ASSOCIATIVE, LIE, AND LEFT-SYMMETRIC ALGEBRAS OF DERIVATIONS

Let P (n) = k[x (1) , x (2) ,aEuro broken vertical bar,x (n) ] be the polynomial algebra over a field k of characteristic zero in the variables x (1) , x (2) ,aEuro broken vertical bar,x (n) and a"' (n) be the left-symmetric Witt algebra of all derivations of P (n) [Bu]. Using the language of a"' (n) , for every derivation D a a"' (n) we define the associative algebra Ass(D), the Lie algebra Lie(D), and the left-symmetric algebra a"' (D) related to the study of the Jacobian Conjecture. For every derivation D a a"' (n) there is a unique n-tuple F = (f (1) , f (2) ,aEuro broken vertical bar,f (n) ) of elements of P (n) such that D = D (F) = f (1) a, (1) + f (2) a, (2) +a <- + f (n) a, (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) ,aEuro broken vertical bar,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 a"' (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.