---
abstract: 'In this note we classify the non-Noetherian generalized Heisenberg algebras ${\mathcal{H}(f)}$ introduced in [@LZ15].  In case $\degg f>1$, we determine all locally finite and also all locally nilpotent derivations of ${\mathcal{H}(f)}$ and describe the automorphism group of these algebras.' 
author:
- 'Samuel A. Lopes[^1]'
title: 'Non-Noetherian generalized Heisenberg algebras'
---

**MSC Numbers (2010)**: Primary 16W25, 16W20; Secondary 16P40, 16S36.**Keywords**: generalized Heisenberg algebra; ambiskew polynomial ring; weak generalized Weyl algebra; automorphism group; derivation; Noetherian. 

Introduction {#S:intro}
============

Fix a polynomial $f\in{\mathbb{C}}[h]$. The *generalized Heisenberg algebra* ${\mathcal{H}(f)}$ is the unital associative ${\mathbb{C}}$-algebra with generators $x$, $y$, $h$ satisfying the relations: $$\label{E:intro:def}
hx=xf(h), \quad yh=f(h)y, \quad yx-xy=f(h)-h.$$ See [@LZ15] and the references therein for information on how these algebras first appeared and on their applications to theoretical physics. 

Ambiskew polynomial rings were introduced by Jordan over a series of papers (see the references in [@dJ00]), but for our purposes the best suited definition is the one found in [@dJ00], which we briefly recall.  Let $\sigma$ be an endomorphism of a commutative ${\mathbb{C}}$-algebra $B$, $c\in B$ and $p\in{\mathbb{C}}$. The ambiskew polynomial ring $R(B, \sigma, c, p)$ is the ${\mathbb{C}}$-algebra generated by $B$ and two indeterminates, $x$ and $y$, subject to the relations $$bx=x\sigma(b),\quad yb=\sigma(b)y, \quad yx-pxy=c,\quad \mbox{for all $b\in B$.}$$ On comparing these relations with those in , one immediately sees that $$\label{E:intro:ambi}
{\mathcal{H}(f)}\cong R({\mathbb{C}}[h], \sigma, f(h)-h, 1),$$ where $\sigma:{\mathbb{C}}[h]\rightarrow{\mathbb{C}}[h]$ is the algebra endomorphism given by $\sigma(h)=f(h)$. In particular, one can see that there is an overlap between the generalized Heisenberg algebras defined above and (generalized) down-up algebras (see Corollary \[C:n:gdua\] below). 

The algebras ${\mathcal{H}(f)}$ can also be seen as weak generalized Weyl algebras over a polynomial algebra in two variables, in the sense of [@LMZ15], a construction which includes the generalized Weyl algebras introduced by V.V. Bavula in [@vB92].  In [@LZ15] the authors determine a basis for ${\mathcal{H}(f)}$ over ${\mathbb{C}}$, compute the center of ${\mathcal{H}(f)}$, solve the isomorphism problem for this family of algebras and classify all the finite-dimensional irreducible representations of ${\mathcal{H}(f)}$.

In this note we show that ${\mathcal{H}(f)}$ is (right or left) Noetherian if and only if $\degg f=1$ and that ${\mathcal{H}(f)}$ is isomorphic to a generalized down-up algebra if and only if $\degg f\leq 1$. For this reason, we then concentrate on the case where $\degg f>1$ and determine the locally nilpotent and the locally finite derivations of ${\mathcal{H}(f)}$, all ${\mathbb{Z}}$-gradings of ${\mathcal{H}(f)}$ and describe the automorphism group of ${\mathcal{H}(f)}$. In particular, we obtain the following results in case $\degg f>1$:

-   ${\mathcal{H}(f)}$ in neither right nor left Noetherian (Proposition \[P:n:n\]);

-   ${\mathcal{H}(f)}$ admits a unique (up to an integer multiple) nontrivial ${\mathbb{Z}}$-grading, in which $x$ has degree $1$, $y$ has degree $-1$ and $h$ has degree $0$ (Corollary \[C:lfd:zgrad\]);

-   the automorphism group of ${\mathcal{H}(f)}$ is abelian: it is isomorphic to ${\mathbb{C}}^*\times \mathsf{C}$, where $\mathsf{C}$ is a finite cyclic group whose order divides $(\degg f) -1$ (Theorem \[T:aut:main\]). 

In Section \[S:n\] of the paper we review some properties of ${\mathcal{H}(f)}$ which have been established in [@LZ15], determine when ${\mathcal{H}(f)}$ is Noetherian and when it is isomorphic to a generalized down-up algebra, while in Section \[S:lfd:sg\] we introduce a useful commutative subalgebra of ${\mathcal{H}(f)}$, which is a maximal commutative subalgebra if $\degg f>1$. Assuming that $\degg f>1$, we then investigate the locally finite and the locally nilpotent derivations of ${\mathcal{H}(f)}$ and also its ${\mathbb{Z}}$-gradings in Section \[S:lfd\], and in the final section, Section \[S:aut\], we describe the automorphism group of ${\mathcal{H}(f)}$ and show that it is always an abelian group generated by the automorphisms which fix $h$ and the automorphisms which fix $x$.

We make use of the commutator notation $[a, b]=ab-ba$. The sets of integers, nonnegative integers and positive integers are denoted by ${\mathbb{Z}}$, ${\mathbb{Z}}_{\geq 0}$ and ${\mathbb{Z}}_{> 0}$, respectively.  The field of complex numbers is denoted by ${\mathbb{C}}$, and the multiplicative group of nonzero complex numbers is denoted by ${\mathbb{C}}^*$. For a polynomial $g\in{\mathbb{C}}[h]$, $\degg g$ will always denote the degree of $g$ as a polynomial in $h$.

Throughout the paper, $\sigma:{\mathbb{C}}[h]\rightarrow{\mathbb{C}}[h]$ is the algebra endomorphism given by $\sigma(h)=f(h)$. For any function $\phi: X\rightarrow X$, we will use the notation $\phi^k$ to mean the $k$-th power of $\phi$ with respect to composition.  In particular, $\phi^0$ denotes the identity on the set $X$.

