representation of $\mathscr{W}_{q}$ given in Proposition 6.5. Let $\delta:A\to
A$ be a derivation of $A$. It extends uniquely to a derivation
$\tilde{\delta}:A_{x}\to A_{x}$ and, conjugating by $\tilde{\iota}$, gives a
derivation on $\mathscr{W}_{q}$: the claim of the corollary thus follows
immediately from the last part of Proposition 6.5. ∎

##  7 Twisted cohomology

In Proposition 2.13 we described the finite subgroups of
$\operatorname{Aut}(A)$ or, equivalently, the finite groups that act
faithfully on our algebra $A$ by algebra automorphisms, and showed that they
are all cyclic. A natural next step is to describe the finite dimensional Hopf
algebras that act on $A$ faithfully in the appropriate sense. In that
generality this is a problem that we do not know how to approach, so we will
settle for the much smaller problem of describing the faithful actions of Taft
Hopf algebras, motivated by the fact that these Hopf algebras can be viewed as
«quantum thickenings» of cyclic groups and are therefore very close to them.
We will do this in the next section.

We will recall the details about Taft algebras later. For now, let us mention
simply that an action of a Taft Hopf algebra on our algebra $A$ is determined
by an algebra automorphism $\phi:A\to A$ of finite order and a $\phi$-twisted
derivation of $A$. We know the automorphisms very well already, and our task
in the present section will be to obtain a description of the twisted
derivations.

Let $\omega\in\Bbbk$ be different from $0$ and $1$, and let
$\phi=\phi_{0,\omega}:A\to A$ be the algebra automorphism such that
$\phi(x)=\omega x$ and $\phi(y)=\omega^{N-1}y$; clearly $\phi$ has finite
order exactly when $\omega$ has finite order, but we do not need to impose
that restriction yet. We write ${}_{\phi}A$ for the $A$-bimodule which
coincides with $A$ as a right $A$-module and whose left action is such that
$a\mathbin{\triangleright}x=\phi(a)x$ for all $a\in A$ and all
$x\in{}_{\phi}A$. As the automorphism $\phi$ is homogeneous, the $A$-bimodule
${}_{\phi}A$ is a graded one.

We are interested in computing the space of $\phi$-twisted derivations: such a
map is a linear function $\partial:A\to A$ such that

$\partial(ab)=\partial(a)\cdot b+\phi(a)\cdot\partial(b)$ (203)

for all $a$ and $b$ in $A$. It is immediate that a $\phi$-twisted derivation
$A\to A$ is exactly the same thing as a derivation $A\to{}_{\phi}A$ into the
bimodule ${}_{\phi}A$, and because of this we will write
$\operatorname{Der}(A,{}_{\phi}A)$ for the space of all $\phi$-twisted
derivations and conflate the two notions.

Now, if $m$ is an element of $A$, then the map
$\operatorname{ad}_{\phi}(m):a\in A\mapsto ma-\phi(a)m\in A$ is an element of
$\operatorname{Der}(A,{}_{\phi}A)$: we say that such a $\phi$-twisted
derivation is an inner one. We can thus split the calculation of
$\operatorname{Der}(A,{}_{\Phi}A)$ in two steps: compute the $\phi$-twisted
derivations modulo inner ones, and then glue back the inner ones into the
result. The first step amounts, precisely, to the calculation of the first
Hochschild cohomology space $\operatorname{H}^{1}(A,{}_{\phi}A)$, and this is
what we will do — and just as in Section 3 we will do this by first computing
the spaces $\operatorname{H}^{0}(A,{}_{\phi}A)$ and
$\operatorname{H}^{2}(A,{}_{\phi}A)$.

The Hochschild cohomology $\operatorname{H}^{*}(A,{}_{\phi}A)$ can be
identified with the cohomology of the complex
$\operatorname{Hom}_{A^{e}}(P_{*},{}_{\phi}A)$, with $P_{*}$ the projective
resolution described in Section 3, and this complex, in turn — just as we
built the complex (83) in that section — is naturally isomorphic to the
complex

${A}$${A\otimes
V^{*}}$${A\otimes\Lambda^{2}V^{*}}$${0}$$\scriptstyle{\delta_{0}}$$\scriptstyle{\delta_{1}}$
(204)

with differentials such that

$\displaystyle\delta_{0}(a)$ $\displaystyle=(\omega xa-
ax)\otimes\hat{x}+(\omega^{N-1}ya-ay)\otimes\hat{y},$ (205)
$\displaystyle\delta_{1}(b\otimes\hat{x}+c\otimes\hat{y})$
$\displaystyle=\left((\omega^{N-1}yb-by)+(cx-\omega
xc)-\sum_{s+1+t=N}\omega^{s}x^{s}bx^{t}\right)\otimes\hat{x}\wedge\hat{y}$
(206)

for all choices of $a$, $b$ and $c$ in $A$. We remark that we are writing
«$A$» here and in (204) and not «${}_{\phi}A$»: we will write the twisted left
action explicitly, as in the formulas above for the differentials — we hope
this is less confusing than the alternative.

Just as when we dealt with the «untwisted» Hochschild cohomology, we start our
calculation with a lemma that tells us how some specific commutators — twisted
commutators now222So we have not broken our promise not to do any more
commutation formulas! — behave. In its statement and in the rest of this
section we will use the convention that if $\mathit{snark}$ is an object
related to $A$ then we will write $\widetilde{\smash{\mathit{snark}}}$ the
‘corresponding’ object related to the localization $A_{x}$.

######  Lemma 7.1.

The map $\alpha:a\in A\mapsto\omega xa-ax\in A$ is injective and has image
equal to $xA$, while the map $\tilde{\alpha}:a\in A_{x}\mapsto\omega xa-ax\in
A_{x}$ is bijective.

###### Proof.

Let $a$ be a non-zero element of $A$. There are $d\in\mathbb{N}_{0}$ and
$a_{0}$,…, $a_{d}\in\Bbbk[x]$ such that $a_{d}\neq 0$ and
$a=\sum_{i=0}^{d}a_{i}y^{i}$, and we have that

$\alpha(a)=\sum_{i=0}^{d}a_{i}(\omega
xy^{i}-y^{i}x)\equiv(\omega-1)a_{d}xy^{d}\mod F_{d-1}.$ (207)

Since $\omega\neq 1$, it follows immediately from this that $\alpha(a)\neq 0$,
so that $\ker\alpha=0$.

Let us now show that $xa$ is in the image of $\alpha$: we will proceed by
induction with respect to the integer $d$. If $d=0$, then $a=a_{0}\in\Bbbk[x]$
and we have that $xa=\alpha(a/(\omega-1))\in\operatorname{img}\alpha$. If
instead $d\geq 1$, then we have, according to (207), that

$xa-\alpha\left(\frac{a_{d}y^{d}}{\omega-1}\right)\in F_{d-1}.$ (208)

As $xA=Ax$, the image of $\alpha$ is certainly contained in $xA$, and
therefore the difference in (208) is both in $F_{d-1}$ and in $xA$: the
inductive hypothesis therefore implies that it is in the image of $\alpha$,
and so is $xa$, as we wanted. With this we have proved the claims of the lemma
about the map $\alpha$.

Let now $a$ be an element of $A_{x}$. There is a non-negative integer
$l\in\mathbb{N}_{0}$ such that $x^{l}a\in A$. If $\tilde{\alpha}(a)=0$, then
$\alpha(x^{l}a)=x^{l}\tilde{\alpha}(a)=0$ and what we have already proved
implies that $x^{l}a=0$, so that $a=0$: this shows that the map
$\tilde{\alpha}$ is injective. On the other hand, as $x^{l+1}a\in xA$, we
already know there is a $b\in A$ such that $\alpha(b)=x^{l+1}a$ and then
$\tilde{\alpha}(x^{-l-1}b)=a$: the map $\tilde{\alpha}$ is also surjective. ∎

With this lemma we can now easily determine the twisted cohomology
$\operatorname{HH}^{*}(A,{}_{\phi}A)$ in degrees $0$ and $2$. Recall that when
$N\geq 2$ the homogeneous components of $A$ are finite-dimensional, so we can
work with Hilbert series in that case.

######  Proposition 7.2.

We have that $\operatorname{HH}^{0}(A,{}_{\phi}A)=0$. If $N\geq 2$, then the
Hilbert series of $\operatorname{HH}^{2}(A,{}_{\phi}A)$ is

$h_{\operatorname{HH}^{2}(A,{}_{\phi}A)}=\begin{dcases*}\hfil t^{-N}&if
$\omega^{N-1}\neq 1$;\\\\[5.0pt] \frac{t^{-N}}{1-t^{N-1}}&if
$\omega^{N-1}=1$.\end{dcases*}$ (209)

If $N=1$, then $\operatorname{HH}^{2}(A,{}_{\phi}A)=0$.

In the proof of the corollary we will exhibit concrete representatives for all
cohomology classes in $\operatorname{HH}^{2}(A,{}_{\phi}A)$.

###### Proof.

Since the map $\alpha$ of the lemma is injective, it follows from the formula
(205) for the differential $\delta_{0}$ that the latter is itself injective
and, therefore, that $\operatorname{HH}_{0}(A,{}_{\phi}A)=0$.

Let us suppose that $N\geq 2$ and that $\omega^{N-1}\neq 1$. If $f\in\Bbbk[y]$
and $v\in A$, then there is $c\in A$ such that

$\omega xc-
cx=\alpha(c)=-(\omega^{N-1}-1)xv-\sum_{s+1+t=N}\omega^{s}x^{s}fx^{t},$ (210)

because the right hand side of this equality is in $xA$ — recall that $xA=Ax$
— and therefore

$\displaystyle\delta_{1}(f\otimes\hat{x}+c\otimes y)$
$\displaystyle=\left((\omega^{N-1}yf-fy)+(cx-\omega
xc)-\sum_{s+1+t=N}\omega^{s}x^{s}fx^{t}\right)\otimes\hat{x}\wedge\hat{y}$
(211) $\displaystyle=(\omega^{N-1}-1)(fy+xv)\otimes\hat{x}\wedge\hat{y}.$
(212)

This tells us that

$(\Bbbk[y]y+xA)\otimes\hat{x}\wedge\hat{y}\subseteq\operatorname{img}\delta_{1}.$
(213)

The subspace $\Bbbk[y]y+xA$ that appears here is the bilateral ideal $I$ of
$A$ generated by $x$ and $y$. It is clear from the formula (206) and our
hypothesis that $N\geq 2$ that the image of $\delta_{1}$ is contained in
$I\otimes\hat{x}\wedge\hat{y}$, so in (213) we actually have an equality. It
follows from this that $\operatorname{HH}^{2}(A,{}_{\phi}A)$ can be identified
with $A/I\otimes\hat{x}\wedge\hat{y}$, so it has dimension $1$, and is
generated by the class of the element $1\otimes\hat{x}\wedge\hat{y}$, which is
homogeneous of degree $-N$. We thus have that
$h_{\operatorname{HH}^{2}(A,{}_{\phi}A)}=t^{-N}$.

Next, we suppose that $N\geq 2$ and that $\omega^{N-1}=1$. According to (206),
if $b$ and $c$ are in $A$ we have now

$\delta_{1}(b\otimes\hat{x}+c\otimes\hat{y})=\left([y,b]+(cx-\omega
xc)-\sum_{s+1+t=N}\omega^{s}x^{s}bx^{t}\right)\otimes\hat{x}\wedge\hat{y}.$
(214)

The three terms in the sum wrapped with big parentheses are clearly in $xA$,
so

$\operatorname{img}\delta_{1}\subseteq xA\otimes\hat{x}\wedge\otimes\hat{y}.$
(215)

As $\delta_{1}(c\otimes 1)=-\alpha(c)\otimes\hat{x}\wedge\hat{y}$ for each
$c\in A$, Lemma 7.1 implies us that in fact we have an equality in (215) and
therefore that $\operatorname{HH}^{2}(A,{}_{\phi}A)$ is in this case
isomorphic to $A/xA\otimes\hat{x}\wedge\hat{y}$. This is isomorphic as a
graded vector space to $\Bbbk[y]\otimes\hat{x}\wedge\hat{y}$, and therefore in
this case the Hilbert series we are after is

$h_{\operatorname{HH}^{2}(A,{}_{\phi}A)}=\frac{t^{-N}}{1-t^{N-1}}.$ (216)

Finally, let us suppose that $N=1$. For all $b$ and $c$ in $A$ we have now
that

$\delta_{1}(b\otimes\hat{x}+c\otimes\hat{y})=\Bigl{(}[y,b]+(cx-\omega
xc)-b\Bigr{)}\otimes\hat{x}\wedge\hat{y}.$ (217)

If $u$ is an element of $A$, then there are $f\in\Bbbk[y]$ and $v\in A$ such
that $u=f+xv$, and there is a $g\in A$ such that $gx-\omega xg=-\alpha(g)=xv$,
so that
$\delta_{1}(-f\otimes\hat{x}+g\otimes\hat{y})=u\otimes\hat{x}\wedge\hat{y}$.
This tells us that the map $\delta_{1}$ is in this case surjective, so that we
have $\operatorname{HH}^{2}(A,{}_{\phi}A)=0$. ∎

We can now compute $\operatorname{HH}^{1}(A,{}_{\phi}A)$ using the same
strategy that we used for the regular cohomology in Section 3.

