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A left-invariant Riemannian metric is preserved by every left translation. Its Killing fields generated by left translations are right-invariant vector fields.

For a matrix Lie group, the left-invariant Maurer-Cartan form is $\rho =g^{-1}dg$. It identifies each tangent space with the Lie algebra and obeys $d\rho +\rho \wedge \rho =0$.

A finite-dimensional Lie algebra is semisimple when it has no nonzero solvable ideals.

The universal enveloping algebra $U(\mathfrak{g})$ is the associative algebra generated by $\mathfrak{g}$ subject to $xy-yx=[x,y]$. Its modules are the same as Lie algebra representations.

The three-dimensional Heisenberg Lie algebra has a basis $a,b,c$ with $[a,b]=c$ and $c$ central. It is a two-step nilpotent Lie algebra.

A Lie algebra homomorphism is a linear map $f$ satisfying $f([x,y])=[f(x),f(y)]$.

The lower central series is defined by ${\gamma }_1(\mathfrak{g})=\mathfrak{g}$ and ${\gamma }_{i+1}(\mathfrak{g})=[\mathfrak{g},{\gamma }_i(\mathfrak{g})]$.

Like the center of a group, the Lie-algebra center consists of elements commuting with every element; here commutation means vanishing of the Lie bracket.

The center is $Z(\mathfrak{g})=\{z:[z,x]=0\text{ for every }x\in \mathfrak{g}\}$.

The derived series is ${\mathfrak{g}}^{(0)}=\mathfrak{g}$ and ${\mathfrak{g}}^{(n+1)}=[{\mathfrak{g}}^{(n)},{\mathfrak{g}}^{(n)}]$.

A vector subspace $I\subseteq \mathfrak{g}$ is an ideal when $[\mathfrak{g},I]\subseteq I$. It is therefore the kernel of a Lie-algebra quotient map.

A Lie algebra is abelian when every Lie bracket vanishes.

The Lie bracket is the product in a Lie algebra: a bilinear map with an Alternating Lie bracket and the Jacobi identity. Bilinearity means, for $a,b\in k$,Unlike matrix multiplication, a Lie bracket need not satisfy associativity. In the sl2 matrix commutator example, $[H,[E,F]]=0$ but $[[H,E],F]=2H$. The replacement compatibility law is the Jacobi identity, which makes bracketing with a fixed element a derivation of a Lie algebra.

For Matrix Lie groups, the Lie bracket is the matrix commutator $[X,Y]=XY-YX$ on their tangent space. For the cross-product Lie algebra, it is $[x,y]=x\times y$. Both products measure a failure of infinitesimal motions to commute. In a real Lie group, a zero Lie bracket means that the two corresponding one-parameter subgroups commute for all real parameters.

If a Lévy process has mean zero and $\mathrm{Var}(X_1)={\sigma }^2<\infty$, then $\mathrm{Var}(X_t)=t{\sigma }^2$ and $X_t^2-t{\sigma }^2$ is a martingale.

The intersection $L^1\cap L^2$ is Banach for $\Vert f{\Vert }_1+\Vert f{\Vert }_2$, but it is generally incomplete when equipped with the $L^1$ norm alone. Truncations of an $L^1$ function outside $L^2$ provide an $L^1$-Cauchy sequence with no limit in the intersection.

For $1\le p\le \infty$, the Lebesgue space $L^p$ is complete in its usual norm.

For a small-noise diffusion $dX=a(X)dt+\sqrt{ϵ}\sigma (X)dW$, the Freidlin--Wentzell action of a smooth path is

A rate function is a lower-semicontinuous map $I$ into $[0,\infty ]$. It is good when every sublevel set $\{x:I(x)\le c\}$ is compact.

Axisymmetric harmonic functions separate into radial powers times Legendre polynomials.

A Dirichlet Laplacian eigenfunction vanishes on the boundary and satisfies $-\Delta u=\lambda u$.

The normalized order-zero function has the representation