The Noetherian property {#S:n}
=======================

Below we record a few results from [@LZ15] which will be useful in the course of this paper.  As usual, $Z({\mathcal{H}(f)})$ denotes the center of ${\mathcal{H}(f)}$.

\[L:n:LZ\] Let $f\in{\mathbb{C}}[h]$. Then:

-   The set $\{ x^i h^j y^k \mid i, j, k\in {\mathbb{Z}}_{\geq 0}  \}$ is a basis of ${\mathcal{H}(f)}$.

-   The algebra ${\mathcal{H}(f)}$ is a domain if and only if $\degg f\geq 1$.

-   The center of ${\mathcal{H}(f)}$ contains the polynomial algebra ${\mathbb{C}}[z]$, where $z=xy-h=yx-f(h)$. If $\degg f\neq 1$, then $Z({\mathcal{H}(f)})={\mathbb{C}}[z]$.

1.   Identifying ${\mathcal{H}(f)}$ with the ambiskew polynomial ring $R({\mathbb{C}}[h], \sigma, f(h)-h, 1)$ as in , it follows that ${\mathcal{H}(f)}$ is conformal, as defined in [@dJ00 Section 2.3], and the corresponding Casimir element is precisely the central element $z=xy-h$ defined above. 

2.   Suppose $f\in{\mathbb{C}}$. Then by considering the generators $-x$, $y$ and $h-f$, we see that ${\mathcal{H}(f)}\cong R({\mathbb{C}}[h], \sigma, h, 1)$, with $\sigma=0$, and from [@dJ00 Theorem 7.10] we conclude that ${\mathcal{H}(f)}$ is a prime ring.  Thus by Lemma \[L:n:LZ\](b), ${\mathcal{H}(f)}$ is a prime ring for any $f\in{\mathbb{C}}[h]$.

3.   Since the center of ${\mathcal{H}(f)}$ contains the polynomial algebra ${\mathbb{C}}[z]$ and ${\mathcal{H}(f)}$ has countable dimension over ${\mathbb{C}}$, it follows from Dixmier’s version of Schur’s Lemma that ${\mathcal{H}(f)}$ is never primitive. 

There is an order two anti-automorphism of ${\mathcal{H}(f)}$, denoted by $\iota$, that fixes $h$ and interchanges $x$ and $y$: $$\label{E:iota}
\iota: {\mathcal{H}(f)}\rightarrow{\mathcal{H}(f)}, \quad\quad x\mapsto y, \quad y\mapsto x, \quad h\mapsto h.$$ Hence ${\mathcal{H}(f)}$ is isomorphic to its opposite algebra ${\mathcal{H}(f)}^{\text{op}}$.

\[P:n:n\] The algebra ${\mathcal{H}(f)}$ is right (respectively, left) Noetherian if and only if $\degg f=1$.

If $\degg f=1$ then ${\mathcal{H}(f)}$ is a generalized Weyl algebra over a polynomial ring in two variables, and thus it is right and left Noetherian.  So assume that $\degg f\neq 1$. In particular, $f(h)-h$ has some root $\alpha\in{\mathbb{C}}$. Let $F(h)=f(h+\alpha)-\alpha$. Then $\degg F=\degg f$ (here we assume the zero polynomial has degree $0$) and $F(h)\in h{\mathbb{C}}[h]$. Moreover, $F(h-\alpha)=f(h)-\alpha$ and then ${\mathcal{H}(f)}\cong \mathcal{H}(F)$ by [@LZ15 Lemma 3].  So there is no loss in assuming that $f(h)\in h{\mathbb{C}}[h]$. By the isomorphism ${\mathcal{H}(f)}\cong {\mathcal{H}(f)}^{\text{op}}$ it will be enough to show that ${\mathcal{H}(f)}$ is not left Noetherian. 

For each $n\in{\mathbb{Z}}_{\geq 0}$ define the left ideal $$I_n=\sum_{i=0}^n {\mathcal{H}(f)}hy^i.$$ Then $I_n\subseteq I_{n+1}$ for all $n\geq 0$ and we finish the proof by showing that these inclusions are strict.  Note that by Lemma \[L:n:LZ\](a), $${\mathcal{H}(f)}=\bigoplus_{j, k\geq 0}x^j{\mathbb{C}}[h]y^k.$$ Given $j, k\geq 0$ and $g(h)\in{\mathbb{C}}[h]$, we have $x^j g(h)y^k hy^i=x^j g(h)\sigma^k(h)y^{k+i}$. Assume, by way of contradiction, that $hy^{n+1}\in I_n$. Then there exist $g_i(h)\in{\mathbb{C}}[h]$, $i=0, \ldots, n$, such that $\displaystyle hy^{n+1}=\sum_{i=0}^n g_i (h)\sigma^{n+1-i}(h)y^{n+1}$. It follows by Lemma \[L:n:LZ\](a) that $$\label{E:P:n:n}
h=\sum_{i=0}^n g_i (h)\sigma^{n+1-i}(h).$$ As by hypothesis $\sigma(h)=f(h)\in h{\mathbb{C}}[h]$, one can deduce that $\sigma^{n+1-i}(h)\in f(h){\mathbb{C}}[h]$ for all $0\leq i\leq n$ and then implies that $h\in f(h){\mathbb{C}}[h]$, which is a contradiction since under our hypothesis either $f(h)=0$ or $\degg f>1$. This proves that $hy^{n+1}\not\in I_n$ for any $n\geq 0$ and hence $\left\{ I_n \right\}_{n\geq 0}$ is a strict ascending chain of left ideals of ${\mathcal{H}(f)}$.

The case $f\in{\mathbb{C}}$ of Proposition \[P:n:n\] follows also from [@dJ00 Corollary 7.3], which applies when $\sigma$ is not injective.  In terms of the endomorphism $\sigma$, Proposition \[P:n:n\] could be restated as: The algebra ${\mathcal{H}(f)}$ is right (respectively, left) Noetherian if and only if $\sigma$ is an automorphism. 

