# High-temperature phonon-mediated superconductivity in monolayer Mg2B4C2

Sobhit Singh<EMAIL_ADDRESS>Department of Physics and Astronomy,
Rutgers University, Piscataway, New Jersey 08854, USA  Aldo H. Romero
<EMAIL_ADDRESS>Department of Physics and Astronomy, West Virginia
University, Morgantown, West Virginia, USA  José D. Mella  Departamento de
Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,
Santiago, Chile  Departamento de Física, Facultad de Ciencias, Universidad de
Chile, Santiago, Chile  Vitalie Eremeev  Instituto de Ciencias Básicas,
Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Santiago, Chile
Enrique Muñoz  Institute of Physics, Pontificia Universidad Católica de Chile,
Santiago, Chile  Anastassia N. Alexandrova  Department of Chemistry and
Biochemistry, University of California, Los Angeles, CA 90095, USA  California
NanoSystems Institute, Los Angeles, CA 90095, USA  Karin M. Rabe  Department
of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854,
USA  David Vanderbilt  Department of Physics and Astronomy, Rutgers
University, Piscataway, New Jersey 08854, USA  Francisco Muñoz
<EMAIL_ADDRESS>Departamento de Física, Facultad de Ciencias, Universidad
de Chile, Santiago, Chile  Center for the Development of Nanoscience and
Nanotechnology (CEDENNA), Santiago, Chile

###### Abstract

A new two-dimensional material – Mg2B4C2, belonging to the family of the
conventional superconductor MgB2, is theoretically predicted to exhibit
superconductivity with critical temperature $T{{}_{c}}$ estimated in the 47–48
K range (predicted using the McMillian-Allen-Dynes formula) without any tuning
of external parameters such as doping, strain, or substrate-induced effects.
The origin of such a high intrinsic $T{{}_{c}}$ is ascribed to the presence of
strong electron-phonon coupling and topological Dirac states (which are absent
in MgB2) yielding a large density of states at the Fermi level. This material
also features a nontrivial electronic band topology exhibiting Dirac points,
practically gapless Dirac nodal lines, and topological nontrivial edge states.
Consequently, it is a potential candidate for realization of topological
superconductivity in 2D. This system is obtained after replacing the
chemically active boron layers in MgB2 by chemically inactive boron-carbon
layers. Hence, the surfaces of this material are inert. Our calculations
confirm the stability of 2D Mg2B4C2. We also find that the key features of
this material remain essentially unchanged when its thickness is increased by
modestly increasing the number of inner MgB2 layers.

## Introduction

The discovery of highly crystalline two-dimensional (2D) superconductors
Uchihashi (2017); Saito  _et al._ (2017); Jiang  _et al._ (2014); Kamihara
_et al._ (2008); Brun  _et al._ (2016), such as NbSe2 monolayer Frindt (1972);
Xi  _et al._ (2015); Ugeda  _et al._ (2016); Li  _et al._ (2019), has provided
new possibilities for van der Waals (vdW) heterostructures nano-engineering of
novel insulator-superconductor interfaces Novoselov  _et al._ (2016) and 2D
Josephson junctions, without the need of an insulating layer Yabuki  _et al._
(2016). One main challenging issue in the realization of 2D superconductivity
is that most of the well-known conventional bulk superconductors either do not
superconduct or poorly superconduct when their dimensions are reduced Frindt
(1972); Xi  _et al._ (2015); Ugeda  _et al._ (2016); Xi  _et al._ (2016); Xu
and Beckman (2016); de la Barrera  _et al._ (2018); Bao  _et al._ (2018); Yan
_et al._ (2019a, 2020). Although numerous 2D phonon-mediated superconductors
have recently been predicted from first-principles calculations, the highest
predicted intrinsic $T_{c}$ stayed around 20 K Penev  _et al._ (2016); Wang
_et al._ (2018); Lei  _et al._ (2017); Dai  _et al._ (2012); Gao  _et al._
(2017); Qu  _et al._ (2019); Yan  _et al._ (2019b, 2020) (19 K for B2C
monolayer Dai  _et al._ (2012), 10.3 K for B2O monolayer Yan  _et al._ (2020),
and 19–25 K for borophenes Gao  _et al._ (2017), to name a few). Though in
some cases $T_{c}$ has been enhanced by means of the chemical doping,
intercalation, strain, and/or substrate proximity effects Rosner  _et al._
(2002a); Pogrebnyakov  _et al._ (2004); Weller  _et al._ (2005); Gauzzi  _et
al._ (2007); Savini  _et al._ (2010); Wu  _et al._ (2016); Zhang and Dong
(2017); Qu  _et al._ (2019); Bekaert  _et al._ (2019); Yan  _et al._ (2020),
it is essential to discover intrinsic 2D superconductors that exhibit
high-$T_{c}$ without any doping or tuning of external parameters (here
high-$T_{c}$ does not refer to unconventional superconductivity as in case of
cuprates or iron-based superconductors Kruchinin (2014)).