######  Proposition 7.3.

If $N\geq 2$, then the Hilbert series of $\operatorname{HH}^{1}(A,{}_{\phi}A)$
is

$h_{\operatorname{HH}^{1}(A,{}_{\phi}A)}(t)=\begin{dcases*}0&if
$\omega^{N-1}\neq 1$;\\\ \frac{1}{t(1-t^{N-1})}&if
$\omega^{N-1}=1$.\end{dcases*}$ (218)

If $N=1$, the $\operatorname{HH}^{1}(A,{}_{\phi}A)=0$.

###### Proof.

Let us suppose first that $N\geq 2$. The Hilbert series of our algebra $A$, as
we saw before, is $h_{A}(t)=(1-t)^{-1}(1-t^{N-1})^{-1}$, and the Euler
characteristic of the complex (204) is

$h_{A}(t)-(t^{-1}+t^{-(N-1)})h_{A}(t)+t^{-N}h_{A}(t)=t^{-N},$ (219)

so the invariance of the Euler characteristic when passing to cohomology
implies that

$h_{\operatorname{HH}^{0}(A,{}_{\phi}A)}(t)-h_{\operatorname{HH}^{1}(A,{}_{\phi}A)}(t)+h_{\operatorname{HH}^{2}(A,{}_{\phi}A)}(t)=t^{-N}.$
(220)

The Hilbert series $h_{\operatorname{HH}^{1}(A,{}_{\phi}A)}(t)$ can be
computed immediately from this and the preceding proposition, and turns out to
be what is described in the statement.

Let us now suppose that $N=1$ and let $\eta=b\otimes\hat{x}+c\otimes\hat{y}$,
with $b$ and $c$ in $A$, be a $1$-cocycle in the complex (204). If $f\in A$,
then

$\eta+\delta_{0}(f)=(b+\omega xf-
xf)\otimes\hat{x}+(\text{something})\otimes\hat{y},$ (221)

and Lemma 7.1 implies that we can choose $f$ so that $b+\omega xf-xf$ is in
$\Bbbk[y]$. This means that, up to replacing $\eta$ by a cohomologous
$1$-cocycle, we can suppose that $b$ itself is in $\Bbbk[y]$. In that case we
have, according to (206), that

$0=\delta_{1}(\eta)=(cx-\omega xc-b)\otimes\hat{x}\wedge\hat{y}.$ (222)

Since $cx-\omega xc\in xA$ and $b\in\Bbbk[y]$, this implies that $cx-\omega
xc=b=0$ and the injectivity of the map $\alpha$ that, in fact, also $c=0$.
This shows that every $1$-cocycle in our complex (204) is cohomologous to zero
and therefore that $\operatorname{HH}^{1}(A,{}_{\phi}A)=0$. ∎

The proposition we have just proved shows that all $\phi$-twisted derivations
$\operatorname{Der}(A,{}_{\phi}A)$ are in fact inner except in the special
case in which $N\geq 2$ and $\omega^{N-1}=1$. In this special case we can make
the following observations. We will work with the localization $A_{x}$ of $A$
at the powers of the normal element $x$, in which we have the following
commutation rule:

$yx^{-1}-x^{-1}y=-x^{N-2}.$ (223)

The canonical map $A\to A_{x}$ is injective, so we view $A$ as a subalgebra of
$A_{x}$. The automorphism $\phi:A\to A$ such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{N-1}y$ clearly extends to an automorphism
$\tilde{\phi}:A_{x}\to A_{x}$, and every $\phi$-twisted derivation
$\partial:A\to A$ extends uniquely to a $\tilde{\phi}$-twisted derivation
$\tilde{\partial}:A_{x}\to A_{x}$ that has
$\tilde{\partial}(x^{-1})=-\omega^{-1}x^{-1}\partial(x)x^{-1}$.

The following technical result will allow us to construct twisted derivations.

######  Proposition 7.4.

Suppose that $N\geq 2$ and $\omega^{N-1}=1$.

  1. (_i_)

There is a $\phi$-twisted derivation $\partial_{0}^{\phi}:A\to A$ such that

$\partial_{0}^{\phi}(x)=1-\omega,\qquad\partial_{0}^{\phi}(y)=x^{N-2},$ (224)

and it is homogeneous of degree $-1$ and not inner. The inner
$\tilde{\phi}$-twisted derivation
$\operatorname{ad}_{\tilde{\phi}}(x^{-1}):A_{x}\to A_{x}$ of $A_{x}$
corresponding to $x^{-1}$ preserves the subalgebra $A$ and, in fact, the map
$\partial_{0}^{\phi}$ is the restriction of
$\operatorname{ad}_{\tilde{\phi}}(x^{-1})$ to $A$.

  2. (_ii_)

There is a derivation $\partial_{N-1}:A\to A$ such that

$\partial_{N-1}(x)=xy,\qquad\partial_{N-1}(y)=\hskip-10.00002pt\sum_{s+2+t=N}(s+1)x^{s+1}yx^{t}+\frac{N-1}{2}\left(y^{2}-Nx^{N-1}y\right).$
(225)

This derivation commutes with $\phi:A\to A$.

  3. (_iii_)

If $\partial:A\to{}_{\phi}A$ is a twisted derivation that is homogeneous of
some degree $l$, then the map

$\partial_{N-1}\circ\partial-\partial\circ\partial_{N-1}:A\to A$ (226)

is also a $\phi$-twisted derivation that is homogeneous, now of degree
$l+N-1$. There is therefore a map

$\operatorname{ad}(\partial_{N-1}):\partial\in\operatorname{Der}(A,{}_{\phi}A)\longmapsto\partial_{N-1}\circ\partial-\partial\circ\partial_{N-1}\in\operatorname{Der}(A,{}_{\phi}A)$
(227)

that is homogeneous of degree $N-1$.

  4. (_iv_)

If $u$ is an element of $A_{x}$ such that the inner $\tilde{\phi}$-derivation
$\operatorname{ad}_{\tilde{\phi}}(u):A_{x}\to A_{x}$ preserves the subalgebra
$A$, so that the restriction $\operatorname{ad}_{\tilde{\phi}}(u)|_{A}:A\to A$
is an element of $\operatorname{Der}(A,{}_{\phi}A)$, then the map
$\operatorname{ad}_{\tilde{\phi}}(\tilde{\partial}_{N-1}(u)):A_{x}\to A_{x}$
also preserves $A$ and

$\operatorname{ad}(\partial_{N-1})\Bigl{(}\operatorname{ad}_{\tilde{\phi}}(u)|_{A}\Bigr{)}=\operatorname{ad}_{\tilde{\phi}}(\tilde{\partial}_{N-1}(u))|_{A}.$
(228)

Here $\tilde{\partial}_{N-1}(u)$ denotes the value at $u$ of the extension
$\tilde{\partial}_{N-1}$ of the derivation $\partial_{N-1}$ of (_ii_) to
$A_{x}$.

###### Proof.

(_i_) Let $\Bbbk\langle X,Y\rangle$ be the free algebra on two generators $X$
and $Y$. There is an automorphism $\hat{\phi}:\Bbbk\langle
X,Y\rangle\to\Bbbk\langle X,Y\rangle$ such that $\hat{\phi}(X)=\omega X$ and
$\hat{\phi}(Y)=Y$, and it preserves the bilateral ideal $I=(YX-XY-X^{N})$,
since $\omega^{N-1}=1$. The quotient $\Bbbk\langle X,Y\rangle/I$ is of course
isomorphic to $A$, and we may thus view $A$ as a $\Bbbk\langle
X,Y\rangle$-bimodule. Since the algebra $\Bbbk\langle X,Y\rangle$ is free,
there is a unique derivation $\hat{\partial}:\Bbbk\langle
X,Y\rangle\to{}_{\hat{\phi}}A$ such that $\hat{\partial}(X)=1-\omega$ and
$\hat{\partial}(Y)=x^{N-2}$. This derivation vanishes on the ideal $I$: to
check this, it is sufficient to show that it vanishes on the generator $YX-
XY-X^{N}$ of that ideal, and that is easy. One need only keep in mind that
$\hat{\partial}$ is a _$\hat{\phi}$ -twisted_ derivation, so that

$\hat{\partial}(X^{N})=(1+\omega+\cdots+\omega^{N-1})(1-\omega)x^{N-1}=(1-\omega)x^{N-1}.$
(229)

It follows from this that $\hat{\partial}$ induces a derivation
$\partial_{0}^{\phi}:A\to{}_{\phi}A$ such that
$\partial_{0}^{\phi}(x)=1-\omega$ and $\partial_{0}^{\phi}(y)=x^{N-2}$. It is
manifestly homogeneous of degree $-1$, and it is not inner because there are
no elements of negative degree in $A$.

Now, we certainly have an inner $\tilde{\phi}$-twisted derivation
$\operatorname{ad}_{\tilde{\phi}}(x^{-1}):A_{x}\to A_{x}$, and we can compute
that

$\displaystyle\operatorname{ad}_{\tilde{\phi}}(x^{-1})(x)$
$\displaystyle=x^{-1}x-\tilde{\phi}(x)x^{-1}=1-\omega,$ (230)
$\displaystyle\operatorname{ad}_{\tilde{\phi}}(x^{-1})(y)$
$\displaystyle=x^{-1}y-\tilde{\phi}(y)x^{-1}=x^{N-2}.$ (231)

Since $\tilde{\phi}(A)=A$, the facts that $x$ and $y$ generate $A$ and that
$\operatorname{ad}_{\tilde{\phi}}(x^{-1})$ is a $\tilde{\phi}$-twisted
derivation imply together that $\operatorname{ad}_{\tilde{\phi}}(x^{-1})$ maps
the subalgebra $A$ into itself. Moreover, its restriction to $A$ is a
$\phi$-twisted derivation that coincides with $\partial^{\phi}_{0}$ at $x$ and
at $y$, so we have that
$\operatorname{ad}_{\tilde{\phi}}(x^{-1})|_{A}=\partial^{\phi}_{0}$.

(_ii_) The derivation $\partial_{N-1}$ described here is simply the one given
by Proposition 5.6 when $l=N-1$, which we know to be homogeneous of degree
$N-1$ and not inner. To make this explicit we have used the fact that
$\Phi_{2}=\frac{1}{2}(y^{2}-Nx^{N-1}y)$. A simple direct calculation shows
that $\partial_{N-1}$ commutes with $\phi$.

(_iii_) This follows from an easy direct calculation. They key fact that makes
things work is that the derivation $\partial_{N-1}:A\to A$ commutes with the
automorphism $\phi:A\to A$.

(_iv_) A direct calculation shows that we have an equality of maps $A_{x}\to
A_{x}$

$\operatorname{ad}(\tilde{\partial}_{N-1})\Bigl{(}\operatorname{ad}_{\tilde{\phi}}(u)\Bigr{)}=\operatorname{ad}_{\tilde{\phi}}\Bigl{(}\tilde{\partial}_{N-1}(u)\Bigr{)}$
(232)

The left hand side preserves $A$ because $\operatorname{ad}_{\tilde{\phi}}(u)$
and $\tilde{\partial}_{N-1}$ do, so the right hand side also preserves it. The
equality (228) that appears in the statement of the proposition is now
immediate. ∎

Combining the different parts of the proposition we have just proved we can
construct $\phi$-twisted derivations $A\to{}_{\phi}A$ of all degrees of the
form $-1+l(N-1)$ with $l\in\mathbb{N}_{0}$.

