Cartan's Magic Formula for Simplicial Complexes

Cartan's magic formula LX = iX d + d iX = (d+iX)^2=D_X2 relates the exterior derivative d, an interior derivative iX and its Lie derivative LX. We use this formula to define a finite dimensional vector space of vector fields X on a finite abstract simplicial complex G. This space has a Lie algebra structure satisfying L[X,Y] = LX LY - LY LX as in the continuum. Any such vector field X defines a coordinate change on the finite dimensional vector space l^2(G) which play the role of translations along the vector field. If iX^2=0, the relation LX=DX² with DX=iX+d mirrors the Hodge factorization L=D^2, where D=d+d^* we can see ft = - LX f defining the flow of X as the analogue of the heat equation ft = - L f and view the Newton type equations f'' = -LX f as the analogue of the wave equation f'' = -L f. Similarly as the wave equation is solved by u(t)=exp(i Dt) u(0) with complex valued u(t)=f(t)-i D^-1 ft(t), also any second order differential equation f'' = -LX f is solved by u(t) = exp(i DX t) u(0) in l^2(G,C}). If X is supported on odd forms, the factorization property LX = DX² extends to the Lie algebra and i[X,Y] remains an inner derivative. If the kernel of LX on p-forms has dimension bp(X), then the general Euler-Poincare formula holds for every parameter field X. Extreme cases are iX=d^*, where bk are the usual Betti numbers and X=0, where bk=fk(G) are the components of the f-vector of the simplicial complex G. We also note that the McKean-Singer super-symmetry extends from L to Lie derivatives. It also holds for LX on Riemannian manifolds. the non-zero spectrum of LX on even forms is the same than the non-zero spectrum of LX on odd forms. We also can deform with DX' = [BX,DX] of DX=d+iX + bX, BX=dX-dX^*+i b_X the exterior derivative d governed by the vector field X.