3 · lie theory

lie groups, lie algebras and actions

A Lie group $G$ is a smooth manifold equipped with a group structure whose multiplication and inversion maps are smooth. Its Lie algebra $\mathfrak{g}=T_e G$ is the tangent space at the identity, endowed with the Lie bracket of left-invariant vector fields. The exponential map $$ \exp:\mathfrak{g}\to G,\qquad \xi\mapsto\gamma_\xi(1), $$ where $\gamma_\xi$ is the one-parameter subgroup generated by $\xi$, linearizes the group law in a neighborhood of the identity. An action of $G$ on a smooth manifold $M$ is a smooth map $$ G\times M\to M,\qquad (g,x)\mapsto g\cdot x $$ satisfying the usual axioms. Differentiating the action at the identity produces the infinitesimal generators $$ \xi_M(x)=\frac{d}{dt}\Big|_{t=0}\exp(t\xi)\cdot x, $$ which are vector fields on $M$ and define a Lie-algebra homomorphism $\mathfrak{g}\to\mathfrak{X}(M)$. The de Rham complex of differential forms on $M$ is naturally $G$-equivariant; its cohomology therefore carries a representation of $G$ and serves as a repository of global topological invariants of the action.

definition (lie group). a smooth manifold $G$ together with a group structure such that multiplication $G\times G\to G$ and inversion $G\to G$ are smooth maps.

definition (left-invariant vector field). a vector field $X$ on $G$ is left-invariant if $T_h L_g\cdot X(h)=X(gh)$ for all $g,h\in G$, where $L_g(h)=gh$. identifying $X(e)$ with an element of $T_e G$ yields a linear isomorphism between left-invariant fields and $\mathfrak{g}$.

definition (lie bracket on $\mathfrak{g}$). for $\xi,\eta\in\mathfrak{g}$ let $X_\xi$, $X_\eta$ be the associated left-invariant fields. then $$ [\xi,\eta]:=[X_\xi,X_\eta](e)\in T_e G=\mathfrak{g}, $$ where $[X_\xi,X_\eta]$ is the ordinary Lie bracket of vector fields on $G$.

theorem (lie algebra axioms). the product $[\cdot,\cdot]$ on $\mathfrak{g}$ is bilinear, skew-symmetric, and satisfies the Jacobi identity $$ [\xi,[\eta,\zeta]]+[\eta,[\zeta,\xi]]+[\zeta,[\xi,\eta]]=0. $$

proof. bilinearity and skew-symmetry are inherited from the Lie bracket of vector fields. the Jacobi identity holds for all smooth vector fields on any manifold (expand $[X,[Y,Z]]$ in coordinates, or use $[X,Y]f=X(Yf)-Y(Xf)$ three times on an arbitrary smooth function $f$). restricting to left-invariant fields and evaluating at $e$ yields the identity on $\mathfrak{g}$. $\square$

definition (smooth action). a smooth left action of $G$ on $M$ is a smooth map $G\times M\to M$, $(g,x)\mapsto g\cdot x$, with $e\cdot x=x$ and $(gh)\cdot x=g\cdot(h\cdot x)$.

definition (infinitesimal generator). for $\xi\in\mathfrak{g}$ the fundamental vector field (infinitesimal generator) $\xi_M\in\mathfrak{X}(M)$ is $$ \xi_M(x)=\frac{d}{dt}\Big|_{t=0}\exp(t\xi)\cdot x. $$

theorem (homomorphism $\mathfrak{g}\to\mathfrak{X}(M)$). for a smooth left action, $$ [\xi_M,\eta_M]=[\xi,\eta]_M $$ for all $\xi,\eta\in\mathfrak{g}$. thus $\xi\mapsto\xi_M$ is a Lie-algebra homomorphism $\mathfrak{g}\to\mathfrak{X}(M)$.

proof. for fixed $x\in M$ define $\Phi_g(x)=g\cdot x$. then $\xi_M(x)=\frac{d}{dt}|_{0}\Phi_{\exp(t\xi)}(x)$. the flow of $\xi_M$ is $\varphi_t^{\xi}(x)=\exp(t\xi)\cdot x$, because $$ \frac{d}{dt}\bigl(\exp(t\xi)\cdot x\bigr) =\xi_M\bigl(\exp(t\xi)\cdot x\bigr) $$ by the chain rule and the group law. the Lie bracket of two complete vector fields is recovered from the commutator of their flows: $$ [\xi_M,\eta_M] =\frac12\frac{\partial^2}{\partial s\,\partial t}\Big|_{s=t=0} \bigl( \varphi_{-s}^{\xi}\circ\varphi_{-t}^{\eta}\circ\varphi_{s}^{\xi}\circ\varphi_{t}^{\eta} \bigr) $$ (standard formula). substituting $\varphi_t^{\xi}=\Phi_{\exp(t\xi)}$ and using $\exp(s\xi)\exp(t\eta)\exp(-s\xi)=\exp\bigl(t\,\mathrm{Ad}_{\exp(s\xi)}\eta\bigr)$ together with $$ \frac{d}{ds}\Big|_{0}\mathrm{Ad}_{\exp(s\xi)}\eta=[\xi,\eta] $$ shows that the infinitesimal commutator is $[\xi,\eta]_M$. (equivalently: differentiate the identity $\Phi_g\circ\varphi_t^{\eta}=\varphi_t^{\mathrm{Ad}_g\eta}\circ\Phi_g$ in $g$ at $e$.) $\square$

remark (exponential near the identity). the map $\exp$ is a local diffeomorphism $\mathfrak{g}\supset U\ni 0\to V\subset G$ about $e$. thus every element of $G$ near the identity is $\exp(\xi)$ for a unique small $\xi$, which is why infinitesimal generators control local continuous group actions.

symmetry groups of differential equations

A local transformation group acting on the space of independent and dependent variables is called a symmetry group of a system of differential equations if it maps the solution set into itself. Equivalently, the graph of every solution is sent to the graph of another solution. The practical determination of such groups proceeds by linearizing the invariance condition, which leads to the notion of prolongation.

definition (local symmetry group). let $\Delta(x,u^{(n)})=0$ be a system of differential equations of order $n$ on independent variables $x$ and dependent variables $u$. a local Lie group $G$ acting on an open set of $(x,u)$-space is a symmetry group of $\Delta=0$ if, whenever $u=f(x)$ is a local solution, every sufficiently small transformed graph $g\cdot(x,f(x))$ is again the graph of a local solution.

remark. for evolutionary systems and ODEs this means the transformed function satisfies the same equation. the infinitesimal form of the condition is that the prolonged generator be tangent to the solution manifold in jet space (next section).

prolongation to jet spaces

The $k$-th jet space $J^k$ is the manifold whose coordinates are the independent variables, the dependent variables, and all partial derivatives up to order $k$. A local diffeomorphism of the base space lifts uniquely to a contact transformation of $J^k$ (the $k$-th prolongation) that preserves the contact ideal. For a vector field $$ \mathbf{v} =\xi^i(x,u)\partial_{x^i}+\varphi^\alpha(x,u)\partial_{u^\alpha} $$ the prolongation formula supplies the coefficients of all higher-order terms: $$ \varphi^{\alpha,J} =D_J\bigl(\varphi^\alpha-\xi^i u^\alpha_i\bigr) +\xi^i u^\alpha_{J,i}, $$ where $D_J$ denotes the total derivative corresponding to the multi-index $J$. A vector field is an infinitesimal symmetry of a system $\Delta=0$ if its prolongation is tangent to the submanifold defined by $\Delta$ and all its differential consequences. The resulting linear system of determining equations for the coefficients $\xi^i$ and $\varphi^\alpha$ is over-determined and, in favourable cases, can be solved explicitly.

definition (first jet space for one independent variable). for an ODE in one dependent variable $u(x)$, coordinates on $J^1$ are $(x,u,p)$ with $p$ standing for $u_x$. the contact form is $$ \theta=du-p\,dx. $$

definition (first prolongation of a vector field on $(x,u)$). for $$ \mathbf{v}=\xi(x,u)\partial_x+\varphi(x,u)\partial_u $$ the first prolongation is $$ \mathrm{pr}^{(1)}\mathbf{v} =\xi\partial_x+\varphi\partial_u+\varphi^x\partial_p, $$ with $$ \varphi^x =D_x(\varphi-\xi p)+\xi p_x =D_x\varphi-p\,D_x\xi, $$ where $D_x=\partial_x+p\partial_u+p_x\partial_p+\cdots$ is the total derivative with respect to $x$.

theorem (infinitesimal criterion for ODEs of order one). let $\Delta(x,u,p)=0$ define a regular first-order ODE $p=F(x,u)$ (or more generally a regular hypersurface in $J^1$). a vector field $\mathbf{v}$ is an infinitesimal symmetry if and only if $$ \mathrm{pr}^{(1)}\mathbf{v}(\Delta)=0 \quad\text{whenever}\quad \Delta=0. $$ in evolutionary form $u'=F(x,u)$ this is equivalent to $$ \varphi^x=F_x\xi+F_u\varphi+F_p\varphi^x $$ holding on $p=F$, which yields a linear determining equation for $\xi$ and $\varphi$.

proof. locally the solution graphs of the ODE are integral curves of the contact condition $\theta=0$ constrained to $\Delta=0$. a one-parameter group of point transformations maps $C^1$ curves to $C^1$ curves. it maps solutions to solutions if and only if it maps the submanifold $\mathcal{S}=\{\Delta=0\}\subset J^1$ into itself while preserving the contact ideal restricted to $\mathcal{S}$. for a one-parameter group $\Phi_t$ generated by $\mathbf{v}$, this is equivalent to the prolonged field $\mathrm{pr}^{(1)}\mathbf{v}$ being tangent to $\mathcal{S}$, i.e. $\mathrm{pr}^{(1)}\mathbf{v}(\Delta)|_{\mathcal{S}}=0$. (preservation of contact is automatic for prolongations of point transformations by construction of $\varphi^x$.) substituting the coordinate formula for $\varphi^x$ and restricting to $\Delta=0$ produces the displayed determining equation. $\square$

remark (higher order). the same tangency criterion with $\mathrm{pr}^{(k)}\mathbf{v}$ applies to systems of order $k$ on $J^k$. the prolongation formula quoted above is the inductive mechanism that produces all higher coefficients from $\xi$ and $\varphi$ alone; no independent higher-order data appear for point symmetries.

differential invariants and reduction

When the symmetry group is solvable, a complete set of functionally independent differential invariants can be constructed by successive quadrature. These invariants serve as new coordinates in which the original equation is reduced in order. For a single ordinary differential equation of order $n$ admitting an $r$-dimensional solvable symmetry algebra, the order can be lowered by $r$, often permitting integration by quadratures. The same invariants yield explicit formulae for the general solution when the reduced equation is solvable.

definition (differential invariant). a function $I$ on an open set of $J^k$ is a differential invariant of the prolonged action of $G$ if $\mathrm{pr}^{(k)}\mathbf{v}(I)=0$ for every infinitesimal generator $\mathbf{v}$ of the action.

theorem (order reduction for a one-parameter symmetry). let $\Delta(x,u,u',\dots,u^{(n)})=0$ be an $n$-th order ODE admitting a nowhere-vanishing infinitesimal point symmetry $\mathbf{v}$. then there exist local coordinates $(y,w)$ on the $(x,u)$-plane in which $\mathbf{v}=\partial_w$, and the ODE reduces to an equation of order $n-1$ for $w$ as a function of $y$ after using the invariant independent variable and the first differential invariants built from the prolonged action.

proof. because $\mathbf{v}\neq 0$ at a given point, rectification of a non-vanishing vector field (chapter 2) yields local coordinates $(y,w)$ with $\mathbf{v}=\partial_w$. in these coordinates the prolonged generators have no $\partial_y$ component at order zero, and the invariance condition forces the original equation, rewritten in jet coordinates of $(y,w)$, to be independent of $w$ itself (only derivatives of $w$ with respect to $y$ enter). setting $z=w_y$ therefore produces an equation of order $n-1$ in $z(y)$. recovering $w$ from $z$ is a single quadrature $w=\int z\,dy+C$. $\square$

remark (solvable algebras). if an $r$-dimensional symmetry algebra admits a chain of subalgebras of dimensions $1,2,\dots,r$ (solvability), one reduces order by one at each step, obtaining a cascade of quadratures when each reduced equation remains solvable. this is Lie's classical integration algorithm for ODEs.

lie structures and special equations

A Lie structure on a manifold is a Lie-algebra homomorphism from a fixed abstract Lie algebra into the Lie algebra of vector fields. Prolongation of such a structure produces a Lie algebra of contact vector fields on the jet space. The classical Riccati equation $$ y'=a(x)+b(x)y+c(x)y^2 $$ is characterized by the property that its contact symmetries form a copy of $\mathfrak{sl}(2,\mathbb{R})$. The associated linear fractional transformations act on the projective line of initial values and account for the nonlinear superposition principle that expresses the general solution in terms of three particular solutions. Similar geometric analyses apply to other equations whose symmetry algebras are low-dimensional classical Lie algebras.

definition (lie structure of type $\mathfrak{g}$). a Lie-algebra homomorphism $\rho:\mathfrak{g}\to\mathfrak{X}(M)$ is a Lie structure of type $\mathfrak{g}$ on $M$.

theorem (riccati superposition). let $y_1,y_2,y_3$ be three particular solutions of a Riccati equation $y'=a+by+cy^2$ with $c\not\equiv 0$, and suppose they are pairwise distinct on an interval. then the general solution is given by the cross-ratio formula $$ \frac{y-y_1}{y-y_2}\Big/\frac{y_3-y_1}{y_3-y_2} =\mathrm{const}. $$ equivalently, the general solution is a fractional linear function of an arbitrary constant involving $y_1,y_2,y_3$.

proof. if $c\not\equiv 0$ the substitution $y=-u'/(c u)$ linearizes the Riccati equation to a second-order linear homogeneous ODE for $u$. three particular solutions of the Riccati equation correspond to three pairs of bases of the two-dimensional solution space of that linear ODE. the projective coordinate on the space of solutions is a fractional linear function of the integration constants. eliminating the linear solution basis between three particular Riccati solutions yields a constant cross ratio among $y,y_1,y_2,y_3$ along any solution $y$. thus $$ \frac{(y-y_1)/(y-y_2)}{(y_3-y_1)/(y_3-y_2)} $$ is independent of $x$, which is the stated formula. $\square$

remark ($\mathfrak{sl}(2)$ action). fractional linear transformations $y\mapsto(\alpha y+\beta)/(\gamma y+\delta)$ with $\alpha\delta-\beta\gamma=1$ realize $\mathrm{PSL}(2)$ on the projective line of values of $y$. they map the family of solutions of a fixed Riccati equation to itself, which is the finite form of the $\mathfrak{sl}(2,\mathbb{R})$ symmetry algebra.

pfaffian systems and derived flags

A Pfaffian system on a manifold $M$ is a locally free submodule $\mathcal{I}\subset\Omega^1(M)$. Its dual is the distribution (vector-field system) $$ \mathcal{D} =\mathcal{I}^\perp =\bigl\{X\in\mathfrak{X}(M)\mid \langle\theta,X\rangle=0\text{ for all }\theta\in\mathcal{I}\bigr\}. $$ The derived system $\mathcal{I}'$ is generated by those forms in $\mathcal{I}$ whose exterior derivatives vanish modulo $\mathcal{I}$. Iteration produces the derived flag $$ \mathcal{I}\supset\mathcal{I}'\supset\mathcal{I}''\supset\cdots. $$ The system is integrable (in the sense of Frobenius) if and only if $\mathcal{I}'=\mathcal{I}$. The ranks of the successive derived systems are differential invariants that classify the local equivalence problem for the distribution.

definition (pfaffian system). a $C^\infty$ locally free submodule $\mathcal{I}\subset\Omega^1(M)$ of constant rank is a Pfaffian system. its annihilator $\mathcal{D}=\mathcal{I}^\perp$ is a distribution of constant rank.

definition (derived system). $$ \mathcal{I}' =\bigl\{\theta\in\mathcal{I}\mid d\theta\equiv 0\pmod{\mathcal{I}}\bigr\}. $$

theorem (frobenius, pfaffian form). a regular Pfaffian system $\mathcal{I}$ is completely integrable (through each point there passes a unique maximal integral manifold of dimension $\mathrm{rank}\,\mathcal{D}$) if and only if $\mathcal{I}'=\mathcal{I}$, equivalently if and only if $d\theta\equiv 0\pmod{\mathcal{I}}$ for every local generator $\theta$ of $\mathcal{I}$. in dual language, $\mathcal{D}$ is integrable if and only if it is involutive: $X,Y\in\Gamma(\mathcal{D})$ implies $[X,Y]\in\Gamma(\mathcal{D})$.

proof. (involutivity $\Leftrightarrow$ ideal condition.) let $\theta_1,\dots,\theta_k$ be local generators of $\mathcal{I}$, and let $X_1,\dots,X_{n-k}$ be a local frame of $\mathcal{D}$. for any vector fields $X,Y$ in $\mathcal{D}$, $$ d\theta(X,Y) =X\langle\theta,Y\rangle-Y\langle\theta,X\rangle-\langle\theta,[X,Y]\rangle =-\langle\theta,[X,Y]\rangle $$ because $\langle\theta,X\rangle=\langle\theta,Y\rangle=0$. thus $d\theta\equiv 0\pmod{\mathcal{I}}$ on $\mathcal{D}\times\mathcal{D}$ for all $\theta\in\mathcal{I}$ if and only if $\langle\theta,[X,Y]\rangle=0$ for all such pairs, i.e. $[X,Y]\in\mathcal{D}$. this is involutivity of $\mathcal{D}$, which is equivalent to $\mathcal{I}'=\mathcal{I}$.

(involutivity $\Rightarrow$ local integral manifolds.) this is the classical Frobenius theorem for distributions: if $\mathcal{D}$ is involutive of rank $m$, there exist local coordinates in which $\mathcal{D}$ is spanned by $\partial/\partial x^1,\dots,\partial/\partial x^m$, so that the plaques $x^{m+1}=\mathrm{const},\dots,x^n=\mathrm{const}$ are integral manifolds. the construction proceeds by successive application of the flow-box theorem for $m$ commuting (or involutive) fields, exactly as in the straightening argument of chapter 2, upgraded by the Jacobi identity to keep the remaining fields tangent to successive slices. uniqueness of maximal integral manifolds is the uniqueness of ODE trajectories for the spanning fields. $\square$

remark (derived flag). when $\mathcal{I}'\neq\mathcal{I}$ the successive ranks $\mathrm{rank}\,\mathcal{I}^{(j)}$ are local invariants of the distribution under diffeomorphisms; they are the first numerical data of the Cartan-Kahler / equivalence analysis of exterior systems.

monge systems and singular characteristics

Monge systems arise as exterior differential systems that encode under-determined ordinary differential equations or control-theoretic constraints of the form $$ \dot x=f(x,u),\qquad y=g(x,u). $$ They are generated by a collection of one-forms and a single two-form. The singular characteristic curves of the closed two-form determine the abnormal extremals of the associated optimal-control problem. Regularization of these curves yields a geometric criterion for controllability: the Lie algebra generated by the control vector fields must span the tangent space at every point (Chow's theorem). Thus the same formal apparatus that classifies symmetries of differential equations also governs the accessibility properties of non-linear control systems.

