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Let P  n  = k[x  1  , x  2  ,…,x  n ] be the polynomial algebra over a field k of characteristic zero in the variables x  1  , x  2  ,…,x  n  and ℒ  n  be the left-symmetric Witt algebra of all derivations of P  n  [Bu]. Using the language of ℒ  n , for every derivation D ∈ ℒ  n  we define the associative algebra AssD, the Lie algebra LieD, and the left-symmetric algebra ℒ  D  related to the study of the Jacobian Conjecture. For every derivation D ∈ ℒ  n  there is a unique n-tuple F = (f  1  , f  2  ,…,f  n ) of elements of P  n  such that D = D  F  = f  1 ∂ 1  + f  2 ∂ 2  +⋯ + f  n ∂ n . In this case, using an action of the Hopf algebra of noncommutative symmetric functions NSymm on P  n , we show that these algebras are closely related to the description of coefficients of the formal inverse to the polynomial endomorphism X + tF, where X = (x  1  , x  2  ,…,x  n ) and t is an independent parameter. We prove that the Jacobian matrix J(F) is nilpotent if and only if all right powers D   F  [r]  of D  F  in ℒ  n  have zero divergence. In particular, if J(F) is nilpotent then DF is right nilpotent. We give one formula for the coefficients of the formal inverse to X + tF as a left-symmetric polynomial in one variable and formulate some open questions.

ASSOCIATIVE, LIE, AND LEFT-SYMMETRIC ALGEBRAS OF DERIVATIONS