######  Corollary 7.5.

Suppose that $N\geq 2$ and $\omega^{N-1}=1$, and let $\phi:A\to A$ be the
automorphism such that $\phi(x)=\omega x$ and $\phi(y)=y$. Let
$\tilde{\phi}:A_{x}\to A_{x}$ and $\tilde{\partial}_{N-1}:A_{x}\to A_{x}$ be
the extensions of $\phi$ and of the $\phi$-twisted derivation
$\partial_{N-1}:A\to A$ of Proposition 7.4. For each $l\in\mathbb{N}_{0}$ the
$\tilde{\phi}$-twisted derivation

$\operatorname{ad}_{\tilde{\phi}}\bigl{(}\tilde{\partial}_{N-1}^{l}(x^{-1})\bigr{)}:A_{x}\to
A_{x}$ (233)

preserves $A$ and its restriction to $A$ is a map

$\partial^{\phi}_{l}:A\to A$ (234)

that is a homogeneous $\phi$-twisted derivation of degree $-1+l(N-1)$ and that
coincides with the map

$\operatorname{ad}(\partial_{N-1})^{l}(\partial_{0}^{\phi})\in\operatorname{Der}(A,{}_{\phi}A).$
(235)

###### Proof.

This can be proved immediately by induction starting with (_i_) of the
proposition and using part (_iv_) for the inductive step. ∎

######  Remark 7.6.

There is a general-nonsense way of looking at our last few results. We will
explain it, as it helps to make sense of them.

Suppose that $N\geq 2$ and that $\omega^{N-1}=1$, and let $\nu$ be the order
of $\omega$ in $\Bbbk^{\times}$, which is a divisor of $N-1$. General
considerations imply that the automorphism $\phi$ acts in a canonical way on
the Hochschild cohomology $\operatorname{HH}^{*}(A)$ of $A$, the invariant
subspace $\operatorname{HH}^{*}(A)^{\phi}$ is a sub-Gerstenhaber algebra of
$\operatorname{HH}^{*}(A)$, and there is a «Gerstenhaber module structure»

$\operatorname{HH}^{*}(A)^{\phi}\otimes\operatorname{HH}^{*}(A,{}_{\phi}A)\to\operatorname{HH}^{*}(A,{}_{\phi}A)$
(236)

that in particular restricts to a Lie module structure

$\operatorname{HH}^{1}(A)^{\phi}\otimes\operatorname{HH}^{1}(A,{}_{\phi}A)\to\operatorname{HH}^{1}(A,{}_{\phi}A),$
(237)

which itself is induced by the map

$f\otimes
g\in\operatorname{Der}(A)\otimes\operatorname{Der}(A,{}_{\phi}A)\mapsto\operatorname{ad}(f)(g)\in\operatorname{Der}(A,{}_{\phi}A),$
(238)

using the notation of part (_iii_) of Proposition 7.4.

If we view $\operatorname{HH}^{1}(A)$ as
$\operatorname{Der}(A)/\operatorname{InnDer}(A)$ then the action of $\phi$ on
$\operatorname{HH}^{1}(A)$ is induced by the map

$\phi^{\sharp}:d\in\operatorname{Der}(A)\mapsto\phi\circ
d\circ\phi^{-1}\in\operatorname{Der}(A),$ (239)

and a direct calculation proves that if $d:A\to A$ is a homogeneous derivation
of degree $l$, then $\phi^{\sharp}(d)=\omega^{l}d$. It follows from this that
the invariant subspace $\operatorname{HH}^{1}(A)^{\phi}$ is spanned by the
classes of the derivations of the form $L_{j\nu}$, with $j\in\mathbb{Z}$ such
that $j\geq-1$, and $\partial_{0}$, using the notation of Section 5. On the
other hand, Proposition 7.3 tells us that the only homogeneous components of
$\operatorname{HH}^{1}(A,{}_{\phi}A)$ that are non-zero are those of degrees
$l$ such that $l\geq-1$ and $l\equiv-1\mod N-1$: as the action (237) is
homogeneous, this implies that the only $L_{l}$ in
$\operatorname{HH}^{1}(A)^{\phi}$ that can act non-trivially on
$\operatorname{HH}^{1}(A,{}_{\phi}A)$ are those with those with $l\equiv 0\mod
N-1$. What Corollary 7.5 does is, essentially, to describe the action of
$L_{N-1}$, the simplest of those: in part (_i_) of Proposition 7.4 we
exhibited an element $\partial_{0}^{\phi}$ in
$\operatorname{Der}(A,{}_{\phi}A)$, and in Corollary 7.5 we produced many
others by using the action (238). We will see below that in this way we obtain
a basis for $\operatorname{HH}^{1}(A,{}_{\phi}A)$. ∎

We are just a hop, skip and jump away from the description of
$\operatorname{H}^{1}(A,{}_{\phi}A)$ in the exceptional case in which it is
non-zero — apart, alas, for one exception.

######  Proposition 7.7.

Suppose that $N\geq 4$ and that $\omega^{N-1}=1$, and let $\phi:A\to A$ be the
automorphism of $A$ such that $\phi(x)=\omega x$ and $\phi(y)=y$. The
cohomology classes of the $\phi$-twisted derivations $\partial^{\phi}_{l}$
with $l\in\mathbb{N}_{0}$ that we described in Corollary 7.5 freely span the
vector space $\operatorname{H}^{1}(A,{}_{\phi}A)$.

This excludes the cases in which $2\leq N\leq 3$. Now if $N=2$ then the
hypothesis on $\omega$ is that $\omega=1$ and, since we are assuming
throughout that $\omega\neq 1$, this case cannot actually occur. The case in
which $N=3$, on the other hand, is a real possibility that is excluded in this
proposition — and for good reason, as we will see below.

###### Proof.

It is enough that we show that the cohomology classes of those $\phi$-twisted
derivations are linearly independent in $\operatorname{H}^{1}(A,{}_{\phi}A)$,
because if that is the case then they span the whole space: this follows
immediately from our calculation of the Hilbert series of
$\operatorname{H}^{1}(A,{}_{\phi}A)$ in Proposition 7.3 and the fact that for
all $l\in\mathbb{N}_{0}$ the twisted derivation $\partial^{\phi}_{l}$ has
degree $-1+l(N-1)$.

Let us suppose that the cohomology classes of the twisted derivations in the
statement are linearly dependent. There are then an integer $d\geq 0$ and
scalars $a_{0}$, …, $a_{d}$ in $\Bbbk$ such that $a_{d}\neq 0$ and the
$\phi$-twisted derivation

$\delta\coloneqq\sum_{i=0}^{d}a_{i}\partial_{i}^{\phi}:A\to A$ (240)

is inner, so that there is an element $u$ in $A$ such that
$\delta=\operatorname{ad}_{\phi}(u)$. We let $P$ be the polynomial
$\sum_{i=0}^{d}a_{i}T\in\Bbbk[T]$ and consider the $\tilde{\phi}$-twisted
derivation

$\operatorname{ad}_{\tilde{\phi}}\bigl{(}P(\tilde{\partial}_{N-1})(x^{-1})\bigr{)}:A_{x}\to
A_{x}$ (241)

It follows from Corollary 7.5 that this map preserves $A$ and its restriction
to $A$ is precisely $\delta$. In particular, the map in (241) and
$\operatorname{ad}_{\tilde{\phi}}(u):A_{x}\to A_{x}$ take the same value on
$x$ and $y$ and therefore, since they are $\tilde{\phi}$-twisted derivations,
they are in fact equal — in other words, we have that

$\operatorname{ad}_{\tilde{\phi}}\Bigl{(}P(\tilde{\partial}_{N-1})(x^{-1})-u\Bigr{)}=0.$
(242)

We know form Lemma 7.1 that the map $\tilde{\alpha}:a\in A_{x}\mapsto\omega
xa-xa\in A_{x}$ is injective, and as a consequence of that

if $v\in A_{x}$ and the map $\operatorname{ad}_{\tilde{\phi}}(v):A_{x}\to
A_{x}$ vanishes at $x$, then $v=0$. (243)

From this and the equality (242) we can conclude that

$P(\tilde{\partial}_{N-1})(x^{-1})=u\in A.$ (244)

Let us next check that for all $l\in\mathbb{N}_{0}$ we have that

$\tilde{\partial}_{N-1}^{l}(x^{-1})\equiv\prod_{i=0}^{l-1}\left(\frac{t(N-1)}{2}-1\right)\cdot
x^{-1}y^{l}\mod\tilde{F}_{l-1}.$ (245)

This is clear when $l=0$, and when $l=1$ we have that

$\tilde{\partial}_{N-1}(x^{-1})=-x^{-1}\tilde{\partial}_{N-1}(x)x^{-1}=-yx^{-1}\equiv-x^{-1}y\mod\tilde{F}_{0},$
(246)

in accordance with the formula we want to prove. The general case follows by
induction and the observation that for all $l\in\mathbb{N}_{0}$ we have

$\displaystyle\tilde{\partial}_{N-1}(x^{-1}y^{l})=\tilde{\partial}_{N-1}(x^{-1})y^{l}+x^{-1}\tilde{\partial}_{N-1}(y^{l})=-yx^{-1}y^{l}+x^{-1}\sum_{i=0}^{l-1}y^{i}\tilde{\partial}_{N-1}(y)y^{l-1-i}$
(247)
$\displaystyle=-yx^{-1}y^{l}+x^{-1}\sum_{i=0}^{l-1}y^{i}\left[\sum_{s+2+t=N}(s+1)x^{s+1}yx^{t}+\frac{N-1}{2}\left(y^{2}-Nx^{N-1}y\right)\right]y^{l-1-i}$
(248)
$\displaystyle\equiv\left(\frac{l(N-1)}{2}-1\right)x^{-1}y^{l+1}\mod\tilde{F}_{l}.$
(249)

Now, using the congruence (245) we have just proved we see that

$P(\tilde{\partial}_{N-1})(x^{-1})\equiv\underbrace{a_{d}\prod_{i=0}^{d-1}\left(\frac{t(N-1)}{2}-1\right)}{}\cdot
x^{-1}y^{d}\mod\tilde{F}_{d-1}$ (250)

As $N\geq 4$, the scalar marked with a brace is non-zero. If we call it
$\gamma$ for brevity, then what we have is that

$A\ni P(\tilde{\partial}_{N-1})(x^{-1})=\gamma\cdot x^{-1}y^{d}+w$ (251)

for some $w\in\tilde{F}_{d}$, and this is absurd. This contradiction proves
the proposition. ∎

When $N=3$, in the situation of Proposition 7.7 we can compute that

$\tilde{\partial}_{2}(x)=xy,\qquad\tilde{\partial}_{2}(x^{-1})=-x^{-1}y+x,\qquad\tilde{\partial}_{2}(y)=x^{4}+y^{2}$
(252)

and using that that

$\tilde{\partial}_{2}^{2}(x^{-1})=-x^{3}\in A.$ (253)

This tells us that the $\phi$-twisted derivation $\partial^{\phi}_{2}:A\to A$
is inner, and implies that in fact the $\phi$-twisted derivation
$\partial^{\phi}_{l}:A\to A$ is inner whenever $l$ is an integer such that
$l\geq 2$. The conclusion of Proposition 7.7 is therefore very false when
$N=3$. We can fix it as follows:

######  Proposition 7.8.

Suppose that $N=3$ and that $\omega^{N-1}=1$, so that $\omega=-1$, and let
$\phi:A\to A$ be the automorphism of $A$ such that $\phi(x)=-x$ and
$\phi(y)=y$. The $\tilde{\phi}$-twisted derivations of $A_{x}$

$\operatorname{ad}_{\tilde{\phi}}(x^{-1}),\qquad\quad\operatorname{ad}_{\tilde{\phi}}(\tilde{\partial}_{2}(x^{-1})),\qquad\quad\operatorname{ad}_{\tilde{\phi}}(\tilde{\partial}_{2}^{l}(x^{-1}y^{2})),\quad
l\geq 0$ (254)

preserve the subalgebra $A$ and the cohomology classes of their restrictions
to $A$ freely span the vector space $\operatorname{H}^{1}(A,{}_{\phi}A)$.