We recall that a generalized down-up algebra $L(g, r, s, \gamma)$, given by the parameters $g\in{\mathbb{C}}[H]$ and $r, s, \gamma\in{\mathbb{C}}$, is defined as the unital associative ${\mathbb{C}}$-algebra generated by $d$, $u$ and $H$, subject to the relations: $$dH-rHd+\gamma d=0,\quad
Hu-ruH+\gamma u=0,\quad
du-sud+g(H)=0.$$ Generalized down-up algebras were defined in [@CS04] as generalizations of the down-up algebras introduced by Benkart and Roby in [@BR98].  Generalized down-up algebras include all down-up algebras, the algebras similar to the enveloping algebra of $\mathfrak{sl}_{2}$ defined by Smith [@spS90], Le Bruyn’s conformal $\mathfrak{sl}_{2}$ enveloping algebras [@lLB95] and Rueda’s algebras similar to the enveloping algebra of $\mathfrak{sl}_{2}$ [@sR02]. 

\[C:n:gdua\] The algebra ${\mathcal{H}(f)}$ is isomorphic to a generalized down-up algebra if and only if $\degg f\leq 1$.

Suppose first that $\degg f\leq 1$, say $f(h)=ah+b$ for $a, b\in{\mathbb{C}}$. Then it is straightforward to verify that ${\mathcal{H}(f)}\cong L(H-f(H), a, 1, -b)$, under an isomorphism that sends $x$, $y$ and $h$ to $u$, $d$ and $H$, respectively.  Conversely, suppose that $\degg f>1$. Then by Proposition \[P:n:n\] and Lemma \[L:n:LZ\](b), ${\mathcal{H}(f)}$ is a non-Noetherian domain.  Hence ${\mathcal{H}(f)}$ cannot be isomorphic to a generalized down-up algebra, as a generalized down-up algebra is a domain if and only if it is Noetherian, by Propositions 2.5 and 2.6 of [@CS04]. 

In view of this result, we will henceforth focus most of our attention on the generalized Heisenberg algebras ${\mathcal{H}(f)}$ with $f\in{\mathbb{C}}[h]$ such that $\degg f>1$.

The commutative algebra ${\mathcal{H}(f)}_0$ {#S:lfd:sg}
============================================

In this short section we record a few useful formulas for computing in ${\mathcal{H}(f)}$ and then explore an interesting commutative subalgebra of ${\mathcal{H}(f)}$.

\[L:c:c\] Let $k\in{\mathbb{Z}}_{\geq 0}$ and $g\in{\mathbb{C}}[h]$. Then the following hold:

-   $[y, x^k]=x^{k-1}(\sigma^k (h)-h)$;

-   $[y^k, x]=(\sigma^k (h)-h)y^{k-1}$;

-   $(x^k g y^k)x=x(x^k \sigma(g) y^k + x^{k-1} (\sigma^k (h)-h)g y^{k-1})$;

-   $y(x^k g y^k)=(x^k \sigma(g) y^k + x^{k-1} (\sigma^k (h)-h)g y^{k-1})y$;

-   $x^k g y^k$ commutes with $x^j \tilde g y^j$ for all $\tilde g\in{\mathbb{C}}[h]$ and all $j\in{\mathbb{Z}}_{\geq 0}$.

Parts (a) and (b) have been established in [@dJ00], formulas (6a)–(6b).  We prove part (c) using (b): $$\begin{aligned}
(x^k g y^k)x&=x^k g xy^k +x^k g [y^k,x] \\
&=x^{k+1} \sigma(g) y^k +x^k g (\sigma^k (h)-h)y^{k-1}\\
&=x(x^{k} \sigma(g) y^k +x^{k-1} (\sigma^k (h)-h)g y^{k-1}).\end{aligned}$$ Formula (d) follows from applying the anti-automorphism $\iota$ of to (c). 

Finally, we prove (e) by induction on $k$, the case $k=0$ being trivial: $$g(x^j \tilde g y^j)=x^j \sigma^j(g)\tilde g y^j=x^j \tilde g \sigma^j(g) y^j=(x^j \tilde g  y^j) g.$$ Now suppose (e) holds for a certain $k\geq 0$. Thus we have: $$\begin{aligned}
 (x^{k+1} g y^{k+1})(x^j \tilde g y^j) &=  (x^{k+1} g y^{k})y(x^j \tilde g y^j)\\
 &=  x(x^{k} g y^{k})(x^j \sigma(\tilde g) y^j + x^{j-1} (\sigma^j (h)-h)\tilde g y^{j-1})y \quad\quad \mbox{by (d)}\\
 &=  x(x^j \sigma(\tilde g) y^j + x^{j-1} (\sigma^j (h)-h)\tilde g y^{j-1})(x^{k} g y^{k})y \quad\quad \mbox{($\ast$)}\\
 &=  (x^j \tilde g y^j)x(x^{k} g y^{k})y \quad\quad \mbox{by (c)}\\
 &=  (x^j \tilde g y^j)(x^{k+1} g y^{k+1}),\end{aligned}$$ where ($\ast$) follows from the induction hypothesis.  So (e) holds for all $k\in{\mathbb{Z}}_{\geq 0}$.

There is an obvious grading of ${\mathcal{H}(f)}$ relative to which $x$ has degree $1$, $y$ has degree $-1$ and $h$ has degree $0$. We denote the corresponding homogeneous subspaces by ${\mathcal{H}(f)}_\ell$, for $\ell\in{\mathbb{Z}}$, so that $$\label{E:zgrading}
{\mathcal{H}(f)}=\bigoplus_{\ell\in{\mathbb{Z}}} {\mathcal{H}(f)}_\ell, \quad \quad \mbox{with \quad $\displaystyle {\mathcal{H}(f)}_\ell=\bigoplus_{i-k=\ell}{\mathbb{C}}\, x^i {\mathbb{C}}[h] y^k$.}$$ We call this the *standard grading* of ${\mathcal{H}(f)}$, and, whenever we mention a homogeneous component or element of ${\mathcal{H}(f)}$, we will always be referring to this standard grading. 

The subalgebra ${\mathcal{H}(f)}_0$ has basis $\{x^k h^j y^k \mid k, j\geq 0\}$ and ${\mathcal{H}(f)}_\ell=x^\ell \, {\mathcal{H}(f)}_0$ if $\ell\geq 0$; ${\mathcal{H}(f)}_\ell= {\mathcal{H}(f)}_0 \, y^{-\ell}$ if $\ell\leq 0$. Thus we have the decomposition $${\mathcal{H}(f)}=\bigoplus_{\ell>0} x^\ell\,{\mathcal{H}(f)}_0 \oplus {\mathcal{H}(f)}_0 \oplus \bigoplus_{\ell>0} {\mathcal{H}(f)}_0 \, y^\ell.$$

\[P:lfd:sg:hoiscom\] The subalgebra ${\mathcal{H}(f)}_0$ is commutative.  If $\degg f>1$, then ${\mathcal{H}(f)}_0$ is a maximal commutative subalgebra of ${\mathcal{H}(f)}$ which strictly contains ${\mathbb{C}}[z, h]$, the polynomial subalgebra of ${\mathcal{H}(f)}$ generated by $h$ and the central element $z=xy-h$.