Among all the Bardeen–Cooper–Schrieffer (BCS) type conventional
superconductors, MgB2 stands out with a record $T_{c}$ of 39 K, the highest
reported $T_{c}$ at zero-pressure Nagamatsu  _et al._ (2001); Choi  _et al._
(2002a, b). Such a high-$T_{c}$ in MgB2 stems from the strong electron-phonon
(el-ph) coupling occurring primarily due to the in-plane stretching of B-B
bonds ($i.e.$, $E_{2g}$ phonon modes), which strongly couple with the self-
doped charge carriers from magnesium to boron atoms An and Pickett (2001);
Bohnen  _et al._ (2001); Choi  _et al._ (2002a, b); Pogrebnyakov  _et al._
(2004). Remarkably, only two ($E_{2g}$) out of a total of nine phonon modes
contribute strongly to the total el-ph coupling in MgB2 An and Pickett (2001);
Bohnen  _et al._ (2001); Iavarone  _et al._ (2002a); Choi  _et al._ (2002a,
b); Pickett (2002); Mazin and Antropov (2003); Xi (2008); Pickett (2008,
2006).  Once the fundamental mechanism of such a high-$T_{c}$ in bulk MgB2 was
understood, which by the way was a subject of intense research for over a
decade period An and Pickett (2001); Yildirim  _et al._ (2001); Iavarone  _et
al._ (2002a); Szabó  _et al._ (2001); Bohnen  _et al._ (2001); Choi  _et al._
(2002a, b); Pickett (2002); Rosner  _et al._ (2002b); Mazin and Antropov
(2003); Choi  _et al._ (2003); Pogrebnyakov  _et al._ (2004); Kortus (2007,
2007); Xi (2008); Pickett (2008, 2006), researchers started proposing novel
ways to augment $T_{c}$ through rational material design approach Pickett  _et
al._ (1981); An and Pickett (2001); Rosner  _et al._ (2002a); Pickett  _et
al._ (2003); Pickett (2006, 2008); Boeri  _et al._ (2021). Pickett and co-
workers proposed that one can, in principle, achieve a much higher $T_{c}$
(than 39 K) by designing a MgB2-like stable material which has a similar Fermi
surface as in MgB2, and in which more than two phonon modes couple to the
electronic states near the Fermi level, thereby, resulting in a sizable total
el-ph coupling Pickett (2006, 2008); Rosner  _et al._ (2002a). This idea has
been employed for the rational design of new bulk superconductors with a good
success rate Verma  _et al._ (2003); Rosner  _et al._ (2002c); An  _et al._
(2002); Choi  _et al._ (2009); Miao  _et al._ (2016); Bersier  _et al._
(2009); Norman (2016); Stanev  _et al._ (2018); Zhai  _et al._ (2019);
Klintenberg and Eriksson (2013); Kolmogorov  _et al._ (2010); Gou  _et al._
(2013). The high-pressure superconductivity observed at 250 K in lanthanum
hydride is one such example Liu  _et al._ (2017); Peng  _et al._ (2017); Liu
_et al._ (2018); Drozdov  _et al._ (2019).