definition (control system of driftless type). on a manifold $M$, a family of vector fields $X_1,\dots,X_m$ defines the driftless control system $$ \dot x=\sum_{i=1}^m u_i(t)\,X_i(x),\qquad u_i\in\mathbb{R}. $$ the reachable set from $x_0$ is the set of points attained by piecewise constant controls in finite time.

theorem (chow, statement). if $M$ is connected and the Lie algebra generated by $X_1,\dots,X_m$ under iterated brackets spans $T_x M$ at every $x$ (the Lie algebra rank condition), then the reachable set from any point is all of $M$. in particular the system is completely controllable.

remark. the proof constructs motions in bracket directions by commutators of flows (as in the proof that $[\xi_M,\eta_M]=[\xi,\eta]_M$ recovers Lie brackets from group commutators). full measure-theoretic and compactness details belong to geometric control; the theorem is the precise geometric criterion previewed by singular characteristics of Monge systems.

theorem (bracket generates new directions). if $X,Y$ are smooth vector fields with complete flows $\varphi_t$, $\psi_s$, then $$ \frac{\partial^2}{\partial s\,\partial t}\Big|_{0} \bigl( \varphi_{-t}\circ\psi_{-s}\circ\varphi_t\circ\psi_s(x) \bigr) =[X,Y](x). $$

proof. write $F(s,t)=\varphi_{-t}\circ\psi_{-s}\circ\varphi_t\circ\psi_s(x)$. then $F(s,0)=F(0,t)=x$ for all small $s,t$, so the first partials at $(0,0)$ vanish. the mixed partial is the Lie derivative of $Y$ along $X$ minus the Lie derivative of $X$ along $Y$, which is the coordinate formula for $[X,Y]$. (expand each flow to order two in its parameter and compose; all first-order terms cancel and the second-order cross term is $[X,Y]$.) $\square$

summary

Lie theory converts the search for symmetries into a systematic algebraic procedure on jet space. Prolongation produces the determining equations, differential invariants effect the reduction, and the dual languages of Pfaffian systems and vector-field distributions furnish the geometric setting in which integrability and controllability are decided. The resulting framework unifies the classical integration methods for ordinary differential equations with the modern geometric theory of control and exterior differential systems.

exercises

exercise 1 (matrix lie group). show that $\mathrm{GL}(n,\mathbb{R})$ is a Lie group, identify $T_I\mathrm{GL}(n)$ with $\mathrm{M}_n(\mathbb{R})$, and verify that the Lie bracket of left-invariant fields corresponds to the matrix commutator $[A,B]=AB-BA$.

exercise 2 (symmetry of a first-order ODE). take $u'=0$ (constant solutions) on the line. show that every vector field of the form $\mathbf{v}=\xi(x)\partial_x+\varphi(u)\partial_u$ with $\varphi$ independent of $x$ and $\xi$ independent of $u$ is an infinitesimal symmetry, and interpret the corresponding finite transformations.

exercise 3 (frobenius in $\mathbb{R}^3$). let $\theta=dz-y\,dx$ on $\mathbb{R}^3$ with coordinates $(x,y,z)$. compute $d\theta$ and decide whether the Pfaffian system $\mathcal{I}=\mathrm{span}\{\theta\}$ is integrable. describe the integral manifolds if it is, or exhibit two sections of $\mathcal{D}$ whose bracket leaves $\mathcal{D}$ if it is not.

exercise 4 (riccati cross ratio). for $y'=y^2$ verify by direct differentiation that if $y_1,y_2,y_3,y$ are solutions then the cross ratio $((y-y_1)/(y-y_2))/((y_3-y_1)/(y_3-y_2))$ is constant on any interval where all four are defined and the denominators do not vanish.