The first statement is a direct consequence of Lemma \[L:c:c\](e).  Assume now that $\degg f>1$. Then $\sigma$ is injective and has infinite order.  For any $i, k\in {\mathbb{Z}}_{\geq 0}$ and $g\in{\mathbb{C}}[h]$, $[h, x^i g y^k]=x^i (\sigma^i(h)-\sigma^k(h))g y^k$. Hence, if $g\neq 0$, we deduce that $[h, x^i g y^k]=0\iff i=k$, and from this it is straightforward to conclude that ${\mathcal{H}(f)}_0$ is the centralizer of $h$, hence a maximal commutative subalgebra of ${\mathcal{H}(f)}$.

The commuting elements $h$ and $z$ are homogeneous of degree $0$ and are easily seen to be algebraically independent, as $z^k-x^ky^k$ is in the span of $\{ x^i g y^i \mid i<k,\ g\in{\mathbb{C}}[h]\}$. Suppose, by contradiction, that there exist $g_k\in{\mathbb{C}}[h]$ such that $xhy=\sum_{k\geq 0} g_k z^k$. Then by the argument above we must have $g_k=0$ for all $k>1$ and $\sigma(g_1)=h$, which is possible only if $\degg f=1$. Therefore $xhy\in{\mathcal{H}(f)}_0\setminus {\mathbb{C}}[z, h]$.

By Lemma \[L:c:c\](c)–(d), it is possible to extend $\sigma$ to a ${\mathbb{C}}$-linear endomorphism $\tilde\sigma$ of ${\mathcal{H}(f)}_0$ so that $\tilde\sigma(x^k g y^k)=x^k \sigma(g) y^k + x^{k-1} (\sigma^k (h)-h)g y^{k-1}$, for all $k\in{\mathbb{Z}}_{\geq 0}$ and $g\in{\mathbb{C}}[h]$. For simplicity, we still denote this endomorphism by $\sigma$ instead of $\tilde\sigma$. By Lemma \[L:c:c\](c)–(d) and Lemma \[L:n:LZ\](a), $\sigma$ is defined by the relations: $$\label{E:lfd:sg:sigma}
\theta x=x \sigma (\theta), \quad\quad y\theta=\sigma(\theta)y, \quad \quad \mbox{for all $\theta\in{\mathcal{H}(f)}_0$.}$$ In particular, implies that $\sigma$ is an algebra endomorphism of ${\mathcal{H}(f)}_0$.

locally finite derivations of ${\mathcal{H}(f)}$ when $\degg f>1$ {#S:lfd}
=================================================================

Henceforth we will assume that $\degg f>1$. By Corollary \[C:n:gdua\] we are assuming that ${\mathcal{H}(f)}$ is not a generalized down-up algebra.  Most of our subsequent results do not hold if $\degg f\leq 1$.

Our goal in this section is to determine all locally finite derivations of ${\mathcal{H}(f)}$. In particular, we will classify all ${\mathbb{Z}}$-gradings of ${\mathcal{H}(f)}$ and show that ${\mathcal{H}(f)}$ has no nontrivial locally nilpotent derivations.  Our methods are akin to those used in [@SAV15]. 

Let $\delta$ be a ${\mathbb{C}}$-linear endomorphism of ${\mathcal{H}(f)}$. We recall the following standard definitions:

-   $\delta$ is a derivation of ${\mathcal{H}(f)}$ if $\delta(ab)=\delta(a)b+a\delta(b)$;

-   $\delta$ is locally finite if for every $a\in{\mathcal{H}(f)}$ the ${\mathbb{C}}$-linear span of $\{ \delta^k(a) \mid k\in{\mathbb{Z}}_{\geq 0}\}$ is finite dimensional;

-   $\delta$ is locally nilpotent if for every $a\in{\mathcal{H}(f)}$ there is $k\in{\mathbb{Z}}_{\geq 0}$ such that $\delta^k(a)=0$;

-   $\delta$ is homogeneous of degree $r\in{\mathbb{Z}}$ if $\delta \left({\mathcal{H}(f)}_\ell\right)\subseteq {\mathcal{H}(f)}_{\ell+r}$ for all $\ell\in{\mathbb{Z}}$.

Assume $\delta$ is any derivation of ${\mathcal{H}(f)}$. Since ${\mathcal{H}(f)}$ is finitely generated, there exist homogeneous derivations $\delta_1, \ldots, \delta_k$ of strictly increasing degrees such that $\delta=\delta_1+\cdots +\delta_k$. Moreover, as seen in [@SAV15 Lemma 1.1], if $\delta$ is locally finite, then so are $\delta_1$ and $\delta_k$, and if $\delta_1$ (respectively, $\delta_k$) is of nonzero degree, then it must be locally nilpotent. 

We need one final definition.  Given a locally nilpotent derivation $\delta$ and $a\in{\mathcal{H}(f)}$, define $$\deg_\delta (a)=\max \{ k\in{\mathbb{Z}}_{\geq 0} \mid \delta^{k}(a)\neq 0\} \quad \mbox{if $a\neq 0$;}$$ define also $\deg_\delta(0)=-\infty$. It can be easily checked (see for example [@lMLnotes]) that for $a, b\in{\mathcal{H}(f)}$, $\deg_\delta(a+b)\leq \max\{\deg_\delta(a), \deg_\delta(b)\}$, with equality if $\deg_\delta(a)\neq\deg_\delta(b)$. Since ${\mathcal{H}(f)}$ is a domain and ${\mathbb{C}}$ has characteristic $0$, we also have, from the Leibniz rule, $\deg_\delta(ab)= \deg_\delta(a)+\deg_\delta(b)$. In particular, ${\mathsf{ker}}\delta$ is factorially closed: if $\delta(ab)=0$ for some nonzero $a, b\in{\mathcal{H}(f)}$, then $\delta(a)=0=\delta(b)$.

\[L:lfd:main\] Assume that $\degg f>1$. Then all locally finite derivations of ${\mathcal{H}(f)}$ are homogeneous of degree $0$.

