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 $A_1$-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \cite{W1}. We also make some comments on the group $G$ from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups.
Motivated by Weyl algebra analogues of the Jacobian conjecture and the Tame Generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of $U_q^+(\mathfrak{so}_5)$.
In this paper, we study the algebra automorphisms and isomorphisms for a family of "symmetric" multiparameter quantized Weyl algebras $\A$ and some related algebras in the generic case. First, we compute the Nakayama automorphism for $\A$ and give a necessary and sufficient condition for $\A$ to be Calabi-Yau. We also prove that $\A$ is cancellative. Then we determine the automorphism group for $\A$ and its polynomial extension $\E$. As an application, we solve the isomorphism problem for $\{\A\}$ and $\{\E\}$. Similar results will be established for the Maltisiniotis multiparameter quantized …
In our previous paper q-alg/9605011 we proposed several algebraic methods for constructing new solutions to the bispectral problem. In the present note the corresponding eigenfunctions are explicitly constructed as multiple Laplace integrals.
In this paper, we study a family of generalized Weyl algebras $\{\A\}$ and their polynomial extensions. We will show that the algebra $\A$ has a simple localization $\A_{\mathbb{S}}$ when none of $p$ and $q$ is a root of unity. As an application, we determine all the height-one prime ideals and the center for $\A$, and prove that $\A$ is cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras $\A$ and their polynomial extensions in the case where none of $p$ and $q$ is a root of unity. We will establish a quantum analogue …
In one of our recent papers, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is the polynomial algebra of a commutative derivation subalgebra $D$ of $A$. In the present paper, a class of the above associative and Lie algebras $A[D]$ with $F$ being a field of characteristic 0, $D$ consisting of locally finite but not locally nilpotent derivations of $A$, are studied. The isomorphism classes and automorphism groups of these as…
We show that endomorphisms of Weyl algebras over fields of characteristic zero are flat and that birational endomorphisms are automorphisms by reducing to positive characteristic. We also give examples showing that endomorphisms of Weyl algebras are not in general flat over fields of positive characteristic.
Generalized Weyl Algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings, including global dimension, rigidity, and simplicity.
Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity.
Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_{1}$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map $θ: R \to C$ by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that $θ$ is inverse to a bijection $ω: C \to R$ constructed in \cite{BW} by a completely different method. The main s…
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 general result. Let $X$ be a smooth affine variety over $\mathbb{C}$, let $D(X)$ denote the ring of algebraic differential operators on $X,$ and let $Γ$ be a finite group. If $D(X)$ is isomorphic to the ring of $Γ$-invariants of a $\mathbb{C}$-domain $R$ on which $Γ$ acts faithfully by $\mathbb{C}$-algebra automorphisms, then $R$ is isomorphic to the ring of diff…
The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the generic algebra A of quantum n by n matrices which are invariant under winding automorphisms of A. More specifically, every such P is the kernel of a map from A to (A^+/P^+) tensor (A^-/P^-) obtained by composing comultiplication, localization, and quotient maps, where A^+ and A^- are special localized quotients of A while P^+ and P^- are prime ideals invariant under winding automorphisms. Further, the algebras A^+ and A^-, which vary with P, can be chosen so that the corre…
To any symmetry of the Cartan matrix of a Generalized Kac-Moody (GKM) algebra we associate a family of automorphisms of the algebra which act in a natural way on the modules of the GKM algebra. We introduce the twining character of a module which is obtained by inserting the action of such an automorphism into the trace that appears in the ordinary character. Twining characters of integrable highest weight modules and Verma modules satisfy a character formula which involves a certain subgroup of the Weyl group. This subgroup is shown to be the Weyl group of another GKM algebra, called the orbi…
Let q be a nonzero complex number that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O_q(M_{m,n}) of quantum matrices. Next, we describe all height one primes of O_q(M_{m,n}); these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O_q(M_{m,n}) whenever O_q(M_{m,n}) has only finitely many height one primes. Finally, we compute the automorphism group of O_q(M_{m,n}) in the case where m is not equal to n. In order to do this, we first study the action of this group on the prime spectrum of O_q(M_{m,n}). Then, by…
Let ${\mathfrak g}$ be a finite dimensional complex simple Lie algebra, and let $r,s\in \mathbb{C}^{\ast}$ be transcendental over $\mathbb{Q}$ such that $r^{m}s^{n}=1$ implies $m=n=0$. We will obtain some basic properties of the two-parameter quantized enveloping algebra $U_{r,s}^{+}(\mathfrak g)$. In particular, we will verify that the algebra $U_{r,s}^{+}(\mathfrak g)$ satisfies many nice properties such as having normal separation, catenarity and Dixmier-Moeglin equivalence. We shall study a concrete example, the algebra $U_{r,s}^{+}(B_{2})$ in detail. We will first determine the normal ele…
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 of $A$. Our main result applies to minimal primitive factors of the quantized enveloping algebra of $U_q(\mathfrak{sl}_2)$ and certain minimal primitive quotients of the positive part of $U_q(\mathfrak{so}_5)$.