Despite the large success with the bulk conventional superconductors, two-
dimensional intrinsic superconductors having a high-$T_{c}$ remained elusive.
Notably, various attempts have been made to realize superconductivity in the
2D analogues of bulk MgB2 Pickett  _et al._ (2003); Naito and Ueda (2004); Xi
(2009); Mazin and Balatsky (2010); Jishi  _et al._ (2011); Xu and Beckman
(2016); Bekaert  _et al._ (2017a, b, 2019). On the one hand, Xu and Beckman
proposed a quasi-2D MgB2 nanosheet with inert surfaces, which turns out to be
a semiconductor with a bandgap of 0.51 eV resulting from the quantum
confinement effects Xu and Beckman (2016). On the other hand, Bekaert et al.
reported that a considerably high-$T_{c}$ of 20 K can be realized in monolayer
MgB2 without surface passivation, $i.e.$, if only such a material with a
highly chemically reactive surface could be made Bekaert  _et al._ (2017a, b).
In a recent study, Bekaert et al. theoretically demonstrated that a MgB2
monolayer can be stabilized by adding hydrogen adatoms. Interestingly, they
find that the hydrogenation process leads to a high-$T_{c}$ of 67 K, which can
be further boosted to over 100 K by means of a biaxial strain on the
hydrogenated MgB2 monolayer Bekaert  _et al._ (2019).  While an experimental
validation of the predicted $T_{c}$ in monolayer MgB2 is still missing, the
aforementioned theoretical works markedly enhance our understanding of
superconductivity in 2D materials.

In this work, we present a novel MgB2-like 2D material – Mg2B4C2, having
charge neutral inert surfaces, which is predicted to superconduct at a
strikingly high-$T_{c}$ in the 47–48 K range (predicted using the McMillian-
Allen-Dynes theory Migdal (1958); Eliashberg (1960); Allen and Mitrović
(1983)), which is among the highest $T_{c}$ yet reported for an intrinsic 2D
material without any doping, strain or substrate-induced effects. The main
advantageous feature in 2D Mg2B4C2 is the fact that, unlike in bulk MgB2, more
than two phonon modes strongly couple to the electronic states near the Fermi
level, thus, resulting in a substantially larger el-ph coupling
($\lambda=1.40$) in monolayer Mg2B4C2 compared to the bulk MgB2
($\lambda_{bulk}=0.73$ Bohnen  _et al._ (2001), and 0.61 Choi  _et al._
(2002b)). We note that the estimated $\lambda$ in monolayer Mg2B4C2 is
comparable to the predicted $\lambda$ (=1.46) in hydrogenated MgB2 monolayer
Bekaert  _et al._ (2019). Moreover, our calculations reveal nontrivial
topological electronic features in Mg2B4C2 exhibiting Dirac cones and
practically gapless Dirac nodal lines at the Fermi level near the corner
points of the hexagonal Brillouin zone (BZ), which enhance the density of
states (DOS) at the Fermi level by almost 30% compared to that of bulk MgB2,
hence, positively contributing towards a higher $T_{c}$. Although we do not
establish any connection between the predicted superconductivity and the
nontrivial topological electronic properties in this work, we note that the
aforementioned two features are often required for realization of topological
superconductivity Kobayashi and Sato (2015), investigation of which is beyond
the scope of the present work.

## Results

Figure 1: Crystal structure. ($\bf{a}$) bulk MgB2, and ($\bf{b}$) side (as
viewed from $\vec{\bf{a}}$) and top views of monolayer Mg2B4C2 (Mg: orange, B:
green, C: black). Solid black lines mark the unit cell boundaries, and shaded
grey areas represent vacuum in the left panel of ($\bf{b}$). The region marked
by dashed black lines in ($\bf{b}$) can be arbitrarily repeated (see text).

Material design strategy. We start by describing our rationale for design of a
stable MgB2-like 2D superconductor having inert surfaces. Generally, layered
vdW materials can be exfoliated to produce their 2D analogues Geim and
Grigorieva (2013). Although bulk MgB2 has a layered structure, it is not a vdW
material. Bulk MgB2 crystallizes in space group $P_{6}/mmm$ (#191) containing
alternating layers of Mg and B atoms stacked along the $\vec{c}$ lattice
direction, as shown in Fig. 1(a) Xi (2008). The bonding between the Mg and B
atoms is purely ionic, which means that Mg atoms donate two electrons to B
atoms, thereby making each Mg $2+$ and each B $1-$. Since a B- is
isoelectronic to a charge neutral carbon atom, a B-B sheet is structurally
analogous to a single layer graphene, but it has a different ordering of bands
than those of graphene. A simple exfoliation of MgB2 into a 2D slab with B (or
Mg) termination would yield a highly reactive electron-rich (or hole-rich)
surface layer that is chemically unstable.