Let $\delta$ be a locally finite derivation of ${\mathcal{H}(f)}$. By the above decomposition $\delta=\delta_1+\cdots +\delta_k$ of $\delta$ into homogeneous derivations of strictly increasing degrees, it will be enough to show that there are no nonzero homogeneous locally nilpotent derivations of ${\mathcal{H}(f)}$ of degree $r\neq 0$.

So assume $\delta$ is a homogeneous locally nilpotent derivation of ${\mathcal{H}(f)}$ of degree $r\neq 0$. Let $d=\deg_\delta(h)$ and suppose that $d>0$. Then $\deg_\delta (f(h))=d\degg f$ and the relation $hx=xf(h)$ yields $$d+\deg_\delta(x)=\deg_\delta(x)+d\degg f,$$ so $\degg f=1$, which contradicts our hypothesis.  Hence $d=0$ and $\delta(h)=0$.

By replacing $\delta$ with $\iota\delta\iota^{-1}$, where $\iota$ is the anti-automorphism defined in , we can assume that $r>0$. Then ${\mathsf{ker}}\delta$ contains some nonzero homogeneous element of positive degree.  Since elements of ${\mathcal{H}(f)}$ of positive degree lie in $x{\mathcal{H}(f)}$ and ${\mathsf{ker}}\delta$ is factorially closed, we deduce that $\delta(x)=0$.

Any derivation maps the center of an algebra into itself, so $\delta$ restricts to a locally nilpotent derivation of ${\mathbb{C}}[z]$, by Lemma \[L:n:LZ\](c), and thus $\delta(z)\in{\mathbb{C}}$. On the other hand, since $z=xy-h$ is homogeneous of degree $0$ and $\delta$ has positive degree, it must be that $\delta(z)=0$, and from $0=\delta(z)=x\delta(y)$, we conclude that $\delta(y)=0$. Then $\delta=0$ and the lemma is proved. 

The next theorem, our main result on derivations of ${\mathcal{H}(f)}$ when $\degg f>1$, shows that the space of locally finite derivations of ${\mathcal{H}(f)}$ is one-dimensional over ${\mathbb{C}}$, spanned by the derivation $\partial$ defined by $$\label{E:lfd:partial}
\partial(x^i h^j y^k)=(i-k)x^i h^j y^k,\quad \mbox{for all $i, j, k\in {\mathbb{Z}}_{\geq 0}$.}$$

\[T:lfd:main\] Assume that $\degg f>1$. If $\delta$ is a locally finite derivation of ${\mathcal{H}(f)}$, then there is $\lambda\in{\mathbb{C}}$ such that $\delta(x)=\lambda x$, $\delta(y)=-\lambda y$ and $\delta(h)=0$.

Let $\delta$ be a locally finite derivation of ${\mathcal{H}(f)}$. By Lemma \[L:lfd:main\], we know that $\delta$ is homogeneous of degree $0$, so there are $\theta_x, \theta_h, \theta_y\in{\mathcal{H}(f)}_0$ so that $$\delta(x)=x\theta_x, \quad  \delta (h)=\theta_h, \quad\mbox{and}\quad \delta(y)=\theta_y y.$$ In particular, since $h$ commutes with $\theta_h$, we have $\delta(g(h))=g'(h)\theta_h$ for all $g(h)\in{\mathbb{C}}[h]$, where $g'(h)$ denotes the derivative of $g(h)$ with respect to $h$.

*Claim 1:* [*$\theta_h=0$ and $\theta_x+\theta_y=0$.*]{}

*Proof of Claim 1:* Write $$\label{E:lfd:main:dt2}
\theta_h=\sum_{k\geq 0}x^kg_k(h) y^k,$$ with $g_k(h)\in{\mathbb{C}}[h]$ and $g_k(h)=0$ except for finitely many indices $k$.

As observed in the proof of Lemma \[L:lfd:main\], $\delta$ restricts to a locally finite derivation of ${\mathbb{C}}[z]$, the center of ${\mathcal{H}(f)}$, and thus $\delta(z)\in{\mathbb{C}}\oplus{\mathbb{C}}z$, say $\delta (z)=\mu z-\lambda$, with $\lambda, \mu\in{\mathbb{C}}$. Since $\mu z-\lambda=\delta (xy-h)=x(\theta_x+\theta_y)y-\theta_h$, we have $$\label{E:lfd:main:z}
\theta_h=x(\theta_x+\theta_y)y-\mu z+\lambda=x(\theta_x+\theta_y -\mu)y+\mu h+\lambda.$$ In particular, $g_0(h)=\mu h+\lambda$.

We now apply $\delta$ to the relation $yh=f(h)y$ and get $\theta_y yh +y\theta_h=f'(h)\theta_h y+f(h)\theta_y y$. As $h$ and $\theta_y$ commute, and $y\theta_h=\sigma(\theta_h)y$, by , we obtain $$\label{E:lfd:main:t2comm}
\sigma(\theta_h)=f'(h)\theta_h .$$ Now combining and we deduce that, for every $k\geq 0$: $$\label{E:lfd:main:t2rels}
\sigma^k(f'(h))g_k(h)=\sigma(g_k(h)) + (\sigma^{k+1} (h)-h)g_{k+1}(h).$$ Setting $k=0$ in we obtain $(f(h)-h)g_{1}(h)=f'(h)g_0(h)-\sigma(g_0(h))$. Since we have already established that $\degg g_0\leq 1$, we deduce now from the latter equation that $\degg (f(h)-h)g_{1}(h)\leq \degg f$, and thus $g_1\in{\mathbb{C}}$. Combining this with the $k=1$ case of , $\sigma(f'(h))g_1(h)=\sigma(g_1(h)) + (\sigma (f(h))-h)g_{2}(h)$, yields $g_2=0$, and in turn the latter gives $g_k=0$ for all $k\geq 2$. Using again the relation $\sigma(f'(h))g_1(h)=\sigma(g_1(h)) + (\sigma (f(h))-h)g_{2}(h)$ with $g_2=0$ and $g_1\in{\mathbb{C}}$ gives $g_1=0$. Therefore we have $$\label{E:lfd:main:g0}
\sigma (g_0)=f'(h)g_0.$$ Suppose $g_0\neq 0$, and let $a$ be the leading coefficient of $f(h)$. Then $\mu\neq 0$ and comparing leading coefficients in yields $\mu a=a(\degg f)\mu$, whence $\degg f=1$, which is a contradiction.  Thus $g_0=0$.

