Cartan's Magic Formula for Simplicial Complexes

Oliver Knill

Cartan's magic formula L_X = i_X d + d i_X = (d+i_X)^2=D_X^2 relates the exterior derivative d, an interior derivative i_X and its Lie derivative L_X. 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] = L_X L_Y - L_Y L_X 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 i_X^2=0, the relation L_X=D_X^2 with D_X=i_X+d mirrors the Hodge factorization L=D^2, where D=d+d^* we can see f_t = - L_X f defining the flow of X as the analogue of the heat equation f_t = - L f and view the Newton type equations f'' = -L_X 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 f_t(t), also any second order differential equation f'' = -L_X f is solved by u(t) = exp(i D_X t) u(0) in l^2(G,C}). If X is supported on odd forms, the factorization property L_X = D_X^2 extends to the Lie algebra and i_[X,Y] remains an inner derivative. If the kernel of L_X on p-forms has dimension b_p(X), then the general Euler-Poincare formula holds for every parameter field X. Extreme cases are i_X=d^*, where b_k are the usual Betti numbers and X=0, where b_k=f_k(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 L_X on Riemannian manifolds. the non-zero spectrum of L_X on even forms is the same than the non-zero spectrum of L_X on odd forms. We also can deform with D_X' = [B_X,D_X] of D_X=d+i_X + b_X, B_X=d_X-d_X^*+i b_X the exterior derivative d governed by the vector field X.

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