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For a Lie group $G$ with Lie algebra $\mathfrak{g}$, the left Maurer-Cartan form is the $\mathfrak{g}$-valued differential 1-formIt translates a tangent vector at any point back to the identity, giving a linear isomorphism $T_{g}G\to \mathfrak{g}$. In particular, $\theta (X^L)=X$ for the left-invariant vector field determined by $X$. This makes $\theta$ left-invariant and supplies a global coframe after choosing a basis of $\mathfrak{g}$.

The exterior derivative on left-invariant vector fields gives $d\theta (X^L,Y^L)=-[X,Y]$, since the two derivative terms of constant coefficients vanish. Thus $d\theta +\frac{1}{2}[\theta ,\theta ]=0$, the Maurer-Cartan equation. For a Matrix Lie group, $\theta =g^{-1}dg$ and this becomes $d\theta +\theta \wedge \theta =0$, where the exterior product includes matrix multiplication. The additive real numbers have $\theta =dx$, while the positive multiplicative real numbers have $\theta =da/a$. For the real affine group, the Maurer-Cartan coframe of the real affine group makes the nonzero Lie bracket visible as $d(db/a)=-(da/a)\wedge (db/a)$.

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 concrete nondegenerate small-noise model, letwhere $a$ is globally Lipschitz, $\Sigma$ is a constant invertible matrix and the time horizon $T$ is finite. The path laws on $C([0,T],{\mathbb{R}}^d)$ satisfy a large deviation principle of speed $1/\epsilon$. Their Freidlin--Wentzell action isfor absolutely continuous paths with $\varphi (0)=x_0$ and finite displayed integral, and $+\infty$ otherwise. Thus the initial condition and admissible path class are part of the rate, not optional qualifications.

The action measures the squared control needed to move the deterministic trajectory away from $\dot{\varphi }=a(\varphi )$. For zero drift and constant scalar noise coefficient $\sigma >0$, the least-action path from $x_0$ to $y$ in time $T$ is the straight line, with cost $(y-x_0{)}^2/(2{\sigma }^{2}T)$, by the Cauchy-Schwarz inequality. The noiseless solution has action zero. Neighborhood probabilities have leading exponential cost $I_T/\epsilon$, in the precise open/closed-set sense of the large deviation principle.

For constant $\Sigma$, continuity of the additive-noise solution map under the uniform norm follows from Gronwall's inequality. The Schilder theorem for scaled Brownian motion and the contraction principle for large deviations explain the path-space rate. With state-dependent diffusion, the integrand uses the inverse covariance matrix $[\sigma (\varphi )\sigma (\varphi {)}^T{]}^{-1}$; regularity and nondegeneracy assumptions must be checked rather than using an inverse for a singular covariance.^[ref]

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.

An axisymmetric harmonic function in three-dimensional Euclidean space is independent of the azimuthal angle $\varphi$. In spherical coordinates its equation isFor a separated solution $R(r)P(\cos \theta )$ smooth at both poles, the angular ordinary differential equation selects the Legendre polynomials $P_{\ell }$, and the radial equation gives $r^{\ell }$ and $r^{-\ell -1}$. Thus on a spherical annulus, a smooth axisymmetric solution has a locally convergent series of modesRegularity in a ball eliminates the singular radial powers. Decay at infinity eliminates the growing powers and the constant term. For example, $r\cos \theta =z$ is regular and harmonic, while $\cos \theta /r^2$ is an exterior dipole potential. These conditions on the domain and poles are essential: the displayed Legendre-polynomial expansion is not an assertion about every solution on a small patch excluding the axis.

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

The normalized order-zero function has the representation

For the Laguerre equationthe ansatz $y={\int }_{\gamma }{e}^{zt}f(t)dt$ givesThe contour must make the endpoint contributionvanish.