# The Lie algebra of a Lie group

Here we define the special vector fields that give Lie groups an associated Lie algebra. The left translation mapping $${L_{g}(h)\equiv gh}$$ is a diffeomorphism on $${G}$$, as is right translation $${R_{g}(h)\equiv hg}$$. A left-invariant vector field $${A}$$ then satisfies

$$\displaystyle \mathrm{d}L_{g}\left(\left.A\right|_{h}\right)=\left.A\right|_{L_{g}\left(h\right)}=\left.A\right|_{gh}$$

for any $${g}$$ and $${h}$$. In words, the vector field at any point can be obtained by the left translation of its value at any other point. Thus the vector field is invariant under a left translation diffeomorphism. In particular, a left-invariant vector field is then completely determined by its value at the identity.

The left-invariant vector fields on $${G}$$ under the Lie commutator form its associated Lie algebra $${\mathfrak{g}}$$ (which is also isomorphic to the right-invariant vector fields). Since each left-invariant vector field is uniquely determined by its value at the identity element (point) $${e}$$, $${\mathfrak{g}}$$ is isomorphic (as a vector space) to the tangent space $${T_{e}G}$$, which of course has the same dimension as $${G}$$. The elements of the Lie algebra $${\mathfrak{g}\cong T_{e}G}$$ are often called the infinitesimal generators of $${G}$$. The above depicts the Lie group $${G}$$ of rotations of a circle, which has an associated Lie algebra $${\mathfrak{g}\cong\mathbb{R}}$$.

We can also define left-invariant forms by demanding invariance under left translated vector arguments, i.e. we require $${L_{g}^{*}\varphi=\varphi}$$ where $${L_{g}^{*}}$$ is the pullback. As with left-invariant vectors, left-invariant forms are uniquely determined by their value at the identity. The term Maurer-Cartan form can be used to refer to left-invariant 1-forms in general, a particular basis of left-invariant 1-forms, or a $${\mathfrak{g}}$$-valued 1-form that is the identity on left-invariant vector fields.

If we choose a basis of left-invariant 1-forms $${\alpha^{\mu}}$$ (the dual to a basis of $${\mathfrak{g}}$$), we can construct a left-invariant volume form $${\alpha^{1}\wedge\dotsb\wedge\alpha^{n}}$$ called a left Haar measure. A volume form constructed from a basis of right-invariant 1-forms is called a right Haar measure, and if it is bi-invariant, i.e. both left- and right-invariant, it is simply called a Haar measure. Haar measures allow one to construct integrals on $${G}$$ that are invariant under left and/or right translation diffeomorphisms.