We propose that one can passivate the charged surface layers in the MgB2 slab
by systematically substituting one boron by one carbon atom at the top and
bottom surfaces of the slab. Fig. 1(b) shows the top and side views of a
Mg2B4C2 monolayer designed using the aforementioned strategy.  Strikingly, we
find that modestly repeating the intermediate Mg-B layers, $i.e.$, the layers
sandwiched between the top and bottom surfaces (highlighted using dashed
rectangle in Fig. 1(b)), thereby making thicker slabs of (MgB2)nC2 while
remaining in the quasi-2D limit, $n$ being the total number of Mg layers,
retains the key features of the Mg2B4C2 monolayer. The electronic
bandstructures calculated up to $n=5$ are shown in the Supplementary Material
(SM) SM . This feature could be particularly useful in the experimental
realization of 2D superconductivity in Mg2B4C2. We note that MgB2 monolayer
can also be passivated by an appropriate hydrogenation process Bekaert  _et
al._ (2019).

Mg2B4C2 monolayer, shown in Fig. 1(b), belongs to the layer group $p\bar{3}m1$
(#72) having DFT (PBE) optimized lattice parameters $a=b=$ 2.87 Å. The
absolute thickness between the top and bottom atomic layers is 7.14 Å,
whereas, the interlayer spacing between the adjacent Mg and B-B, and Mg and
B-C (C-B) layers is $\sim$1.8 Å, and $\sim$1.7 Å, respectively. We note that
the inversion symmetry is preserved due to the inverted ordering of the top
and bottom layers in the structure shown in Fig. 1(b). However, one could
break the inversion symmetry by replicating the top and bottom layers, $i.e.,$
by making the top and bottom layers alike, either both as B-C or both as C-B.
Our calculations suggest that the structure with inversion symmetry is
energetically more favorable (5 meV/f.u.) than the structure with broken
inversion symmetry; although both structures are dynamically, elastically, and
mechanically stable since they exhibit all positive phonon frequencies,
positive elastic constants, and satisfy the Born-Huang mechanical stability
criteria (see SM SM ). The only qualitative difference in the electronic
properties of the structure with broken inversion symmetry is a small lifting
of some band degeneracies at the $K$ high symmetry point (see SM SM ). This
effect is analogous to the application of a perpendicular electric field to a
bilayer graphene Li  _et al._ (2012).

In this article, hereafter, we focus only on ground state structure of a
monolayer Mg2B4C2 with preserved inversion symmetry. We note, all other
possible atomic configurations of this composition are higher in energy (see
SM SM ). Furthermore, our exfoliation energy calculations (see Table S2 SM )
suggest that the reported monolayer Mg2B4C2 belongs to the “easily exfoliable”
category, as classified by Mounet et al. Mounet  _et al._ (2018).

Figure 2: Atomic orbitals projected electronic band structure of ($\bf{a}$)
bulk MgB2, and ($\bf{b}$) monolayer Mg2B4C2 calculated without spin-orbit
coupling (SOC) along the high symmetry direction of BZ. Cyan, red, and blue
colors represent the contribution from the s, $p_{x,y}$, and $p_{z}$ orbitals,
respectively. ($\bf{c}$) Calculated Fermi surface of monolayer Mg2B4C2. Light
pink/green, and grey colors depict hole/electron, and intertwined electron-
hole pockets, respectively. ($\bf{d}$) Energy bandgap ($E_{gap}$) plotted in
color scale (eV units) in the vicinity of a K high-symmetry point. The dashed
circle marks the $k$-loop along which Berry phase was computed. ($\bf{e}$) The
local electronic density of states of the (100) and (010) edge states
spectrum. Red/White color denotes the states near the edge/interior of the 2D
system. Topological nontrivial edge states are marked using arrows.

Topological electronic properties of Mg2B4C2 monolayer.  After describing the
crystal structure and its stability, we now focus on the topological
electronic properties of Mg2B4C2 monolayer. We begin by summarizing the key
features of the electronic structure of bulk MgB2 Xi (2008) from which Mg2B4C2
monolayer is derived. As shown in Fig. 2(a), the Fermi surface of MgB2 is
composed of boron $p$ orbitals, where $p_{x,y}$ orbitals hybridize with $s$
orbitals to form strong covalent in-plane $\sigma$ bonds at the zone center,
while the unhybridized $p_{z}$ orbitals form relatively weak out-of-plane
$\pi$ bonds at zone boundaries (Mg acts as electron donor). Due to such a
distinct Fermi-surface geometry, two superconducting gaps exists in bulk MgB2:
(i) the stronger $\sigma$ gap of $\sim$7 meV, and (ii) the weaker $\pi$ gap of
$\sim$2-3 meV Szabó  _et al._ (2001); Choi  _et al._ (2002a, b); Iavarone  _et
al._ (2002a, b); Margine and Giustino (2013); Mou  _et al._ (2015); Aperis
_et al._ (2015); Kortus  _et al._ (2001). Different symmetries of the $\sigma$
and $\pi$ bonds largely suppress the impurity scattering in MgB2 Iavarone  _et
al._ (2002a); Xi (2008); Mazin and Antropov (2003); Kortus  _et al._ (2001).

Since the basic structure and charge neutrality of MgB2 is preserved in
monolayer Mg2B4C2, the electronic band structure of monolayer Mg2B4C2
qualitatively resembles with that of the bulk MgB2, as shown in Fig. 2(a,b),
but with some additional features. For instance, there is a new set of
degenerate $\sigma$ bands ($\sigma_{outer}$) present at $\Gamma$ below the
Fermi level arising from the $p_{x,y}$ orbitals of the outer boron-carbon
layers. The other set of degenerate $\sigma$ bands ($\sigma_{inner}$) at
$\Gamma$ that cross the Fermi level (also present in MgB2) are formed by the
$p_{x,y}$ orbitals of the inner boron-boron layer. These two sets of $\sigma$
bands are almost parallel and split by $\sim 1.6$ eV at $\Gamma$. Since the
$\sigma_{outer}$ bands are completely occupied, they should, in principle,
have no contribution in superconductivity, unless there is a large external
field applied in a FET-like geometry Ye  _et al._ (2012).

In addition to a new set of $\sigma$ bands at $\Gamma$, we notice the presence
of Dirac-like band crossings at the K point, as well as along the high-
symmetry directions near the K point of monolayer Mg2B4C2. Regardless of their
topological nature, these band crossings at the Fermi level, highlighted using
a dashed magenta box in Fig. 2(b), yield a large DOS at the Fermi level
(almost 30% larger than in bulk MgB2), which contributes substantially to the
total el-ph coupling in the studied monolayer. We note that the Dirac-like
crossing at K is also present in bulk MgB2, but it is situated well-above the
Fermi level Jin  _et al._ (2019).  The Dirac-like band crossings in Mg2B4C2
monolayer are formed by highly dispersing $p_{z}$ orbitals of carbon and boron
atoms (see SM SM  for details). Thus, the Fermi surface of Mg2B4C2 monolayer,
shown in Fig. 2(c), embodies three main features: (i) two hole pockets at
$\Gamma$ (one circular and another that takes the shape of the BZ) composed of
$\sigma$ bonded boron $p_{x,y}$ orbitals, (ii) an electron pocket at M formed
by boron $p_{z}$ orbitals, and (iii) intertwined electron and hole pockets at
the K point and along K–M high-symmetry line, formed by $\pi$ bonded carbon
and boron $p_{z}$ orbitals. We note that all these pockets show very strong
coupling to the phonon modes, and, as a result, they play the key role in
governing superconductivity in Mg2B4C2 monolayer, as we discuss later.
Furthermore, the sharp and well-defined (almost flat) boundaries of the
charge-carrier pockets at the Fermi surface set up the stage for the possible
realization of Kohn-like divergencies Kohn (1959), and charge-density wave
ordering Grüner (1988); Zhu  _et al._ (2015) in this 2D system, which is
beyond the scope of present work and calls for a more comprehensive attention
in the future.

By plotting the energy bandgap (Egap) distribution in the vicinity of the K
points, we discover presence of a triangular nodal line in the vicinity of
each K point, as shown in Fig. 2(d). However, this is not a truly gapless
nodal line since a small Egap ($\sim$5 meV) exists due to the subtle breaking
of $M_{z}$ mirror symmetry. It is worth noting that the Dirac point at K is
protected by the $C_{3v}$ rotation, inversion, and time-reversal symmetries;
a small gap opens at Dirac points when the inversion symmetry is broken by
making the top and bottom B-C layers identical SM ; Gibson  _et al._ (2015).
Although there are theoretical proposals suggesting the possibility of
topological superconductivity in Dirac semimetals Kobayashi and Sato (2015),
we think that the so-far studied models are quite simple, and this topic
requires a more thorough examination before any exotic effects can be
confidently claimed here.