Let g be a complex simple Lie algebra of type B_2 and q be a non-zero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of the positive part U^+(g) of the enveloping algebra of g, whose Gelfand-Kirillov dimension is equal to 2, are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U_q^+(g) of the quantized enveloping algebra of g and then we study the simple factor algebras of U_…
Let $O_τ(Γ)$ be a family of algebras \textit{quantizing} the coordinate ring of $\mathbb{C}^2 / Γ$, where $Γ$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$, and let $G_Γ$ be the automorphism group of $O_τ$. We study the natural action of $G_Γ$ on the space of right ideals of $O_τ$ (equivalently, finitely generated rank $1$ projective $O_τ$-modules). It is known that the later can be identified with disjoint union of algebraic (quiver) varieties, and this identification is $G_Γ$-equivariant. In the present paper, when $Γ\cong \mathbb{Z}_2$, we show that the $G_Γ$-action on each quiver var…
Let $V$ be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group $G$ of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for $G$ and the irreducible modules for $V^G$ which are contained in $V$ where $V^G$ is the space of $G$-invariants of $V.$ We also prove a concomitant Galois correspondence between vertex operator subalgebras of $V$ which contain $V^G$ and closed Lie subgroups of $G$ in the case that $G$ is abelian.
We investigate the general structure of the automorphism group and the Lie algebra of derivations of a finitely generated vertex operator algebra. The automorphism group is isomorphic to an algebraic group. Under natural assumptions, the derivation algebra has an invariant bilinear form and the ideal of inner derivations is nonsingular.
We study Jacobi pairs in details and obtained some properties. We also study the natural Poisson algebra structure $(\PP,[...,...],...)$ on the space $\PP:=\C[y]((x^{-\frac1N}))$ for some sufficient large $N$, and introduce some automorphisms of $(\PP,[...,...],...)$ which are (possibly infinite but well-defined) products of the automorphisms of forms $e^{\ad_H}$ for $H\in x^{1-\frac1N}\C[y][[x^{-\frac1N}]]$ and $τ_c:(x,y)\mapsto(x,y-cx^{-1})$ for some $c\in\C$. These automorphisms are used as tools to study Jacobi pairs in $\PP$. In particular, starting from a Jacobi pair $(F,G)$ in $\C[x,y]$…
Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantum matrices, $q$-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras $U(L_+)$ of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the $X$-inner automorphisms of such algebras. The theorems and examples in our paper show that for algebras $R$ of this type, the non-identity $X$-inner automorphisms of $R$ tend to have infinite order. Thus if $G$ is a finite group of automorphi…
We classify the centers of the quantized Weyl algebras that are PI and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this …
This is a footnote to earlier joint work with Yu. Berest, which constructed a bijection between the space of ideal classes of the Weyl algebra and a union of Calogero-Moser varieties. A key property of this bijection is that it is equivariant with respect to the action of the automorphism group of the Weyl algebra: the main result of the present note is that it is uniquely determined by that property.
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$, $X\mapsto x$, $Y\mapsto y$, be a $K$-algebra homomorphism, i.e. $[y,x]=1$. It is proved that the set of eigenvalues of the inner derivation $\ad (yx)$ of the Weyl algebra $A_1$ is $\Z$ and the eigenvector algebra of $\ad (yx)$ is $K< x,y> $ (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: {\em is an algebra endomorphism of $A_1$ an automorphism…
Let R be a Noetherian domain and let (σ, δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F = t^2 - v is normal in the Ore extension R[t; σ, δ] where v ε R. We show F is prime in R[t; σ, δ] if and only if F is irreducible in Q[t; σ, δ] if and only if there does not exist w ε Q such that v = σ (w)w - δ (w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.