From the above we conclude that $\theta_h=\sum_{k\geq 0}x^kg_k(h) y^k=0$ and finally by we get $\theta_x+\theta_y=0$, establishing Claim 1. 

So far we have shown that $$\delta(x)=x\theta_x, \quad  \delta (h)=0, \quad\mbox{and}\quad \delta(y)=-\theta_x y,$$ so it remains to be inferred that $\theta_x\in{\mathbb{C}}$.

*Claim 2:* [*$\delta(\theta)=0$, for all $\theta\in{\mathcal{H}(f)}_0$.*]{}

*Proof of Claim 2:* Since $\delta(g)=0$ for all $g\in{\mathbb{C}}[h]$, it suffices to show that if $\theta\in{\mathcal{H}(f)}_0$ and $\delta (\theta)=0$, then also $\delta (x\theta y)=0$. This follows easily using the fact that ${\mathcal{H}(f)}_0$ is commutative, as proved in Proposition \[P:lfd:sg:hoiscom\]. 

From Claim 2 it follows that, for all $k\geq 0$, $\delta (\theta_x^k)=0$, which implies that $\delta^k(x)=x\theta_x^k$. As $\delta$ is locally finite, the span of $\{ \theta_x^k \mid k\in{\mathbb{Z}}_{\geq 0}\}$ must then be finite dimensional.  This is possible only if $\theta_x\in{\mathbb{C}}$, thus finishing the proof of the theorem. 

Since locally nilotent derivations are locally finite, we derive the following corollary. 

Assume that $\degg f>1$. Then ${\mathcal{H}(f)}$ has no nonzero locally nilpotent derivations. 

Suppose that ${\mathcal{H}(f)}=\bigoplus_{\alpha\in {\mathbb{C}}} V_\alpha$ is a grading.  Define the ${\mathbb{C}}$-linear endomorphism $\delta$ of ${\mathcal{H}(f)}$ by $\delta (v_\alpha)=\alpha v_\alpha$ for all $v_\alpha \in V_\alpha$ and all $\alpha\in {\mathbb{C}}$. It is immediate to check that $\delta$ is a diagonalizable derivation of ${\mathcal{H}(f)}$ whose eigenvalues are those $\alpha\in {\mathbb{C}}$ such that $V_\alpha\neq (0)$. Conversely, if $\delta$ is a diagonalizable derivation, then $\delta$ determines a grading where $V_\alpha$ is the $\alpha$-eigenspace of $\delta$. Furthermore, diagonalizable derivations are clearly locally finite. 

Thus, we deduce from Theorem \[T:lfd:main\] that, except for the trivial grading in which every element of ${\mathcal{H}(f)}$ has degree $0$, ${\mathcal{H}(f)}$ only admits the standard grading defined in , up to scaling by some integer.  More precisely, we have:

\[C:lfd:zgrad\] Assume that $\degg f>1$. Then for any ${\mathbb{Z}}$-grading of ${\mathcal{H}(f)}$, there is an integer $\ell\in{\mathbb{Z}}$ such that, relative to that grading, $x$ has degree $\ell$, $y$ has degree $-\ell$ and $h$ has degree $0$.

Automorphisms of ${\mathcal{H}(f)}$ when $\degg f>1$ {#S:aut}
====================================================

When $\deg f=1$ the algebra ${\mathcal{H}(f)}$ is a Noetherian generalized down-up algebra, by Corollary \[C:n:gdua\], and the automorphisms of the latter have been investigated in [@CL09].  We continue to assume that $\degg f>1$ and note again that our results do not generalize to the cases with $\degg f\leq 1$.

Since ${\mathcal{H}(f)}$ has no nonzero locally nilpotent derivations, it seems natural to conjecture that the automorphism group of ${\mathcal{H}(f)}$ is somewhat small.  However, over ${\mathbb{C}}$ we can consider also the exponential of a diagonalizable derivation.  Specifically, let $c\in{\mathbb{C}}$ and let $\partial$ be the derivation of ${\mathcal{H}(f)}$ defined in .  Then the expression $$\exp(c\partial) := \sum_{k=0}^\infty  \frac{(c\partial)^k}{k!}$$ defines an automorphism of ${\mathcal{H}(f)}$ satisfying $$\exp(c\partial) (x) = \sum_{k=0}^\infty  \frac{c^k}{k!}x=\exp(c)x, \quad \exp(c\partial) (y)=\exp(-c)y, \quad \exp(c\partial) (h) = h,$$ with inverse $\exp(-c\partial)$.

The above motivates the following definition.  For each $\lambda\in{\mathbb{C}}^*$, let $\phi_\lambda$ be the automorphism of ${\mathcal{H}(f)}$ defined by $$\label{E:aut:defpl}
\phi_\lambda(x)=\lambda x, \quad \phi_\lambda(y)=\lambda^{-1} y, \quad \phi_\lambda(h)=h.$$

The group of algebra automorphisms of ${\mathcal{H}(f)}$ will be denoted by $\aut$. We have a first description of $\aut$ below. 

\[P:aut:H\] Assume $\degg f> 1$. Then the following hold:

-   Any automorphism of ${\mathcal{H}(f)}$ restricts to an automorphisms of ${\mathbb{C}}[h]$, and $x$ and $y$ are eigenvectors. 

-   $\displaystyle \{\phi\in\aut \mid \phi(h)=h\}=\{\phi_\lambda\mid \lambda\in{\mathbb{C}}^*\}\cong{\mathbb{C}}^*$, and this is a central subgroup of $\aut$.

-   $\displaystyle \{\phi\in\aut \mid \phi(x)=x\}$ is a finite cyclic subgroup whose order divides $(\degg f)-1$.

Let $\phi$ be an automorphism of ${\mathcal{H}(f)}$. Then as argued in Claim 4 of the proof of [@LZ15 Theorem 5], the relation $\phi(h)\phi(x)=\phi(x)f(\phi(h))$ with $\degg f > 1$ implies that $\phi(h)\in{\mathbb{C}}[h]$; applying this result to $\phi^{-1}$ gives that $\phi(h)=ah+b$, for some $a, b\in{\mathbb{C}}$ with $a\neq 0$.

