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56. Definition. 57. Theorem. If B is an even superskewsymmetric matrix with coefficients in K then B is superHamiltonian. Proof. 44), = 0 for any even R E G'N. 58. Lemma. Let A : G N -+ G N be an even operator with coefficients in K. Then [DO,A](R) = 0 for any even R E G,N. Proof. 19). 59. Remark. If 9 is a finite-dimensional Lie algebra over a field then the ring of polynomial functions on the dual space 9* to 9 possesses a natural Hamiltonian structure whose associated Poisson bracket can be defined very simply by: (a) being a derivation with respect to each argument; and (b) CHAPTER!

Then there exists H E Proof. e such that -R = 8H 8q. Define a map At : C --- Crt] by At(J) homogeneous elements f E = tdeg(f) f for degree- C, and extend it by additivity on C. 69) H =L • qi At (Ri) dt. 0 8H We shall show that 8q - = R. 17), it is xt R, V X enough to show that X(H) "-' E Dev. Obviously, it is enough to consider only Z2-homogeneous X's and R's, and degree-homogeneous R's. 70) = LX, At(R,)dt+ o 1 X( At( R,)) dt . 68) then so does At Ol). 70). 68), we have consequently :LX(Qi)qi = :Lgc)l'(X)).

A homomorphism is canonical if and only if where (q,)) and ¥ = (, = I acts on operators matrix elements-wise by the 45 SUPERHAMILTONIAN FORMALISM Proof. 61) for even H's. 61) is equivalent to 4>XH (q) = X(H)4>(q), or 4>(XH) = X(H) (4)). 64) is satisfied then 4> is canonical. JqjXj,Xj E K. 65). 66. Remark. 67) 4>( {H, F}) = {4>(H), 4>(F)}, VH, FE C. 68) 4>({H,F})'" {4>(H),4>(F)}, VH,FEC'. Let us show that both definitions are equivalent. 68) in the form CHAPTER! 61) ( for this H).

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