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 Ass_D, the Lie algebra Lie_D, 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 ^[r] _F 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.