Now writing $\phi(x)$ as a sum of terms of the form $x^i g_{i, j}y^j$ with $i, j\in{\mathbb{Z}}_{\geq 0}$ and $g_{i, j}\in{\mathbb{C}}[h]$, and comparing the corresponding expressions for $\phi(h)\phi(x)$ and $\phi(x)f(\phi(h))$, we obtain $\phi(x)\in{\mathcal{H}(f)}_1$. Similarly, $\phi(y)\in{\mathcal{H}(f)}_{-1}$, so $\phi$ is homogeneous of degree $0$. Thus, there exist $\theta_x, \theta_y\in{\mathcal{H}(f)}_0$ such that $\phi(x)=x\theta_x$ and $\phi(y)=\theta_y y$. Applying the same reasoning to $\phi^{-1}$, we deduce that $\theta_x, \theta_y\in{\mathbb{C}}^*$, which proves (a). 

Now assume $\phi\in\aut$ and $\phi(h)=h$. By (a) there exist $\lambda, \mu\in{\mathbb{C}}^*$ such that $\phi(x)=\lambda x$ and $\phi(y)=\mu y$. Applying $\phi$ to the relation $[y, x]=f(h)-h$ yields $\lambda\mu=1$, so $\phi=\phi_\lambda$. This proves the equality in (b), and the isomorphism $\{\phi_\lambda\mid \lambda\in{\mathbb{C}}^*\}\cong{\mathbb{C}}^*$ is clear, as $\phi_\lambda\circ\phi_\mu=\phi_{\lambda\mu}$ for all $\lambda, \mu\in{\mathbb{C}}^*$.

Next, we show that the subgroup $\{\phi_\lambda\mid \lambda\in{\mathbb{C}}^*\}$ is central in $\aut$. Let $\lambda\in{\mathbb{C}}^*$, and suppose $\psi\in\aut$ is arbitrary.  By (a) we know that $\psi(h)\in{\mathbb{C}}[h]$, which implies that $\phi_\lambda\circ\psi(h)=\psi\circ\phi_\lambda(h)$. But as $x$ and $y$ are eigenvalues for any automorphism of ${\mathcal{H}(f)}$, $\phi_\lambda\circ\psi$ and $\psi\circ\phi_\lambda$ also agree on these generators, and thus $\phi_\lambda\circ\psi=\psi\circ\phi_\lambda$.

To prove part (c), suppose that $\phi\in\aut$ and $\phi(x)=x$. We know already that $\phi(h)=ah+b$ and $\phi(y)=cy$, for some $a, b, c\in{\mathbb{C}}$ with $a, c\neq 0$. Then $xf(ah+b)=x\phi(f(h))=\phi(h)x=(ah+b)x=x(af(h)+b)$, and we obtain $$\label{E:aut:H:f}
f(ah+b)=af(h)+b.$$ Therefore, $$\begin{aligned}
 c(f(h)-h) &=c[y, x]=\phi([y, x])=\phi(f(h)-h)=af(h)+b-(ah+b)\\&=a(f(h)-h),\end{aligned}$$ and we conclude that $c=a$.

Write $f(h)=\sum_{k=0}^{n}a_k h^k$, where $n=\degg f$ and $a_k\in{\mathbb{C}}$. Applying the derivation $\frac{d}{dh}$ to $n-1$ times yields $a^{n-1}f^{(n-1)}(ah+b)=af^{(n-1)}(h)$, as $n-1\geq 1$. As $f^{(n-1)}(h)=(n-1)!(na_n h+a_{n-1})$, we obtain $$\label{E:aut:H:ab}
a^{n-1}=1 \quad \mbox{and}\quad b=\frac{(a-1)a_{n-1}}{na_n}.$$

Let $\mathsf{U}_{n-1}=\{ \zeta\in{\mathbb{C}}^*\mid \zeta^{n-1}=1 \}$ be the cyclic group of order $n-1$, and define a map $$\Psi: \{\phi\in\aut \mid \phi(x)=x\}\longrightarrow \mathsf{U}_{n-1}, \quad \phi\mapsto a,\ \ \mbox{where $\phi(h)=ah+b$.}$$ Then $\Psi$ is well defined by , and it is a group homomorphism.  If $\Psi(\phi)=1$ for some $\phi\in\aut$ with $\phi(x)=x$, then the above shows that $\phi(y)=y$ and $\phi(h)=h+b$. Again by we deduce that $b=0$, so $\phi$ is the identity on ${\mathcal{H}(f)}$. This shows that $\Psi$ is an injective group homomorphism and thus $\{\phi\in\aut \mid \phi(x)=x\}$ is isomorphic to a subgroup of $\mathsf{U}_{n-1}$; hence it is a finite cyclic group whose order divides $n-1$.

It is now an easy matter to determine the structure of $\aut$. The symbol $\dot{\times}$ used below denotes the internal direct product of subgroups of a group. 

\[T:aut:main\] Assume $\degg f>1$. Then $$\label{E:aut:main}
\aut=\{\phi_\lambda\mid \lambda\in{\mathbb{C}}^*\}\dot{\times}\{\phi\in\aut \mid \phi(x)=x\}$$ is an abelian group, where:

-   $\{\phi_\lambda\mid \lambda\in{\mathbb{C}}^*\}\cong{\mathbb{C}}^*$ and $\phi_\lambda$ is defined in ;

-   $\{\phi\in\aut \mid \phi(x)=x\}$ is a finite cyclic group whose order divides $(\degg f)-1$ and which, as a set, can be identified with $\{ (a, b)\in{\mathbb{C}}^*\times{\mathbb{C}}\mid f(ah+b)=af(h)+b \}$ via the correspondence $\phi\mapsto (a, b)$, where $\phi(h)=ah+b$.

Since we have already seen in Proposition \[P:aut:H\] that $\{\phi_\lambda\mid \lambda\in{\mathbb{C}}^*\}$ is central, in order to prove the direct product decomposition in , it remains to show that the two subgroups have trivial intersection, which is clear, and generate $\aut$. Let $\psi\in\aut$. Then there is $\lambda\in{\mathbb{C}}^*$ such that $\psi(x)=\lambda x$, whence $\phi^{-1}_\lambda\circ\psi\in\{\phi\in\aut \mid \phi(x)=x\}$, and this shows the latter claim.  Moreover, since $\{\phi\in\aut \mid \phi(x)=x\}$ is abelian, by Proposition \[P:aut:H\](c), the group $\aut$ must also be abelian. 