###### Proof.

This can be proved in essentially the same way as the previous proposition,
and we therefore omit the details. ∎

With this we have concluded the calculation of the twisted Hochschild
cohomology $\operatorname{H}^{*}(A,{}_{\phi}A)$. One nice observation we can
immediately make with the result at hand is:

######  Proposition 7.9.

Suppose that $N\geq 1$ and that $\omega\in\Bbbk^{\times}\setminus\\{1\\}$, and
let $\phi:A\to A$ be the automorphism of $A$ such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{N-1}y$. Every $\phi$-twisted derivation $A\to{}_{\phi}A$ is
the restriction to $A$ of an inner $\tilde{\phi}$-twisted derivation
$A_{x}\to{}_{\tilde{\phi}}A_{x}$ that preserves $A$. ∎

By analogy with the notion of X-inner automorphisms of Harčenko
[Harchenko:1][Harchenko:2], we can rephrase this proposition saying that all
$\phi$-twisted derivations of $A$ are X-inner.

When $N=0$, so that the algebra $A$ is the first Weyl algebra, the calculation
of the twisted Hochschild cohomology $\operatorname{H}^{*}(A,{}_{\phi}A)$ for
an automorphism $\phi:A\to A$ such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{-1}y$ for some $\omega\in\Bbbk^{\times}$ was done in [AFLS]
by Jacques Alev, Thierry Lambre, Marco Farinati and Andrea Solotar: the end
result is that

$\dim\operatorname{H}^{p}(A,{}_{\phi}A)\cong\begin{cases*}1&if $p=0$ and
$\omega=1$, or if $p=2$ and $\omega\neq 1$;\\\ 0&in any other
case.\end{cases*}$ (255)

This is rather different from what we have found above for $N\geq 1$. It
should be noticed that while the automorphism group of the first Weyl algebra
is considerably larger than that of our algebras $A$ with $N\geq 1$, but every
one of its elements of finite order is conjugated to one of the form of the
automorphism $\phi$ described here. On the other hand, there are non-cyclic
finite groups of automorphisms of the first Weyl algebra.

##  8 Actions of Taft algebras

Let now $n$, $m\in\mathbb{N}$ be such that $1<m$ and $m\mid n$, and let
$\lambda\in\Bbbk^{\times}$ be a primitive $m$th root of unity. The generalized
Taft algebra $T=T_{n}(\lambda,m)$ is the algebra freely generated by two
letters $g$ and $\xi$ subject to the relations

$g^{n}=1,\qquad\xi^{m}=0,\qquad g\xi=\lambda\xi g.$ (256)

This algebra was originally studied by David Radford in [Radford]. It is
finite dimensional of dimension $nm$, and the set $\\{\xi^{i}g^{j}:0\leq
i<m,0\leq j<n\\}$ is one of its bases. It is a Hopf algebra with
comultiplication $\Delta:T\to T\otimes T$ and augmentation
$\varepsilon:T\to\Bbbk$ such that

$\Delta(g)=g\otimes g,\qquad\Delta(\xi)=\xi\otimes
1+g\otimes\xi,\qquad\varepsilon(g)=1,\qquad\varepsilon(\xi)=0.$ (257)

When $n=m$ and $\theta=0$ this is the classical Taft Hopf algebra constructed
by Earl Taft in [Taft] with the purpose of exhibiting finite dimensional Hopf
algebras with antipode of arbitrarily large order. We will use freely the
Heyneman–Sweedler notation for the coproduct of $T$ and even omit the sum: we
will write $\Delta(a)$ in the form $a_{1}\otimes a_{2}$.

We are interested in left $T$-module-algebra structures on our algebra $A$,
that is, left $T$-module structures $\mathbin{\triangleright}:T\otimes A\to A$
such that the multiplication and unit map $A\otimes A\to A$ and $\Bbbk\to A$
are both $T$-linear — this means that
$h\mathbin{\triangleright}1_{A}=\varepsilon(h)1_{A}$ and
$h\mathbin{\triangleright}ab=(h_{1}\mathbin{\triangleright}a)(h_{2}\mathbin{\triangleright}b)$
for all choices of $a$ and $b$ in $A$ and $h$ in $T$. We refer the reader to
[Montgomery]*Chapter 4 for general information about module-algebras over Hopf
algebras.

We will further restrict our attention to $T$-module algebra structures on $A$
that are inner-faithful — a notion introduced by T. Banica and J. Bichon in
[BB] — as these correspond to faithful group actions on $A$: the condition is
that there be no non-zero Hopf ideal $I$ in $T$ such that
$I\mathbin{\triangleright}A=0$. According to [Cline]*Corollary 3.7, in the
specific case in which the Hopf algebra is our generalized Taft algebra $T$ we
have a handy criterion:

a $T$-module-algebra structure on $A$ is inner-faithful if and only if the
group $G(T)=\langle g\rangle$ of group-like elements of $T$ acts faithfully on
$A$ and $\xi\mathbin{\triangleright}A\neq 0$. (258)

As the algebra $T$ is generated by $g$ and $\xi$ subject to the relations in
(256), giving a $T$-module structure on $A$ is the same as giving the two maps
$\phi:a\in A\mapsto g\mathbin{\triangleright}a\in A$ and $\partial:a\in
A\mapsto\xi\mathbin{\triangleright}a\in A$ such that
$\phi^{n}=\mathsf{id}_{A}$, $\partial^{m}=0$ and
$\phi\partial=\lambda\partial\phi$, and that structure will be a $T$-module-
algebra structure exactly when $\phi$ is an automorphism of $A$ and $\partial$
a $\phi$-twisted derivation $A\to A$. The automorphism $\phi$ will have finite
order: according to Proposition 2.13, up to conjugating the whole module-
algebra structure by an algebra automorphism of $A$ we can suppose then that
there is a scalar $\omega\in\Bbbk$ such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{N-1}y$, and then the group of group-like elements
$G(T)=\langle g\rangle$ of $T$, which is cyclic of order $n$, will clearly act
faithfully on $A$ if and only if the scalar $\omega$ is a primitive $n$th root
of unity.

We are left with the task of understanding the possibilities for the map
$\partial$. Since it is a $\phi$-twisted derivation $A\to A$, we now from
Proposition 7.9 that the map $\partial$ will be, in fact, the restriction of
an inner $\tilde{\phi}$-twisted derivation of the localization $A_{x}$ that
preserves $A$. The following lemma imposes significant restrictions on what
can actually happen.

######  Lemma 8.1.

Let $\omega$ be a primitive $n$th root of unity in $\Bbbk$, let $\phi:A\to A$
be the algebra automorphism such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{N-1}y$, and let $\tilde{\phi}:A_{x}\to A_{x}$ be the unique
extension of $\phi$ to the localization $A_{x}$. Let $u$ be a non-zero element
of $A_{x}$ such that the inner $\tilde{\phi}$-twisted derivation
$\operatorname{ad}_{\tilde{\phi}}(u):A_{x}\to A_{x}$ preserves the subalgebra
$A$ and let

$\partial\coloneqq\operatorname{ad}_{\tilde{\phi}}(u)|_{A}:A\to A$ (259)

be its restriction to $A$, which is a $\phi$-twisted derivation.

  1. (_i_)

The map $\partial$ is non-zero.

  2. (_ii_)

If $\lambda$ is a scalar, then we have that $\phi\partial=\lambda\partial\phi$
exactly when $\tilde{\phi}(u)=\lambda u$, and when that is the case we have
that $\lambda^{n}=1$.

  3. (_iii_)

If there are a scalar $\lambda$ and a positive integer $m$ such that
$\tilde{\phi}(u)=\lambda u$ and $\partial^{m}=0$, then $n\leq m$.

###### Proof.

(_i_) We have $\partial(x)=ux-\omega xu=-\tilde{\alpha}(u)\neq 0$, according
to Lemma 7.1.

(_ii_) If $\lambda\in\Bbbk$ is such that $\phi\partial=\lambda\partial\phi$,
then

$\displaystyle\tilde{\phi}(u)\omega x-\omega^{2}x\tilde{\phi}(u)$
$\displaystyle=\tilde{\phi}(ux-\omega xu)=\phi(ux-\omega
xu)=\phi(\partial(x))=\lambda\partial(\phi(x))$ (260)
$\displaystyle=\lambda\partial(\omega x)=\lambda(u\omega x-\omega^{2}xu),$
(261)

and therefore

$\tilde{\alpha}\bigl{(}\tilde{\phi}(u)-\lambda u\bigr{)}=\omega
x\bigl{(}\tilde{\phi}(u)-\lambda u\bigr{)}-\bigl{(}\tilde{\phi}(u)-\lambda
u\bigr{)}x=0,$ (262)

so that $\tilde{\phi}(u)=\lambda u$ because of Lemma 7.1. Conversely, if
$\lambda$ is a scalar such that $\tilde{\phi}(u)=\lambda u$, then a simple and
direct calculation shows that we have that $\phi\partial=\lambda\partial\phi$.
Moreover, since the automorphism $\tilde{\phi}$ is manifestly diagonalizable
and all its eigenvalues are powers of $\omega$, in that case we have that
$\lambda^{n}=1$.

(_iii_) Let us suppose that there are $\lambda\in\Bbbk$ and $m\in\mathbb{N}$
such that $\tilde{\phi}(u)=\lambda u$ and $\partial^{m}=0$. Since $u\neq 0$,
there is an integer $d\in\mathbb{N}_{0}$ such that
$u\in\tilde{F}_{d}\setminus\tilde{F}_{d-1}$. An induction shows that if
$l\in\mathbb{N}_{0}$, $a\in A$ and $t\in\Bbbk$ are such that $a\in
F_{l}\setminus F_{l-1}$ and $\phi(a)=ta$, then for all $k\in\mathbb{N}_{0}$ we
have that

$\partial^{k}(a)\equiv\prod_{i=0}^{k-1}(1-t\lambda^{i})\cdot
au^{k}\mod\tilde{F}_{l+kd-1}$ (263)

and, in particular, since $\partial^{m}(a)=0$, that

$\prod_{i=0}^{m-1}(1-t\lambda^{i})\cdot au^{m}\in\tilde{F}_{l+md-1}.$ (264)

As the graded algebra $\operatorname{gr}A_{x}$ for the filtration
$(\tilde{F}_{i})_{i\geq-1}$ is an integral domain, this tells us that, in
fact,

$\prod_{i=0}^{m-1}(1-t\lambda^{i})=0$ (265)

and, then, that $t\in\\{\lambda^{-i}:0\leq i<m\\}$. In particular, for each
$j\in\\{0,\dots,n-1\\}$ we can take $a=x^{j}$, that has $\phi(a)=\omega^{j}a$,
and conclude that the $n$ pairwise different scalars $\omega^{0}$, …,
$\omega^{n-1}$ all belong to the set $\\{\lambda^{-i}:0\leq i<m\\}$. This set
has cardinal at most $m$, so $n\leq m$, as the lemma claims. ∎

The obvious question after having proved Lemma 8.1 is: when is the map
$\partial$ appearing there such that $\partial^{n}=0$? The answer is simple:
never. With the objective of proving this, let us recall some standard
notations from the theory of $q$-variants. If $q$ is a variable, then for each
$n\in\mathbb{N}_{0}$ we let $[n]_{q}\coloneqq 1+q+\cdots+q^{n-1}$ and
$[n]_{q}!\coloneqq[1]_{q}[1]_{q}\cdots[n]_{q}$, and consider for each choice
of $k$ and $i$ in $\mathbb{N}_{0}$ such that $0\leq i\leq k$ the $q$-binomial
or Gaussian binomial coefficient

$\binom{k}{i}_{q}\coloneqq\frac{[k]_{q}!}{[i]_{q}!\cdot[k-i]_{q}!},$ (266)

which is an element of $\mathbb{Z}[q]$.

######  Lemma 8.2.

Let $\omega$ and $\lambda$ be primitive $n$th roots of unity in $\Bbbk$, let
$\phi:A\to A$ be the algebra automorphism such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{N-1}y$, and let $\tilde{\phi}:A_{x}\to A_{x}$ be the unique
extension of $\phi$ to the localization $A_{x}$. If $u$ is a non-zero element
of $A_{x}$ such that $\tilde{\phi}(u)=\lambda u$ and the inner
$\tilde{\phi}$-twisted derivation
$\operatorname{ad}_{\tilde{\phi}}(u):A_{x}\to A_{x}$ preserves the subalgebra
$A$, then the restriction
$\partial\coloneqq\operatorname{ad}_{\tilde{\phi}}(u)|_{A}:A\to A$, which is a
$\phi$-twisted derivation, has $\partial^{n}\neq 0$.

