ## Property 4 For a _complex matrix Lie Group_ $iX\in\mathfrak{g} \,\,\, \forall X\in\mathfrak{g}$. ## Property 5 $e^X$ is an element of the identity component $G_0$ of $\mbox{G}$. i.e. we can map to $e^0$ since $t$ in $e^{tX}$ can be $0$. ## Properties of the elements of matrix Lie Algebras Where $\mbox{G}$ is a matrix Lie group with Lie algebra $\mathfrak{g}$, the following hold for all elements $X$ and $Y$ in $\mathfrak{g}:$ 1. $AXA^{-1} \in \mathfrak{g}, \;\; A\in \mbox{G}$ ^c626d7 2. $sX \in \mathfrak{g} \;\; \forall s \in \mathbb{R}$ ^52c79c 3. $X+Y \in \mathfrak{g}$ ^1976fd 4. $[X,Y] = XY - YX \in \mathfrak{g}$ ^5b8af5 ([Proofs](Lie%20algebras.md#Proof%20of%20the%20properties%20of%20the%20elements%20of%20Matrix%20Lie%20Algebras)) ## Lie algebras that correspond to Abelian Lie groups If $\mbox{G}$ is commutative then its corresponding [Lie algebra](Lie%20algebras.md) $\mathfrak{g}$ is also commutative ([proof](Lie%20algebras.md#Proof%20of%20property%207%20of%20Lie%20algebras%20Lie%2020algebras%20md%20Property%20207)). --- # Proofs and examples ## Proof of the properties of the elements of Matrix Lie Algebras     ## Proof that the Lie Algebra that corresponds with an Abelian Lie Group is also Abelian  --- # Recommended reading The definition of [Lie algebras](Lie%20algebras.md) given here is drawn from * [Hall, B., _Lie Groups Lie Algebras and Representations_, Springer, 2nd edition, 2015.](Hall,%20B.,%20Lie%20Groups%20Lie%20Algebras%20and%20Representations,%20Springer,%202nd%20edition,%202015..md) pg 49. Here the definition is presented for matrix Lie Algebras and following this an example proof of the validity of the cross products as a possible Lie bracket is given. Here Lie brackets are referred to _bracket operations on $\mathfrak{g}$_. * [Woit, Peter. Quantum Theory, Groups and Representations An Introduction, Springer, 2017](Woit,%20Peter.%20Quantum%20Theory,%20Groups%20and%20Representations%20An%20Introduction,%20Springer,%202017.md) (pg 58) Here the definition of Lie Algebras is given in terms of the notion of _abstract_ Lie Algebras. That is, it is introduced in a context for which we don't consider whether there's a matrix representation for a given algebra. * [Hall, Brian. _Quantum Theory for Mathematicians_, Springer (2013).](%5BGraduate%20Texts%20in%20Mathematics%20267%5D%20Brian%20C.%20Hall%20(auth.)%20-%20Quantum%20Theory%20for%20Mathematicians%20(2013,%20Springer-Verlag%20New%20York)%20-%20libgen.lc.pdf) (pg 338). Here the definition for Lie Algebras is given without considering whether there exists a matrix representation as one would when generalizing to _abstract_ Lie Algebras. The broader context of this work is as an introduction to quantum physics aimed at mathematicians. %%Find source that explains that Lie algebras are non-associative%% #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras