The Structure of Di®erential Invariant Algebras of Lie Groups and Lie Pseudo{Groups Peter J. Olver University of Minnesota http://www.math.umn.edu/ » olver =) Juha Pohjanpelto Jeongoo Cheh à 0 Sur la th¶eorie, si importante sans doute, mais pour nous si obscure, des ¿groupes de Lie in¯nis, nous ne savons rien que ce qui trouve dans les m¶emoires de Cartan, premiere exploration a travers une jungle presque imp¶en¶etrable; mais celle-ci menace de se refermer sur les sentiers d¶eja trac¶es, si l'on ne procede bient^ot a un indispensable travail de d¶efrichement. | Andr¶e Weil, 1947 à 1 Pseudo-groups in Action ² Lie | Medolaghi | Vessiot ² Cartan : : : Guillemin, Sternberg ² Kuranishi, Spencer, Goldschmidt, Kumpera, : : : ² Relativity ² Noether's Second Theorem ² Gauge theory and ¯eld theories Maxwell, Yang{Mills, conformal, string, : : : ² Fluid Mechanics, Metereology Euler, Navier{Stokes, boundary layer, quasi-geostropic , : : : ² Linear and linearizable PDEs ² Solitons (in 2 + 1 dimensions) K{P, Davey-Stewartson, : : : ² Image processing ² Numerical methods | geometric integration ² Kac{Moody symmetry algebras ² Lie groups! à 2 What's New? Direct constructive algorithms for: ² Invariant Maurer{Cartan forms ² Structure equations ² Moving frames ² Di®erential invariants ² Invariant di®erential operators ² Constructive Basis Theorem ² Syzygies and recurrence formulae ² GrÄobner basis constructions ² Further applications =) Symmetry groups of di®erential equations =) Vessiot group splitting =) Gauge theories =) Calculus of variations à 3 Di®erential Invariants G | transformation group acting on p-dimensional submani- folds N = fu = f(x)g ½ M G(n) | prolonged action on the submanifold jet space Jn = Jn(M; p) Di®erential invariant I : Jn ! R I(g(n) ¢ (x; u(n))) = I(x; u(n)) =) curvature, torsion, : : : Invariant di®erential operators: D1; : : : ; Dp =) arc length derivative ? ? I(G) | the algebra of di®erential invariants ? ? à 4 The Basis Theorem Theorem. The di®erential invariant algebra I(G) is gener- ated by a ¯nite number of di®erential invariants I1; : : : ; I`, meaning that every di®erential invariant can be locally expressed as a function of the generating invariants and their invariant derivatives: DJ I· = Dj1Dj2 ¢ ¢ ¢ DjnI·: =) Tresse, Kumpera } } functional independence } } ? ? Constructive Version ? ? =) Computational algebra & GrÄobner bases à 5 The Algebra of Di®erential Invariants Key Issues: ² Minimal basis of generating invariants: I1; : : : ; I` ² Commutation formulae for the invariant di®erential operators: p i [ Dj; Dk ] = Aj;k Di iX=1 =) Non-commutative di®erential algebra ² Syzygies (functional relations) among the di®erentiated invariants: ©( : : : DJ I· : : : ) ´ 0 =) Gauss{Codazzi relations Applications: ² Equivalence and signatures of submanifolds and characteri- zation of moduli spaces ² Computation of invariant variational problems: L( : : : DJ I· : : : ) d! Z ² Group splitting of PDEs à 6 Basic Themes ² The structure of a (connected) pseudo-group is ¯xed by its Lie algebra of in¯nitesimal generators: g ² The in¯nitesimal generators satisfy an overdetermined system of linear partial di®erential equations | the determining equations: F = 0 ² The basic structure of an overdetermined system of PDEs is ¯xed by the algebraic structure of its symbol module: I ² The structure of the di®erential invariant algebra I(G) is ¯xed by the prolonged in¯nitesimal generators: g(1) ² These satisfy an overdetermined system of partial di®er- ential equations | the prolonged determining equa- tions: H = 0 ² The basic structure of the prolonged determining equa- tions is ¯xed by the algebraic structure of its prolonged symbol module: J à 7 Jets and Prolongation Jet = Taylor polynomial/series (Ehresmann) th jnv | n order jet of a vector ¯eld Jn = Jn(M; p) | jets of p-dimensional submanifolds N ½ M v(n) | prolonged vector ¯eld on Jn The prolongation map takes (jets of) vector ¯elds on M to vector ¯elds on the jet space Jn. (1) p : j1v 7¡! v There is an induced dual prolongation map p¤ : J ¡! I on the symbol modules, which is algebraic (at su±ciently high order). With this \algebraic prolongation" in hand, the structure of the prolonged symbol module, and hence the di®erential invariants, is algorithmically determined by the structure of the pseudo-group's symbol module. à 8 Pseudo-groups De¯nition. A pseudo-group is a collection of local di®eomorphisms ' : M ! M such that ² Identity: 1M 2 G, ² Inverses: '¡1 2 G, ² Restriction: U ½ dom ' =) ' j U 2 G, ² Composition: im ' ½ dom à =) à ± ' 2 G. De¯nition. A Lie pseudo-group G is a pseudo-group whose transformations are the solutions to an involutive system of partial di®erential equations: F (z; '(n)) = 0: ² Nonlinear determining equations =) analytic (Cartan{KÄahler) ? ? Key