###### Proof.

Let $u$ be a non-zero element of $A_{x}$ such that $\tilde{\phi}(u)=\lambda u$
and $\operatorname{ad}_{\tilde{\phi}}(u)(A)\subseteq A$, and let us consider
the $\phi$-twisted derivation
$\partial\coloneqq\operatorname{ad}_{\tilde{\phi}}(u)|_{A}:A\to A$.

Let $a$ in $A$ and $t\in\Bbbk$ be such that $\phi(a)=ta$. For all
$k\in\mathbb{N}_{0}$ we have that

$\partial^{k}(a)=\sum_{i=0}^{k}(-1)^{k-i}\lambda^{\binom{i}{2}}\binom{k}{i}_{\lambda}t^{i}u^{k-i}au^{i},$
(267)

as can be proved by an obvious induction using the well-known analogue of
Pascal’s identity,

$\binom{k}{i}_{q}=\binom{k-1}{i}_{q}+q^{k-i}\binom{k-1}{i-1}_{q},$ (268)

valid in $\mathbb{Z}[q]$ whenever $0<i<k$. In particular, taking $k=n$ in
(267) we find that

$\partial^{n}(a)=\sum_{i=0}^{n}(-1)^{n-i}\lambda^{\binom{i}{2}}\binom{n}{i}_{\lambda}t^{i}u^{n-i}au^{i}.$
(269)

Now, according to the Cauchy $q$-binomial theorem we have that

$\prod_{i=0}^{n}(y+zq^{i})=\sum_{i=0}^{n}q^{\binom{i}{2}}\binom{n}{i}_{q}y^{n-i}z^{i}$
(270)

in the polynomial ring $\mathbb{Z}[q,y,z]$. Extending scalars to $\Bbbk$ and
specializing $z$ at $-1$ and $q$ at $\lambda$, the left hand side of this
equality becomes $y^{n}-1$, so by looking at the coefficients of the powers of
$y$ in the right hand side we see that $\binom{n}{i}_{\lambda}=0$ when
$0<i<n$. The equality (269) therefore violently simplifies to

$\partial^{n}(a)=\lambda^{\binom{n}{2}}au^{i}+(-1)^{n}u^{n}a,$ (271)

since $t^{n}=1$, as $t$ is an eigenvalue of the automorphism $\phi$ and thus a
power of $\omega$. As $\lambda$ is a primitive $n$th root of unity, we have
that $\lambda^{\binom{n}{2}}=(-1)^{n+1}$ and, therefore, that

$\partial^{n}(a)=(-1)^{n}[u^{n},a].$ (272)

This equality holds whenever $a$ is an eigenvector of $\phi$ in $A$ and, since
the automorphism $\phi$ is diagonalizable, it then follows immediately that
the equality holds in fact for all $a$ in $A$. In particular, the map
$\partial^{n}$ is zero exactly when the element $u^{n}$ of $A_{x}$ commutes
with all elements of $A$: this happens if and only if $u^{n}$ is central in
$A_{x}$, so an element of $\Bbbk$ according to Proposition 1.5, and clearly
this can happen if and only if $u\in\Bbbk$. Now, as $\lambda\neq 1$ and
$\phi(u)=\lambda u$, we certainly have that $u$ is not in $\Bbbk$, so
$\partial^{n}\neq 0$, as the lemma claims. ∎

######  Remark 8.3.

The vanishing of the Gaussian binomial coefficients at well-chosen roots of
unity that we used in this proof is a very special case of the $q$-Lucas
theorem, and it was in fact in that way that we originally proceeded. The
simpler recourse to the $q$-binomial theorem that we used above was suggested
by a comment of Richard Stanley on MathOverflow [A]. Let us note that the
$q$-Lucas theorem was first proved by Gloria Olive in [Olive], where the
theorem appears as Equation (1.2.4). This beautiful result, which generalizes
the classical Lucas theorem for binomial coefficients, has also been proved by
Jacques Désarménien in [Desarmenien], by Volker Strehl in [Strehl], by Bruce
Sagan in [Sagan], and Donald E. Knuth and Herbert S. Wilf describe in [KnW] a
factorization of the Gaussian binomial coefficient $\binom{n}{i}_{q}$ with
$0<i<n$ which certainly includes the $n$th cyclotomic polynomial among the
factors. ∎

After all this work we can state and proof the following markedly
disappointing result:

######  Proposition 8.4.

Let $n$ and $m$ be integers such that $1<m$ and $m\mid n$, and let
$\lambda\in\Bbbk^{\times}$ be a primitive $m$th root of unity in $\Bbbk$.
There is no inner-faithful action of the generalized Taft algebra
$T_{n}(\lambda,m)$ on $A$.

###### Proof.

Let us suppose, in order to reach a contradiction, that there is an inner-
faithful module-algebra action of the Hopf algebra $=T_{n}(\lambda,m)$ on $A$,
and let us consider the maps $\phi:a\in A\mapsto g\mathbin{\triangleright}a\in
A$ and $\partial:a\in A\mapsto\xi\mathbin{\triangleright}a\in A$. The
relations that define the algebra $T$ imply that $\phi^{n}=\mathsf{id}_{A}$,
$\partial^{m}=0$ and $\phi\partial=\lambda\partial\phi$, and the fact that
what we have is a module-algebra structure that $\phi$ is an automorphism and
$\partial$ a $\phi$-twisted derivation.

The criterion (258) for inner-faithfulness from Cline’s paper [Cline] implies
that the order of $\phi$ is exactly $n$, since $g$ has order $n$ in $T$, and
then, according to Proposition 2.13, up to conjugating the action of $T$ on
$A$ by an algebra automorphism we can suppose that there is a primitive $n$th
root of unity $\omega$ in $\Bbbk$ such that $\phi(x)=\omega x$ and
$\phi(y)=\omega^{N-1}y$. Proposition 7.9 now tells us that there is an element
$u$ in the localization $A_{x}$ such that the $\tilde{\phi}$-twisted
derivation $\operatorname{ad}_{\tilde{\phi}}(u):A_{x}\to A_{x}$ preserves the
subalgebra $A$ of $A_{x}$ and the map $\partial$ coincides with its
restriction $\operatorname{ad}_{\tilde{\phi}}(u)|_{A}$ to $A$. The second part
of Lemma 8.1 tells us tat $\tilde{\phi}(u)=\lambda u$ and, since
$\partial^{m}=0$, the third part of that lemma that $n\leq m$. Of course,
since $m$ divides $n$ we have in fact that $n=m$, and we are therefore in the
situation of Lemma 8.2: this is absurd, for the lemma tells us that
$\partial^{n}\neq 0$. ∎

In [Radford] Radford considers a more general class of generalized Taft
algebras: given two positive integers $n$, $m\in\mathbb{N}$ such that $1<m$
and $m\mid n$, a primitive $m$th root of unity $\lambda$ in $\Bbbk$, and an
arbitrary scalar $\tau\in\Bbbk$, he lets $T_{n}(\lambda,m,\tau)$ be the
algebra freely generated by two letters $g$ and $\xi$ subject to the relations

$g^{n}=1,\qquad\xi^{m}=\tau(g^{m}-1),\qquad g\xi=\lambda\xi g,$ (273)

which is a Hopf algebra with respect to the comultiplication and counit such
that

$\Delta(g)=g\otimes g,\qquad\Delta(\xi)=\xi\otimes
1+g\otimes\xi,\qquad\varepsilon(g)=1,\qquad\varepsilon(\xi)=0.$ (274)

When $\tau=0$ we obtain the algebras we considered above, since
$T_{n}(\lambda,m,0)=T_{n}(\lambda,m)$, and the connection between the two
families is that $T_{n}(\lambda,m,0)$ is the graded Hopf algebra corresponding
to the coradical filtration of the pointed Hopf algebra
$T_{n}(\lambda,m,\tau)$ — we refer to Section 5.2 in Montgomery’s book
[Montgomery] for information of this. It is a natural guess, after our last
proposition, that there is also no inner-faithful module-algebra action of
these more general Hopf algebras on our algebra $A$, although a new idea is
needed to verify this. For example, the twisted derivation $\partial$ that
corresponds to the element $\xi$ in an action of $T_{n}(\lambda,m,\tau)$ is
locally finite — in the sense that every element of $A$ is contained in a
finite-dimensional subspace that is invariant under $\partial$ — so
classifying locally-finite twisted derivations of $A$ could well be helpful.

## References


# Tachyonic Instability of Reissner-Nordström-Melvin Black Holes in Einstein-
Maxwell-Scalar Theory

Hengyu Xu1<EMAIL_ADDRESS>Shao-Jun Zhang1,2<EMAIL_ADDRESS>1
College of Science, Zhejiang University of Technology, Hangzhou 310032, China
2Institute for Theoretical Physics $\&$ Cosmology, Zhejiang University of
Technology, Hangzhou 310032, China

###### Abstract

In the framework of Einstein-Maxwell-scalar theory, we studied scalar field
perturbations of Reissner-Nordström-Melvin (RNM) black holes, which describes
the RN black holes immersed in a uniform magnetic field. Due to the coupling
to the Maxwell term, the scalar field acquires an effective mass whose square,
in the presence of the magnetic field, will become negative somewhere outside
the horizon for either sign of the coupling constant $\alpha$, thus triggering
the tachyonic instability and leading to spontaneous scalarization when
$\alpha$ is large enough. The magnetic field has significant influences on the
waveforms and the onset of the instability, which differs for different sign
of $\alpha$. Effects of the black hole charge on the instability are also
studied.

##  I Introduction

Recently, phenomenon of spontaneous scalarization over black holes (BHs) has
been attracting lots of attention. It provides a novel mechanism to endow BHs
with additional scalar hair, thus circumventing the famous BH no-hair theorem
Bekenstein:1998aw ; Robinson:2004zz ; Chrusciel:2012jk  while keeping
consistent with the current available astrophysical observations. Actually,
this phenomenon was first observed long time ago by Damour and Esposito-Farese
in neutron stars in the framework of a scalar-tensor theory Damour:1993hw . In
this theory, stars with vanishing scalar field are solutions, over which the
scalar field perturbation acquires a negative effective mass square due to its
coupling to the surrounding matter, thus triggering the tachyonic instability
and leading to the spontaneous scalarization over the stars. This phenomenon
is then realized in the scenario of BHs. In Refs. Doneva:2017bvd ;
Silva:2017uqg ; Antoniou:2017acq ; Cunha:2019dwb , the authors considered the
Einstein-scalar-Gauss-Bonnet (EsGB) theory where the scalar field is coupled
to the Gauss-Bonnet (GB) invariant non-minimally through some appropriate
coupling function. As general relativity (GR), EsGB also allows the standard
Kerr BH with no scalar hair. However, differently, the Kerr metric is not the
only one describing a rotating BH in EsGB. Rather, there exists another branch
of BH solutions carrying scalar hair in this theory. The bald Kerr BH can be
spontaneously scalarized and transformed into the hairy BH branch. In such
process, due to the coupling between the scalar field and the GB invariant,
the effective mass square of the scalar field perturbations becomes negative
in the strong curvature regime, thus triggers the tachyonic instability, and
addresses the Kerr BH with scalar hair.