The remaining parts of the theorem have already been proved, except for the observation that $\{\phi\in\aut \mid \phi(x)=x\}$ can be identified with the set $\{ (a, b)\in{\mathbb{C}}^*\times{\mathbb{C}}\mid f(ah+b)=af(h)+b \}$. Indeed, if $\phi(x)=x$, then we have seen in the proof of Proposition \[P:aut:H\] that $\phi(h)=ah+b$ and $\phi(y)=ay$, for some $a, b\in{\mathbb{C}}$ with $a\neq 0$, and must hold.  This shows that the correspondence $\phi\mapsto (a, b)$ is well defined and one-to-one.  Conversely, given $(a, b)\in{\mathbb{C}}^*\times{\mathbb{C}}$ satisfying $f(ah+b)=af(h)+b$, it is routine to check that there is an automorphism of ${\mathcal{H}(f)}$ defined by the conditions $\phi(x)=x$, $\phi(y)=ay$, $\phi(h)=ah+b$, and this shows the correspondence is onto. 

Any pair $(a, b)\in{\mathbb{C}}^*\times{\mathbb{C}}$ satisfying $f(ah+b)=af(h)+b$ must also satisfy , where $n=\degg f$, although the conditions in are not sufficient (see the examples below).  Thus, for each $(n-1)$-th root of unity $a$, the corresponding scalar $b$ is determined by , but one still needs to check the relation $f(ah+b)=af(h)+b$ for the pair $(a, b)$.

-   If $\degg f=2$, then $n=2$ in , so $a=1$ and $b=0$, and the pair $(1, 0)$ corresponds to the identity map.  It follows that the group $\{\phi\in\aut \mid \phi(x)=x\}$ is trivial and $\aut\cong{\mathbb{C}}^*$.

-   Let $f(h)=h^3+h$. Then $n=3$ in , so $a=\pm1$. If $a=1$, then $b=0$, and the corresponding automorphism is the identity.  If $a=-1$, then $b=0$, and in fact $f(-h)=-f(h)$. Therefore there is an automorphism $\phi$ of ${\mathcal{H}(f)}$ such that $\phi(x)=x$, $\phi(y)=-y$, $\phi(h)=-h$, and $\aut\cong{\mathbb{C}}^*\times{\mathbb{Z}}_2$.

-   Let $f(h)=h^3+h+1$. Then $n=3$ in , so $a=\pm1$. If $a=-1$, then yields $b=0$, but $f(-h)\neq-f(h)$, so the group $\{\phi\in\aut \mid \phi(x)=x\}$ is trivial and $\aut\cong{\mathbb{C}}^*$.

-   Let $f(h)=h^n$, for $n>1$. Then says that $a$ is a $(n-1)$-th root of unity and $b=0$. Moreover, $f(ah)=a^nf(h)=af(h)$ for any $(n-1)$-th root of unity $a$. Hence $\aut\cong{\mathbb{C}}^*\times{\mathbb{Z}}_{n-1}$.

-   Let $f(h)=h^n+1$, for $n>1$. Then, as before, $a$ is a $(n-1)$-th root of unity and $b=0$. However, in this case, $f(ah)=ah^n+1$ whereas $af(h)=ah^n+a$, so equality holds if and only if $a=1$. Hence $\aut\cong{\mathbb{C}}^*$.

-   Let $f(h)=h^n+h^{k+1}$, for some $n\geq 4$, and take any $1\leq k< n-1$ such that $k\mid n-1$. Then $a$ is a $(n-1)$-th root of unity and $b=0$. In this case $f(ah)=ah^n+a^{k+1}h^{k+1}$ whereas $af(h)= ah^n+ah^{k+1} $, so equality holds if and only if $a^k=1$. By the hypothesis that $k\mid n-1$, we deduce that $\aut\cong{\mathbb{C}}^*\times{\mathbb{Z}}_{k}$.

[10]{}

V. V. Bavula.  Generalized [W]{}eyl algebras and their representations. ,  4(1):75–97, 1992. 

G. Benkart and T. Roby.  Down-up algebras. ,  209(1):305–344, 1998. 

Paula A. A. B. Carvalho and Samuel A. Lopes.  Automorphisms of generalized down-up algebras. ,  37(5):1622–1646, 2009. 

Thomas Cassidy and Brad Shelton.  Basic properties of generalized down-up algebras. ,  279(1):402–421, 2004. 

David A. Jordan.  Down-up algebras and ambiskew polynomial rings. ,  228(1):311–346, 2000. 

Lieven Le Bruyn.  Conformal [${\rm sl}_2$]{} enveloping algebras. ,  23(4):1325–1362, 1995. 

Rencai L[ü]{}, Volodymyr Mazorchuk, and Kaiming Zhao.  Simple weight modules over weak generalized [W]{}eyl algebras. ,  219(8):3427–3444, 2015. 

Rencai L[ü]{} and Kaiming Zhao.  Finite-dimensional simple modules over generalized [H]{}eisenberg algebras. ,  475:276–291, 2015. 

L. Makar-Limanov.  Locally nilpotent derivations, a new ring invariant and applications.  Available at <http://math.wayne.edu/~lml/lmlnotes.pdf>. 

Sonia Rueda.  Some algebras similar to the enveloping algebra of [$\rm sl(2)$]{}. ,  30(3):1127–1152, 2002. 

S. P. Smith.  A class of algebras similar to the enveloping algebra of [${\rm
  sl}(2)$]{}. ,  322(1):285–314, 1990. 

Mariano Su[á]{}rez-Alvarez and Quimey Vivas.  Automorphisms and isomorphisms of quantum generalized [W]{}eyl algebras. ,  424:540–552, 2015. 

<span style="font-variant:small-caps;">Samuel A. Lopes</span>\
*CMUP, Faculdade de Ciências, Universidade do Porto\
Rua do Campo Alegre 687\
4169-007 Porto, Portugal*\
`slopes@fc.up.pt`

[^1]: The author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.