# Vladimir V. Bavula

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Let d be the left homological dimension or weak dimension. In the present paper we give for a large class of algebras the answer to the question: when the dimension of the tensor product of algebras is the sum of dimensions of the multiples, Tensor d-minimal algebras are introduced and it is proved that a large class of algebras are tensor d-minimal(More)
The algebra Sn in the title is obtained from a polynomial algebra Pn in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of Pn. Ignoring non-Noetherian property, the algebra Sn belongs to a family of algebras like the Weyl algebra An and the polynomial algebra P2n. The group of automorphisms Gn of the algebra Sn(More)
Let K be a perfect field of characteristic p > 0, A1 := K〈x, ∂ | ∂x−x∂ = 1〉 be the first Weyl algebra and Z := K[X := x, Y := ∂] be its centre. It is proved that (i) the restriction map res : AutK(A1) → AutK(Z), σ 7→ σ|Z , is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z) | J (τ) = 1} where J (τ) is the Jacobian of τ (note that AutK(Z) = K ∗ ⋉ Γ and if K(More)
Let Pn := K[x1, . . . , xn] be a polynomial algebra over a field K of characteristic zero. The Jacobian algebra An is the subalgebra of EndK(Pn) generated by the Weyl algebra An := D(Pn) = K〈x1, . . . , xn, ∂1, . . . , ∂n〉 and the elements (∂1x1) −1, . . . , (∂nxn) −1 ∈ EndK(Pn). The algebra An appears naturally in study of the group of automorphisms of Pn.(More)
Let An be the n’th Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {An⊗Pm} is proved: an algebra A admits a finite set δ1, . . . , δs of commuting locally nilpotent derivations with generic kernels and ∩i=1ker(δi) = K iff A ≃ An ⊗ Pm for some n and m with 2n + m(More)
For the ring of differential operators on a smooth affine algebraic variety X over a field of characteristic zero a finite set of algebra generators and a finite set of defining relations are found explicitly. As a consequence, a finite set of generators and a finite set of defining relations are given for the module DerK(O(X)) of derivations on the algebra(More)
Let K be a field of characteristic p > 0. It is proved that the group Autord(D(Ln)) of order preserving automorphisms of the ring D(Ln) of differential operators on a Laurent polynomial algebra Ln := K[x ±1 1 , . . . , x n ] is isomorphic to a skew direct product of groups Z n p⋊AutK(Ln) where Zp is the ring of p-adic integers. Moreover, the group(More)
There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the Jacobian set – the set of all algebra endomorphisms of a polynomial algebra with the(More)