complication: 6 9 Abstract object G ? ? à 9 A Non-Lie Pseudo-group Acting on M = R2: X = '(x); Y = '(y) where ' 2 D(R). Ä Cannot be characterized by a system of partial di®erential equations ¢(x; y; X(n); Y (n)) = 0 Theorem. (Johnson, Itskov) Any non-Lie pseudo-group can be completed to a Lie pseudo-group with the same di®erential invariants. Completion of previous example: X = '(x); Y = Ã(y) where '; à 2 D(R). à 10 In¯nitesimal Generators g | Lie algebra of in¯nitesimal generators of the pseudo-group G z = (x; u) | local coordinates on M Vector ¯eld: m @ p @ q @ v = ³a(z) = »i + '® @za @xi @u® aX=1 iX=1 ®X=1 Vector ¯eld jet: (n) b jnv 7¡! ³ = ( : : : ³A : : : ) #A b k b b @ ³ @ ³ ³A = = @zA @za1 ¢ ¢ ¢ @zak In¯nitesimal (Linearized) Determining Equations L(z; ³(n)) = 0 (¤) Remark: If G is the symmetry group of a system of di®erential equations ¢(x; u(n)) = 0, then (¤) is the (involutive completion of) the usual Lie determining equations for the symmetry group. à 11 Symmetry Groups | Review System of di®erential equations: (n) ¢º(x; u ) = 0; º = 1; 2; : : : ; k Prolonged vector ¯eld: p @ q n @ v(n) = »i + ' ® @xi J @u® iX=1 ®X=1 #XJ=0 J where b p p ® ® ® i ® i ' J = DJ ' ¡ ui » + uJ;i » à ! iX=1 iX=1 b ® (n) (n) (n) ´ ©J (x; u ; » ; ' ) In¯nitesimal invariance: (n) v (¢º) = 0 whenever ¢ = 0: In¯nitesimal determining equations: L(x; u; »(n); '(n)) = 0 i ® i ® L( : : : ; x ; : : : ; u ; : : : ; »A; : : : ; 'A; : : : ) = 0 =) involutive completion à 12 The Korteweg{deVries equation ut + uxxx + uux = 0 Symmetry generator: @ @ @ v = ¿(t; x; u) + »(t; x; u) + '(t; x; u) @t @x @u Prolongation: @ @ @ v(3) = v + 't + 'x + ¢ ¢ ¢ + 'xxx @ut @ux @uxxx where t 2 ' = 't + ut'u ¡ ut¿t ¡ ut ¿u ¡ ux»t ¡ utux»u x 2 ' = 'x + ux'u ¡ ut¿x ¡ utux¿u ¡ ux»x ¡ ux»u xxx ' = 'xxx + 3ux'u + ¢ ¢ ¢ In¯nitesimal invariance: (3) t xxx x v (ut + uxxx + uux) = ' + ' + u ' + ux ' = 0 on solutions à 13 ut + uxxx + uux = 0 In¯nitesimal determining equations: ¿x = ¿u = »u = 't = 'x = 0 2 2 ' = »t ¡ 3 u¿t 'u = ¡ 3 ¿t = ¡ 2 »x ¿tt = ¿tx = ¿xx = ¢ ¢ ¢ = 'uu = 0 General solution: ¿ = c1 + 3c4t; » = c2 + c3t + c4x; ' = c3 ¡ 2c4u: Basis for symmetry algebra: @t; @x; t @x + @u; 3t @t + x @x ¡ 2u @u: The symmetry group GKdV is four-dimensional (x; t; u) 7¡! (¸3t + a; ¸x + ct + b; ¸¡2u + c) à 14 Moving Frames For a ¯nite-dimensional Lie group action, a moving frame is de¯ned to be a (right) equivariant map ½(n) : Jn ¡! G where ½(n)(g(n) ¢ z(n)) = ½(n)(z(n)) ¢ g¡1 Moving frames are explicitly constructed by choosing a cross-section to the group orbits, and solving the normaliza- tion equations for the group parameters. The existence of a moving frame requires that the group action be free, i.e., that there is no isotropy, or (locally) that the group orbits have the same dimension as the group itself. à 15 Moving Frames for Pseudo{Groups ² The role of the group parameters is played by the pseudo- group jet coordinates, while the group is replaced by a certain principal bundle H(n) ' Jn £ G(n) ¡! Jn. ² There are an in¯nite number of pseudo-group parameters to normalize, which are to be done order by order. (Or else use the Taylor series approach.) However, at each ¯nite order, the algebraic manipulations in the moving frame normalization procedure are identical to the ¯nite- dimensional calculus. ² Since the pseudo-group G is in¯nite-dimensional, classical freeness is impossible! Rather, a pseudo-group action is said to act freely at a jet z(n) 2 Jn if the only pseudo- group elements which ¯x z(n) are those whose n-jet coincides with the identity n-jet. Theorem. If G acts freely on Jn for n ¸ 1, then it acts freely on all Jk for k ¸ n. n? | order of freeness. à 16 Normalization Ä To construct a moving frame : I. Choose a cross-section to the pseudo-group orbits: ®· (n) uJ· = c·; · = 1; : : : ; rn = ¯ber dim G II. Solve the normalization equations ®· (n) (n) FJ· (x; u ; g ) = c· for the pseudo-group parameters g(n) = ½(n)(x; u(n)) III. Invariantization maps di®erential functions to di®erential invariants: ¶ : F (x; u(n)) 7¡! I(x; u(n)) = F ( ½(n)(x; u(n)) ¢ (x; u(n)) ) =) an algebra morphism and a projection: ¶ ± ¶ = ¶ I(x; u(n)) = ¶( I(x; u(n)) ) à 17 Invariantization A moving frame induces an invariantization process, denoted ¶, that projects functions to invariants, di®erential operators to invariant di®erential operators; di®erential forms to invari- ant di®erential forms, etc. Geometrically, the invariantization of an object is the unique invariant version that has the same cross-section values. Algebraically, invariantization amounts to replacing the group parameters in the transformed object by their moving frame formulas. à 18 Invariantization In particular, invariantization of the jet coordinates leads to a complete system of functionally independent