This phenomenon is also found to exist in other modified gravity theories,
such as the dynamical Chern-Simon gravity (dCSG) in which the scalar field is
coupled instead to the gravitational Chern-Simons invariant Gao:2018acg ;
Myung:2020etf ; Doneva:2021dcc ; Zhang:2021btn ; Chatzifotis:2022mob .
Inspired by the work of Damour and Esposito-Farese mentioned above,
spontaneous scalarization over BH backgrounds induced by the surrounding
matter fields instead of the curvature is also realized in Einstein-Maxwell-
scalar (EMs) theory Herdeiro:2018wub , where the standard bald Reissner-
Nordström (RN) BH can be spontaneously scalarized due to the non-minimal
coupling between the scalar field and the Maxwell term. Inspired by these
work, amounts of effort have been devoted to this direction by considering
variants of scalarization models, various forms of coupling functions,
properties and astrophysical implications of the scalarized BHs, the so-called
spin-induced spontaneous scalarization, spontaneous scalarization in binary
systems, etc. For more details, see a recent review Doneva:2022ewd  and
references therein. The possible existences of scalarized BHs challenge the
Kerr hypothesis of GR, which claims that the most general BH should be
characterized only by the mass, angular momentum and the charge. With the
great advances of astrophysical observations in recent years, especially in
gravitational wave (GW) detection LIGOScientific:2016aoc ;
LIGOScientific:2016sjg ; LIGOScientific:2017bnn ; LIGOScientific:2017ycc ;
LIGOScientific:2020iuh ; LIGOScientific:2021usb  and at a variety of
electromagnetic wavelengths observations GRAVITY:2020gka ;
EventHorizonTelescope:2019dse ; EventHorizonTelescope:2021srq , one can expect
to examine this phenomenon and thus test GR in the near future.

However, most of the studies on spontaneous scalarization to date have
considered rather ideal situations in which BHs live in a clean vacuum. In the
realistic universe, BHs always live in rather complicated astrophysical
environments, such as the magnetic fields, accretion disks, dark matter halos,
cosmological expansion, etc. These environments can influence physical
processes around BHs considerably Blandford:1977ds ; Barausse:2014tra ;
Brito:2014nja . Then, it is natural and important to consider the impact of
the realistic astrophysical environments on the BH scalarization. Among these
various environmental factors, our interest in this work is in the magnetic
fields. According to available astrophysical observations, magnetic field is
believed to pervade our universe with different amplitudes on different scales
Crocker:2010xc ; Olausen:2013bpa . Moreover, astrophysical observations of
recent years also support the existences of BHs permeating in the magnetic
backgrounds. For example, it has been observed that the supermassive BH
Sagittarius $A^{\ast}$ in our galaxy is accompanied by the magnetar SGR
J$1745$-$2900$ Mori:2013yda ; Kennea:2013dfa ; Eatough:2013nva ;
Olausen:2013bpa , and there is a strong magnetic field around $M87^{\ast}$
EventHorizonTelescope:2021srq .

There are already some preliminary studies considering the influences of
magnetic fields on spontaneous scalarization. In Refs. Annulli:2022ivr ;
Hod:2022qir , in the framework of EsGB, the authors considered the possible
existence of tachyonic instability and spontaneous scalarization over the
Kerr-Newmann-Melvin (KNM) BHs. This kind of BHs describes a Kerr-Newmann BH
immersed in a uniform magnetic field Wald:1974np ; Ernst:1976mzr ; Ernst-
Wild:1976 ; Gibbons:2013yq , and can be taken as a simple model to draw a
qualitative picture of the interactions between BHs and the magnetic field
surrounding it. They found that the presence of the magnetic field will push
up the spin threshold for the tachyonic instability. The work is then extended
to dCSG theory in Ref. Zhang:2022sgt . By performing time evolutions of the
scalar field perturbations over the Reissner-Nordström-Melvin (RNM) BHs, the
authors found that the tachyonic instability and spontaneous scalarization can
be induced by the magnetic field. Moreover, the magnetic field will change the
waveforms and the ringdown modes dramatically. Interestingly, it was found in
Ref. Brihaye:2021jop  that even in the absence of compact objects (like BHs),
scalar field can still condensate in the Melvin magnetic universe when it is
coupled non-minimally either to the curvature or to the magnetic field.

In this work, we would like to extend the above studies to EMs theory where
the spontaneous scalarization over BHs will be triggered by the Maxwell field
instead of the curvature. We assume the RNM BHs as the background to study the
influences of magnetic fields. We will show that for different spontaneous
scalarization models, influences of magnetic fields will be rather different.
The work is organized as follows. In Sec. II, we give a brief introduction of
the EMs theory and the RNM BH, and derive the scalar field perturbation
equations. In Sec. III, we describe the numerical strategy to solve the scalar
field perturbation equation and report our numerical results in detail for
different sign of coupling constant. The final section is devoted to the
Summary and Discussion.

In this work, we use the units $c=G=4\pi\epsilon_{0}=1$, where
$c,G,\epsilon_{0}$ are the the speed of light in vacuum, the Newton
gravitational constant and the vacuum permittivity respectively.

##  II The model

The general action of Einstein-Maxwell-scalar (EMs) theory is

$\displaystyle S$ $\displaystyle=$ $\displaystyle-\int
dx^{4}\sqrt{-g}\left[R-h(\phi)F^{2}-2\partial_{\mu}\phi\partial^{\mu}\phi\right],$
(1)

where the scalar field is non-minimally coupled to the Maxwell term
$F^{2}\equiv F_{\mu\nu}F^{\mu\nu}$ through the coupling function $h(\phi)$. By
varying the action, one can derive the equations of motion

$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi=\frac{1}{4}\frac{dh}{d\phi}F^{2},$
(2) $\displaystyle\nabla(hF^{ab})=0,$ (3) $\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=hT^{EM}_{\mu\nu}+T^{\phi}_{\mu\nu},$ (4)
$\displaystyle
T^{EM}_{\mu\nu}=2F_{\mu\rho}F_{\nu}^{~{}\rho}-\frac{1}{2}g_{\mu\nu}F^{2},$
$\displaystyle T^{\phi}_{\mu\nu}=2\nabla_{\mu}\phi\nabla_{\nu}\phi-
g_{\mu\nu}\nabla_{\rho}\phi\nabla^{\rho}\phi.$

As Ref. Herdeiro:2018wub , we take the form of the coupling function as

$\displaystyle h(\phi)=e^{-\alpha\phi^{2}},$ (5)

where $\alpha$ is a coupling constant. With this choice, the theory admits the
Reissner-Nordström-Melvin (RNM) solution with vanishing scalar field $\phi=0$
Ernst:1976mzr ; Gibbons:2013yq

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
H\left[-fdt^{2}+f^{-1}dr^{2}+r^{2}d\theta^{2}\right]+H^{-1}r^{2}\sin^{2}\theta(d\varphi-\Omega
dt)^{2},$ (6) $\displaystyle A_{\mu}dx^{\mu}$ $\displaystyle=$
$\displaystyle\Phi_{0}dt+\Phi_{3}(d\varphi-\Omega dt),$ (7)

where

$\displaystyle f$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ $\displaystyle H$ $\displaystyle=$
$\displaystyle
1+\frac{1}{2}B^{2}\left(r^{2}\sin^{2}\theta+3Q^{2}\cos^{2}\theta\right)+\frac{1}{16}B^{4}\left(r^{2}\sin^{2}\theta+Q^{2}\cos^{2}\theta\right)^{2},$
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle-\frac{2QB}{r}+\frac{QB^{3}r}{2}\left(1+f\cos^{2}\theta\right),$
$\displaystyle\Phi_{0}$ $\displaystyle=$
$\displaystyle-\frac{Q}{r}+\frac{3}{4}QB^{2}r\left(1+f\cos^{2}\theta\right),$
$\displaystyle\Phi_{3}$ $\displaystyle=$
$\displaystyle\frac{2}{B}-H^{-1}\left[\frac{2}{B}+\frac{B}{2}\left(r^{2}\sin^{2}\theta+3Q^{2}\cos^{2}\theta\right)\right].$
(8)

This GR electro-vacuum solution describes a RN black hole with charge $Q$
immersed in a uniform magnetic field $B$ aligned along the symmetry axis.
Although this BH solution is not asymptotically flat but resembles a magnetic
Melvin-like universe Gibbons:2013yq ; Melvin:1963qx , we can take it as a
simple model to draw a qualitative picture of the interactions between the BH
and the magnetic field surrounding it. The event horizon is located at
$r=r_{+}=M+\sqrt{M^{2}-Q^{2}}$. On this background, we would like to study the
scalar field perturbation whose dynamics is governed by Eq. (2) evaluated on
the background, that is

$\displaystyle\nabla^{2}\phi=m^{2}_{\rm eff}\phi,$ (9)

where the scalar field acquires an effective mass whose square is $m^{2}_{\rm
eff}=-\frac{1}{2}\alpha F^{2}$ with the Maxwell term as

$F^{2}=-\frac{2(\partial_{\theta}\Phi_{0}-\Phi_{3}\partial_{\theta}\Omega)^{2}}{fH^{2}r^{2}}+\frac{2(\partial_{\theta}\Phi_{3})^{2}}{r^{4}\sin^{2}\theta}-\frac{2(\partial_{r}\Phi_{0}-\Phi_{3}\partial_{r}\Omega)^{2}}{H^{2}}+\frac{2f(\partial_{r}\Phi_{3})^{2}}{r^{2}\sin^{2}\theta}.$
(10)

Note that $m^{2}_{\rm eff}$ is position-dependent and depends on parameters
$(M,Q,B)$. Following are several properties of the Maxwell term (10) which are
useful for our discussions below:

  * •

It is even in the $\theta$-direction with respect to $\theta=\frac{\pi}{2}$

$\theta\rightarrow\pi-\theta\quad\Rightarrow\quad F^{2}\rightarrow F^{2}.$
(11)

  * •

The full expression of $F^{2}$ is rather involved. We can get a first sense of
the influence of the magnetic field on $F^{2}$ from the small-$B$ expansion,

$F^{2}=-\frac{2Q^{2}}{r^{4}}+\left(2+\frac{39Q^{4}\cos^{2}\theta}{r^{4}}-\frac{Q^{2}(13+23\cos
2\theta)+8Mr\sin^{2}\theta}{2r^{2}}\right)B^{2}+{\cal O}(B^{4}).$ (12)

  * •

In the limit $B=0$, the RNM BH reduces to the RN BH. And the Maxwell term
reduces to

$F^{2}=-\frac{2Q^{2}}{r^{4}},\qquad B\rightarrow 0,$ (13)

which is always negative for $Q\neq 0$. In this case, only for $\alpha<0$ can
$m^{2}_{\rm eff}$ become negative and the tachyonic instability be triggered.
This case has been discussed thoroughly in past few years Herdeiro:2018wub .

While in the limit $Q=0$, the RNM BH reduces to the Schwarzschild-Melvin BH
Ernst:1976mzr  and the Maxwell term becomes

$F^{2}=\frac{512B^{2}(r-2M\sin^{2}\theta)}{r(B^{2}r^{2}\sin^{2}\theta+4)^{4}},\qquad
Q\rightarrow 0$ (14)

which is always positive for $B\neq 0$. In this case, only for $\alpha>0$ can
$m^{2}_{\rm eff}$ become negative and the tachyonic instability be triggered.

Figure 1: (color online) Minimum value (left panel) and maximum value (right
panel) of the Maxwell term $F^{2}$ outside the horizon in the
$\\{Q/M,BM\\}$-phase space.

  * •

For the general case $B\neq 0$ and $Q\neq 0$, $F^{2}$ is positive somewhere
while negative elsewhere. In Fig. 1, we plot the extremal values of the
Maxwell term $F^{2}$ outside the horizon in the $\\{Q/M,BM\\}$-phase space.
From the figure, one can see that the minimum value of $F^{2}$ is always
negative while the maximum value of $F^{2}$ is always positive. This means
that, when the magnetic field is present, for either sign of $\alpha$ can
$m^{2}_{\rm eff}$ become negative somewhere thus triggering the tachyonic
instability when the coupling constant $\alpha$ is large enough. This is
different from the $B=0$ case and $Q=0$ case.

In the following section, we will study carefully the time evolution of the
scalar field perturbation by solving Eq. (9) and obtain object pictures of the
influences of the magnetic field $B$, the coupling constant $\alpha$ and also
the BH charge $Q$ on wave dynamics.

