Related papers: The starred Dixmier's conjecture

Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $\alpha$ be the exchange involution on $A_1(K)$ given by $\alpha(x)= y$ and $\alpha(y)= x$. The Dixmier conjecture of…

The well-known Dixmier conjecture asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the $\gamma,\delta$ conjecture, and…

Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove that the Dixmier conjecture for the first Weyl algebra is true, i.e. each algebra endomorphism of the algebra $A_1$ is an…

The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_1$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P, Q \in A_1$ then $A_1 = K \langle P, Q \rangle$. The Weyl algebra…

Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution…

Let $A_1:=K\langle x, \frac{d}{dx} \rangle$ be the Weyl algebra and $\mI_1:= K\langle x, \frac{d}{dx}, \int \rangle$ be the algebra of polynomial integro-differential operators over a field $K$ of characteristic zero. The Conjecture/Problem…

The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $\alpha$ be the exchange involution on $K[x,y]$: $\alpha(x)= y$ and $\alpha(y)=…

Let $A_1=K < X, Y | [Y,X]=1>$ be the (first) Weyl algebra over a field $K$ of characteristic zero. It is known that the set of eigenvalues of the inner derivation $\ad (YX)$ of $A_1$ is $\Z$. Let $ A_1\ra A_1$, $X\mapsto x$, $Y\mapsto y$,…

We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms…

The two-dimensional Jacobian Conjecture says that a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ that has an invertible Jacobian is an automorphism. We show that if a $\mathbb{C}$-algebra endomorphism…

We study algebra endomorphisms and derivations of some localized down-up algebras $\A$. First, we determine all the algebra endomorphisms of $\A$ under some conditions on $r$ and $s$. We show that each algebra endomorphism of $\A$ is an…

In the paper {\em The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic}, J. Pure Appl. Algebra 212 (2008), no. 10, 2320-2337, see also arXiv:math/0604477, Vladimir Bavula states…

Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal…

Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right…

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F)…

Let S and R be the rings of regular functions on affine algebraic varieties over a field of characteristic 0, R be embedded as a subring in S, and F : S --> S be an endomorphism such that F(R) subset R. Suppose that every ideal of height 1…

The first (associative) Weyl algebra is formally rigid in the classical sense. In this paper, we show that it can however be formally deformed in a nontrivial way when considered as a so-called hom-associative algebra, and that this…

Let $K$ be a field of characteristic zero, $K[x,y]$ be the polynomial ring in two variables. Let $\phi=(f, g)$ be an endomorphism of $K[x,y]$. It is proved that if $\phi$ maps each coordinate to a generator of some proper retract, then it…

We prove that the Weyl algebra over $\mathbb{C}$ cannot be a fixed ring of any domain under a nontrivial action of a finite group by algebra automorphisms, thus settling a 30-year old problem. In fact, we prove the following much more…

The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical…