The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Abstract

Let A n be the nth Weyl algebra and P m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A n ⊗ P m } is proved: an algebra A admits a finite set δ 1 , . . . , δ s of commuting locally nilpotent derivations with generic kernels and ∩ s i=1 ker(δ i ) = K iff A A n ⊗ P m for some n and m with 2n + m = s, and vice versa. The inversion formula for automorphisms of the algebra A n ⊗ P m (and for P m :has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given σ ∈ Aut K (P m ), then deg σ −1 ≤ (deg σ ) m−1 (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given σ ∈ Aut K (A n ⊗ P m ), then deg σ −1 ≤ (deg σ ) 2n+m−1 . Any automorphism σ ∈ Aut K (P m ) is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for σ ∈ Aut K (A n ⊗ P m ). One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242.[6]] problem 1) into a single question, (JD): is a K -algebra endomorphism σ : A n ⊗ P m → A n ⊗ P m an algebra automorphism provided σ (P m ) ⊆ P m and det( ∂σ (x i )

The groups of automorphisms of the Lie algebras of triangular polynomial derivations

“…In combination with Theorem 4.13. (1), this fact yields the main result of the paper G n = T n ⋉(UAut K (P n ) n ⋊(F ′ n ×E n )) (Theorem 5.3. (1)), where…”

“…• (Theorem 3.8) 1. G n = TAut K (P n ) n F n = F n TAut K (P n ) n and TAut K (P n ) n ∩ F n = Sh n−1 .…”

Factorization of Monomorphisms of a Polynomial Algebra in One Variable

“…Proof. Using induction on the degree deg(σ ) and (1), it suffices to show that there are finitely many distinct decompositions for s = 2, i.e. σ = σ 1 σ 2 .…”

“…If a monomorphism δ ∈ is decomposable, δ = σ τ , then one can write formulae for the monomorphisms σ and τ via the coefficients of the polynomial δ(x) and the degree deg(σ ) of σ (Theorem 3.2). In order to do so, we use the inversion formula [1] for an automorphism of a polynomial algebra P n : = K[x 1 , . .…”

“…In [1], the theorem is proved for a field K of characteristic zero but the proof goes without a change for an arbitrary commutative ‫-ޑ‬algebra K. The formulae for σ and τ in δ = σ τ . Let δ = σ τ be as in Theorem 2.2.…”