##  III Numerical strategy and results

We will apply the numerical method as Refs. Krivan:1996da ; PazosAvalos:2004rp
; Dolan:2011dx ; Doneva:2020nbb ; Doneva:2020kfv ; Schiesser ; Zhang:2022sgt
to solve the scalar field perturbation equation (9). As Ref. Zhang:2022sgt ,
we introduce the tortoise coordinate $x$ to map the radial domain
$r\in(r_{+},+\infty)$ to $x\in(-\infty,+\infty)$, and a Kerr-like azimuthal
coordinate $\tilde{\varphi}$ to remove unphysical pathology induced by the
coordinate $\varphi$ near the horizon Krivan:1996da . The two new coordinates
are defined respectively as

$dx\equiv dr/f,\quad d\tilde{\varphi}=d\varphi+\Omega dx.$ (15)

Then the scalar field perturbation equation (9) evaluated on the background
becomes

$\displaystyle
r^{2}\left(\partial_{t}^{2}-\partial_{x}^{2}\right)\phi-2fr\partial_{x}\phi+2r^{2}\Omega\left(\partial_{t}\partial_{\tilde{\varphi}}-\partial_{x}\partial_{\tilde{\varphi}}\right)\phi$
$\displaystyle-\partial_{x}(r^{2}\Omega)\partial_{\tilde{\varphi}}\phi-\frac{f}{\sin\theta}\partial_{\theta}\left(\sin\theta\partial_{\theta}\phi\right)-\frac{fH^{2}}{\sin^{2}\theta}\partial_{\tilde{\varphi}}^{2}\phi=-m_{\rm
eff}^{2}\ fHr^{2}\phi.$ (16)

By decomposing the scalar field perturbation,
$\phi(t,x,\theta,\tilde{\varphi})=\sum_{m}\Psi(t,x,\theta)e^{im\tilde{\varphi}}$,
and introducing an auxiliary variable $\Pi\equiv\partial_{t}\Psi$, the
perturbation equation can be cast eventually into a form of two coupled first-
order partial differential equations

$\displaystyle\partial_{t}\Psi=$ $\displaystyle\Pi,$
$\displaystyle\partial_{t}\Pi=$
$\displaystyle-2im\Omega\Pi+\partial_{x}^{2}\Psi+2\left(im\Omega+\frac{f}{r}\right)\partial_{x}\Psi+\frac{f}{r^{2}}\left(\partial_{\theta}^{2}+\cot\theta\partial_{\theta}\right)\Psi$
(17)
$\displaystyle+\left(\frac{im}{r^{2}}\partial_{x}(r^{2}\Omega)-\frac{m^{2}fH^{2}}{r^{2}\sin^{2}\theta}-m_{\rm
eff}^{2}\ fH\right)\Psi,$

which can be solved by the method of line Schiesser . To solve the
perturbation equation, physical boundary conditions are needed. We impose
ingoing wave condition at the horizon following Ref. Ruoff2000 , while impose
Dirichlet boundary condition at a large radial cutoff with order of $x\sim
1/B$ Brito:2014nja . Moreover, at the poles $\theta=0$ and $\pi$, physical
boundary conditions $\Psi|_{\theta=0,\pi}=0$ for $m\neq 0$ while
$\partial_{\theta}\Psi|_{\theta=0,\pi}=0$ for $m=0$ Dolan:2011dx  are imposed.
For more details, please refer to Refs. Krivan:1996da ; PazosAvalos:2004rp ;
Dolan:2011dx ; Doneva:2020nbb ; Doneva:2020kfv ; Schiesser ; Zhang:2022sgt .

As Refs. Zhang:2020pko ; Zhang:2021btn ; Zhang:2022sgt , taking into account
the “mode-mixing phenomenon” Thuestad:2017ngu ; Zenginoglu:2012us ;
Burko:2013bra  and for simplicity, we consider $m=0$ and assume the initial
scalar field perturbation as

$\displaystyle\Psi(t=0,x)$ $\displaystyle\sim$ $\displaystyle
e^{-\frac{(x-x_{c})^{2}}{2\sigma^{2}}},$ (18) $\displaystyle\Pi(t=0,x)$
$\displaystyle=$ $\displaystyle 0,$ (19)

which is a Gaussian wave-packet localized outside the horizon at $x=x_{c}$
with width $\sigma$ and has time symmetry. Also, we set $M=1$ so that all
quantities are measured in units of $M$. Without loss of generality, observers
are assumed to locate at $x=10M$ and $\theta=\frac{\pi}{5}$.

###  III.1 $\alpha<0$

Figure 2: (color online) Time evolutions of the scalar field perturbation for
$Q=0.8,B=0.1$ and various values of $\alpha$. From left to right and top to
bottom, $|\alpha|$ increases from $0$ to $10.0$.

Let us first consider the $\alpha<0$ case. Without loss of generality, we fix
$Q=0.8$ and $B=0.1$ 111Typical value of the magnetic field we considered in
this work is $BM\sim 0.1$. In our units $c=G=4\pi\epsilon_{0}=1$, we have the
following relation  $\frac{1}{M}\simeq 2.36\times
10^{19}\left(\frac{M_{\odot}}{M}\right)\textrm{Gauss},$ where $M_{\odot}$ is
the solar mass. So $BM\sim 0.1$ corresponds to a magnetic field $B\sim
2.36\times 10^{18}\left(\frac{M_{\odot}}{M}\right)\textrm{Gauss}$; For
stellar-mass BHs with $M\sim 10M_{\odot}$, $B\sim 10^{17}\textrm{Gauss}$,
while for supermassive BHs with $M\sim 10^{6}M_{\odot}$ (for example the
Sagittarius A∗), $B\sim 10^{12}\textrm{Gauss}$. In general, for BHs with mass
$M>10^{2}M_{\odot}$, the magnetic field considered will be smaller than the
ever-measured strongest magnetic field $B\sim 10^{16}\textrm{Gauss}$
Olausen:2013bpa . For smaller magnetic field, the tachyonic instability and
spontaneous scalarization can still be triggered as long as the coupling
constant $\alpha$ is large enough.. In Fig. 2, time evolutions of the scalar
field perturbations for various values of the coupling constant $\alpha$ are
shown. For the figure, one can see that there exists a threshold value of the
coupling constant $|\alpha|=|\alpha_{c}|\simeq 6.9$ above which the tachyonic
instability is triggered which is a signal of spontaneous scalarization. With
the increase of $|\alpha|$, the instability appears earlier and becomes more
violent. This can be understood as larger $|\alpha|$ will yield a more
negative $m_{\rm eff}^{2}$.

Moreover, when $|\alpha|<|\alpha_{c}|$, from the waveforms one can observe an
interesting phenomenon: two different types of ringdown modes appear at
different stages. The first ringdown modes, appearing at early times, are
being in fact similar to the quasinormal modes (QNMs) of RN BH under a massive
scalar field perturbation Galtsov:1978ag ; Konoplya:2007yy ; Konoplya:2008hj .
After time of order $t\sim\frac{1}{B}$, the “Melvin-like” modes with smaller
and smaller amplitudes are excited as a result of an effective ”wall” induced
by the magnetic field Brito:2014nja . This interesting phenomenon has already
been observed in GR Brito:2014nja  and dCSG theory Zhang:2022sgt . Similar
phenomenon was also observed in the context of ”dirty” BHs Barausse:2014tra .
When $|\alpha|>|\alpha_{c}|$, unstable tachyonic modes are excited which will
occupy the energy of the system. Then the Melvin-like modes are suppressed and
even disappear when $|\alpha|$ is large enough.

Figure 3: (color online) Time evolutions of the scalar field perturbation for
Left: $Q=0.7,\alpha=-20$ and various values of $B$ and Right:
$B=0.22,\alpha=-20$ and various values of $Q$.

Figure 4: (color online) Profiles of the effective mass square
$m_{\textrm{eff}}^{2}=-\frac{1}{2}\alpha F^{2}$ for fixed
$\theta=\frac{\pi}{10}$. Left: $Q=0.7$ while $B$ is varied. Right: $B=0.22$
while $Q$ is varied.

To study the influences of the magnetic field $B$ and BH charge $Q$ on the
instability, we fix $\alpha$ and plot the time evolutions of the scalar field
perturbations for various values of $B$ and $Q$ in Fig. 3. From the left
panel, one can see that there exists a threshold $B=B_{c}\simeq 0.22$ below
which the tachyonic instability is triggered. And the tachyonic instability
appears later and becomes more mild as $B$ is increased. This means that the
presence of the magnetic field suppresses the tachyonic instability and even
quenches it when $B$ is large enough. This can be understood from the left
panel of Fig. 4, from which one can see that larger $B$ makes the $m_{\rm
eff}^{2}$ less negative near the horizon. Moreover, for $B>B_{c}$, the Melvin-
like modes appears earlier for larger $B$, which can be understood as the
effective ”wall” becomes closer to the horizon.

From the right panel of Fig. 3, one can see that the influence of the BH
charge $Q$ on the tachyonic instability is opposite to that of $B$: larger $Q$
makes the instability appears earlier and becomes more violent. Also, this can
be understood from the right panel of Fig. 4, from which one can see that
larger $Q$ makes the $m_{\rm eff}^{2}$ more negative near the horizon.

###  III.2 $\alpha>0$

Figure 5: (color online) Time evolutions of the scalar field perturbation for
$Q=0.2,B=0.2$ and various values of $\alpha$. From left to right and top to
bottom, $\alpha$ increases from $0$ to $0.6$.

Figure 6: (color online) Time evolutions of the scalar field perturbation for
Left: $Q=0.4,\alpha=0.4$ and various values of $B$ and Right:
$B=0.2,\alpha=0.38$ and various values of $Q$.

Figure 7: (color online) Profiles of the effective mass square
$m_{\textrm{eff}}^{2}=-\frac{1}{2}\alpha F^{2}$ for fixed
$\theta=\frac{\pi}{10}$. Left: $Q=0.4$ while $B$ is varied. Right: $B=0.2$
while $Q$ is varied.

Now we turn to the $\alpha>0$ case. In Fig. 5, time evolutions of the scalar
field perturbations are shown for fixed $Q=0.2,B=0.2$ and various values of
$\alpha$. As the $\alpha<0$ case, the tachyonic instability will occur when
$\alpha$ exceeds a threshold value. The influences of $B$ and $Q$ on the wave
dynamics are shown in Fig. 6, from which one see that larger $B$ or smaller
$Q$ makes the instability appears earlier and becomes more violent. This is
different from the $\alpha<0$ case, and can also be understood qualitatively
from the behaviors of $m_{\rm eff}^{2}$. From Fig. 7, one can see that $m_{\rm
eff}^{2}$ is positive near the horizon but becomes negative away from the
horizon. Larger $B$ or smaller $Q$ makes $m_{\rm eff}^{2}$ more negative
indicating more violent instability. This also implies that the unstable
tachyonic modes are excited away from the horizon rather than near the
horizon.

##  IV Summary and Discussions

In this work, we study the scalar field perturbations of the RNM BHs in the
framework of EMs theory by performing time simulations. In this framework, the
tacyonic instability and associated spontaneous scalarization are induced by
the matter field rather than the curvature. Our focus is on the influences of
the magnetic field on the wave dynamics. In the absence of the magnetic field,
the tachyonic instability can only be triggered for negative coupling constant
$\alpha$. However, while the magnetic field is present, it can occur for
either sign of $\alpha$. Moreover, for different sign of $\alpha$, the
influences of the magnetic field on the tachyonic instability will be rather
different. For $\alpha<0$, the effective mass square $m_{\textrm{eff}}^{2}$
becomes negative near the horizon and the unstable tachyonic modes should be
excited there, and larger $B$ will make the instability appears later and more
mild or even disappears when $B$ is large enough; While for $\alpha>0$, the
effective mass square $m_{\textrm{eff}}^{2}$ is positive near the horizon but
becomes negative away from the horizon thus the unstable modes are instead
excited away from the horizon, and the influences of the magnetic field are
opposite to that in $\alpha<0$ case. Comparing to the studies in EsGB
Annulli:2022ivr  and dSCG Zhang:2022sgt , one can have the conclusion that the
influences of the magnetic field are model-dependent.

As the presence of the magnetic field changes the asymptotic structures of the
spacetime, the waveforms are shown to be greatly different from that of $B=0$.
Beyond the usual QNMs modes appearing in early times, a new type of ringdown
modes–the “Melvin-like” modes–are excited at late times. The influences of the
BH charge $Q$ on the instability are also studied in detail, and also differs
for different sign of $\alpha$.

In this work, we only perform the time simulations of the wave dynamics at
linear level. To get a complete picture of the process of spontaneous
scalarization, a full non-linear time evolution of the system including the
spacetime is called for. Also, it is interesting and important to obtain the
final scalarized BH metric and study its various properties, such as its
astrophysical implications. We would like to leave these questions for further
investigations.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NNSFC) under Grant No. 12075207.

