ASSOCIATIVE, LIE, AND LEFT-SYMMETRIC ALGEBRAS OF DERIVATIONS

Abstract

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 [] 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.