## References

  * (1) J. D. Bekenstein, Black holes: Classical properties, thermodynamics and heuristic quantization, [arXiv:gr-qc/9808028 [gr-qc]].
  * (2) D. Robinson, Four decades of black holes uniqueness theorems, in The Kerr Spacetime: Rotating Black Holes in General Relativity, eds. D.L. Wiltshire and M. Visser and S. M. Scott (Cambridge University Press, 2009).
  * (3) P. T. Chrusciel, J. Lopes Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond, Living Rev. Rel. 15, 7 (2012). [arXiv:1205.6112 [gr-qc]].
  * (4) T. Damour and G. Esposito-Farese, Nonperturbative strong field effects in tensor - scalar theories of gravitation, Phys. Rev. Lett. 70, 2220-2223 (1993)
  * (5) D. D. Doneva and S. S. Yazadjiev, New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories, Phys. Rev. Lett. 120, no.13, 131103 (2018) [arXiv:1711.01187 [gr-qc]].
  * (6) H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling, Phys. Rev. Lett. 120, no.13, 131104 (2018) [arXiv:1711.02080 [gr-qc]].
  * (7) G. Antoniou, A. Bakopoulos and P. Kanti, Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories, Phys. Rev. Lett. 120, no.13, 131102 (2018) [arXiv:1711.03390 [hep-th]].
  * (8) P. V. Cunha, C. A. Herdeiro and E. Radu, Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor–Gauss-Bonnet Gravity, Phys. Rev. Lett. 123, no.1, 011101 (2019) [arXiv:1904.09997 [gr-qc]].
  * (9) Y. Gao, Y. Huang and D. Liu, Scalar perturbations on the background of Kerr black holes in the quadratic dynamical Chern-Simons gravity, Phys. Rev. D 99, no.4, 044020 (2019) [arXiv:1808.01433 [gr-qc]].
  * (10) Y. S. Myung and D. C. Zou, Onset of rotating scalarized black holes in Einstein-Chern-Simons-Scalar theory, Phys. Lett. B 814, 136081 (2021) [arXiv:2012.02375 [gr-qc]].
  * (11) D. D. Doneva and S. S. Yazadjiev, Spontaneously scalarized black holes in dynamical Chern-Simons gravity: dynamics and equilibrium solutions, Phys. Rev. D 103, no.8, 083007 (2021) [arXiv:2102.03940 [gr-qc]].
  * (12) S. J. Zhang, Massive scalar field perturbation on Kerr black holes in dynamical Chern–Simons gravity, Eur. Phys. J. C 81, no.5, 441 (2021) [arXiv:2102.10479 [gr-qc]].
  * (13) N. Chatzifotis, P. Dorlis, N. E. Mavromatos and E. Papantonopoulos, Scalarization of Chern-Simons-Kerr black hole solutions and wormholes, Phys. Rev. D 105, no.8, 084051 (2022) [arXiv:2202.03496 [gr-qc]].
  * (14) C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual and J. A. Font, Spontaneous Scalarization of Charged Black Holes, Phys. Rev. Lett. 121, no.10, 101102 (2018) [arXiv:1806.05190 [gr-qc]].
  * (15) D. D. Doneva, F. M. Ramazanoğlu, H. O. Silva, T. P. Sotiriou and S. S. Yazadjiev, Scalarization, [arXiv:2211.01766 [gr-qc]].
  * (16) B. P. Abbott et al. [LIGO Scientific and Virgo], Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, no.6, 061102 (2016) [arXiv:1602.03837 [gr-qc]].
  * (17) B. P. Abbott et al. [LIGO Scientific and Virgo], GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116, no.24, 241103 (2016) [arXiv:1606.04855 [gr-qc]].
  * (18) B. P. Abbott et al. [LIGO Scientific and VIRGO], GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett. 118, no.22, 221101 (2017) [erratum: Phys. Rev. Lett. 121, no.12, 129901 (2018)] [arXiv:1706.01812 [gr-qc]].
  * (19) B. P. Abbott et al. [LIGO Scientific and Virgo], GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett. 119, no.14, 141101 (2017) [arXiv:1709.09660 [gr-qc]].
  * (20) R. Abbott et al. [LIGO Scientific and Virgo], GW190521: A Binary Black Hole Merger with a Total Mass of $150M_{\odot}$, Phys. Rev. Lett. 125, no.10, 101102 (2020) [arXiv:2009.01075 [gr-qc]].
  * (21) R. Abbott et al. [LIGO Scientific and VIRGO], GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, [arXiv:2108.01045 [gr-qc]].
  * (22) R. Abuter et al. [GRAVITY], Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole, Astron. Astrophys. 636, L5 (2020) [arXiv:2004.07187 [astro-ph.GA]].
  * (23) K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett. 875, L1 (2019) [arXiv:1906.11238 [astro-ph.GA]].
  * (24) K. Akiyama et al. [Event Horizon Telescope], First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon, Astrophys. J. Lett. 910, no.1, L13 (2021) [arXiv:2105.01173 [astro-ph.HE]].
  * (25) E. Barausse, V. Cardoso and P. Pani, Can environmental effects spoil precision gravitational-wave astrophysics?, Phys. Rev. D 89, no.10, 104059 (2014) [arXiv:1404.7149 [gr-qc]].
  * (26) R. Brito, V. Cardoso and P. Pani, Superradiant instability of black holes immersed in a magnetic field, Phys. Rev. D 89, no.10, 104045 (2014) [arXiv:1405.2098 [gr-qc]].
  * (27) R. D. Blandford and R. L. Znajek, Electromagnetic extractions of energy from Kerr black holes, Mon. Not. Roy. Astron. Soc. 179, 433-456 (1977)
  * (28) R. M. Crocker, D. Jones, F. Melia, J. Ott and R. J. Protheroe, A lower limit of 50 microgauss for the magnetic field near the Galactic Centre, Nature 468, 65 (2010) [arXiv:1001.1275 [astro-ph.GA]].
  * (29) S. A. Olausen and V. M. Kaspi, The McGill Magnetar Catalog, Astrophys. J. Suppl. 212, 6 (2014) [arXiv:1309.4167 [astro-ph.HE]].
  * (30) K. Mori, E. V. Gotthelf, S. Zhang, H. An, F. K. Baganoff, N. M. Barriere, A. Beloborodov, S. E. Boggs, F. E. Christensen and W. W. Craig, et al. NuSTAR discovery of a 3.76-second transient magnetar near Sagittarius A∗, Astrophys. J. Lett. 770, L23 (2013) [arXiv:1305.1945 [astro-ph.HE]].
  * (31) J. A. Kennea, D. N. Burrows, C. Kouveliotou, D. M. Palmer, E. Gogus, Y. Kaneko, P. A. Evans, N. Degenaar, M. Reynolds and J. M. Miller, et al. Swift Discovery of a New Soft Gamma Repeater, SGR J1745-29, near Sagittarius A$\ast$, Astrophys. J. Lett. 770, L24 (2013) [arXiv:1305.2128 [astro-ph.HE]].
  * (32) R. P. Eatough, H. Falcke, R. Karuppusamy, K. J. Lee, D. J. Champion, E. F. Keane, G. Desvignes, D. H. F. M. Schnitzeler, L. G. Spitler and M. Kramer, et al. A strong magnetic field around the supermassive black hole at the centre of the Galaxy, Nature 501, 391-394 (2013) [arXiv:1308.3147 [astro-ph.GA]].
  * (33) L. Annulli, C. A. R. Herdeiro and E. Radu, Spin-induced scalarization and magnetic fields, Phys. Lett. B 832, 137227 (2022) [arXiv:2203.13267 [gr-qc]].
  * (34) S. Hod, Nonequatorial scalar rings supported by magnetized Schwarzschild-Melvin black holes, Phys. Rev. D 105, no.8, 084056 (2022) [arXiv:2204.01748 [gr-qc]].
  * (35) R. M. Wald, Black hole in a uniform magnetic field, Phys. Rev. D 10, 1680-1685 (1974)
  * (36) F. J. Ernst, Black holes in a magnetic universe, J. Math. Phys. 17, no.1, 54-56 (1976)
  * (37) F.J. Ernst and W.J. Wild, Kerr black holes in a magnetic universe, J. Math. Phys. 17, 182 (1976), https://doi.org/10.1063/1.522875.
  * (38) G. W. Gibbons, A. H. Mujtaba and C. N. Pope, Ergoregions in Magnetised Black Hole Spacetimes, Class. Quant. Grav. 30, no.12, 125008 (2013) [arXiv:1301.3927 [gr-qc]].
  * (39) S. J. Zhang, B. Wang, E. Papantonopoulos and A. Wang, Magnetic-induced Spontaneous Scalarization in Dynamcial Chern-Simons Gravity, [arXiv:2209.02268 [gr-qc]].
  * (40) Y. Brihaye, R. Capobianco and B. Hartmann, Spontaneous scalarization of self-gravitating magnetic fields, Phys. Rev. D 103, 124020 (2021) [arXiv:2103.09307 [gr-qc]].
  * (41) M. A. Melvin, Pure magnetic and electric geons, Phys. Lett. 8, 65-70 (1964)
  * (42) D. D. Doneva, L. G. Collodel, C. J. Krüger and S. S. Yazadjiev, Black hole scalarization induced by the spin: 2+1 time evolution, Phys. Rev. D 102, no.10, 104027 (2020) [arXiv:2008.07391 [gr-qc]].
  * (43) D. D. Doneva, L. G. Collodel, C. J. Krüger and S. S. Yazadjiev, Spin-induced scalarization of Kerr black holes with a massive scalar field, Eur. Phys. J. C 80, no.12, 1205 (2020) [arXiv:2009.03774 [gr-qc]].
  * (44) W.E. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations (Academic Press, New York, 1991). 41\.
  * (45) W. Krivan, P. Laguna and P. Papadopoulos, Dynamics of scalar fields in the background of rotating black holes, Phys. Rev. D 54, 4728-4734 (1996) [arXiv:gr-qc/9606003 [gr-qc]].
  * (46) E. Pazos-Avalos and C. O. Lousto, Numerical integration of the Teukolsky equation in the time domain, Phys. Rev. D 72, 084022 (2005) [arXiv:gr-qc/0409065 [gr-qc]].
  * (47) S. R. Dolan, L. Barack and B. Wardell, Self force via $m$-mode regularization and 2+1D evolution: II. Scalar-field implementation on Kerr spacetime, Phys. Rev. D 84, 084001 (2011) [arXiv:1107.0012 [gr-qc]].
  * (48) J. Ruoff, The Numerical Evolution of Neutron Star Oscillations, PhD thesis, University of Tuebingen, 2000.
  * (49) S. J. Zhang, B. Wang, A. Wang and J. F. Saavedra, Object picture of scalar field perturbation on Kerr black hole in scalar-Einstein-Gauss-Bonnet theory, Phys. Rev. D 102, no.12, 124056 (2020) [arXiv:2010.05092 [gr-qc]].
  * (50) I. Thuestad, G. Khanna and R. H. Price, Scalar Fields in Black Hole Spacetimes, Phys. Rev. D 96, no.2, 024020 (2017) [arXiv:1705.04949 [gr-qc]].
  * (51) A. Zenginoğlu, G. Khanna and L. M. Burko, Intermediate behavior of Kerr tails, Gen. Rel. Grav. 46, 1672 (2014) [arXiv:1208.5839 [gr-qc]].
  * (52) L. M. Burko and G. Khanna, Mode coupling mechanism for late-time Kerr tails, Phys. Rev. D 89, no.4, 044037 (2014) [arXiv:1312.5247 [gr-qc]].
  * (53) D. V. Galtsov and V. I. Petukhov, Black Hole in an External Magnetic Field, Zh. Eksp. Teor. Fiz. 74, 801-818 (1978)
  * (54) R. A. Konoplya and R. D. B. Fontana, Quasinormal modes of black holes immersed in a strong magnetic field, Phys. Lett. B 659, 375-379 (2008) [arXiv:0707.1156 [hep-th]].
  * (55) R. A. Konoplya, Magnetic field creates strong superradiant instability, Phys. Lett. B 666, 283-287 (2008) [arXiv:0801.0846 [hep-th]].