In order to prove the nontrivial topological nature of Dirac points, we
compute the Berry phase along a $k$-loop enclosing the gapless point at K, as
marked using dashed lines in Fig. 2(d). Our calculations yield a nontrivial
Berry phase of $\pi$ for Dirac points at K. We note, this exercise could not
be performed for the Dirac nodal line near K because enclosing the nodal line
residing in the $k_{x}$-$k_{y}$ plane would required a $k$-loop encircling
along $k_{z}$ and $k_{z}$ is not defined for a 2D system. Nevertheless, the
presence of time-reversal and spatial-inversion symmetries of Mg2B4C2
monolayer enables us to determine the $Z_{2}$ topological invariants using the
Fu-Kane criterion Fu and Kane (2007).  The inversion parity eigenvalues of the
electronic wavefunction of all 12 occupied bands at four time-reversal
invariant momenta (TRIM) points are given in Table 1. The product of all
parity eigenvalues ($\delta$) at each TRIM is also listed in Table 1. We find
that the $Z_{2}$ topological index is nontrivial due to $\delta$ = -1 at three
TRIM points. Here, we note that bulk MgB2 has a weak $Z_{2}$ topological index
(0; 001) due to the band-inversions occurring at the $\Gamma$ and $A$ (0, 0,
0.5) high-symmetry points of 3D hexagonal BZ Jin  _et al._ (2019). Robust
topological surface states have recently been experimentally observed in bulk
MgB2 Zhou  _et al._ (2019).

Table 1: Parity eigenvalues of all occupied bands and their products at four
TRIM points
TRIM | Parity eigenvalues | $\delta$
---|---|---
$\Gamma$ (0, 0, 0) | $+-+-+-++--++$ | $-1$
$M_{1}$ (0.5, 0.0, 0.0) | $-+-+-++-+---$ | $-1$
$M_{2}$ (0.0, 0.5, 0.0) | $-+-+-++-+---$ | $-1$
$M_{3}$ (0.5, 0.5, 0.0) | $-+-+-++-+--+$ | $+1$

Since the nontrivial topology in 2D systems is often manifested in the gapless
1D edge states, we further confirm the nontrivial topological features of
monolayer Mg2B4C2 by computing the local density of states at (100) and (010)
edges of 60 unit cell thick nano-ribbons. Topologically nontrivial 1D edge
states connecting band-crossing points were obtained at both (100) and (010)
edges, as shown in Fig. 2(e), thus, proving the nontrivial topology of the
Mg2B4C2 monolayer.

Figure 3:  ($\bf{a}$) Calculated phonon spectrum of Mg2B4C2 monolayer with
phonon linewidths $\lambda(\mathbf{q},n)$ plotted using shaded blue color. To
avoid large overlap of $\lambda(\mathbf{q},n)$ with the phonon spectra, we
have divided the intensity by a factor of two. The colored circles mark the
three out-of-plane nondegenerate $A_{1g}$ modes (indices 14, 18, and 19), and
the yellow diamond marks one in-plane doubly degenerate $E_{g}$ mode at
$\Gamma$. These modes exhibit dominant el-ph coupling. The atomic displacement
patterns corresponding to these modes are shown in ($\bf{c-f}$). The
nondegenerate $A_{1u}$ mode (index 15) marked using symbol ‘$\times$’ does not
contribute to the total el-ph coupling, although it appears to be buried in
the large $\lambda(\mathbf{q},n)$ of the $E_{g}$ mode. The numerals 14, 15,
16, 17, 18, and 19 denote the phonon mode index as counted from the lowest to
the highest frequency modes ($i.e.,$ 1-3 for acoustic modes). Mg atoms are
omitted in ($\bf{c}$) for the sake of clarity. The Eliashberg spectral
function $\alpha^{2}F(\omega)$ along with the el-ph coupling constant
$\lambda$ in plotted in the right panel of ($\bf{a}$). ($\bf{b}$) Estimated
$T_{c}$ as a function of the $\mu^{*}$ parameter.

Electron-phonon coupling and superconductivity in Mg2B4C2.  We find that the
roots of superconductivity in Mg2B4C2 monolayer are same as in bulk MgB2 An
and Pickett (2001); Bohnen  _et al._ (2001); Iavarone  _et al._ (2002a); Choi
_et al._ (2002a, b); Pickett (2002); Mazin and Antropov (2003); Xi (2008);
Pickett (2008, 2006). However, the main advantageous factor in Mg2B4C2 is
that, in addition to the doubly degenerate $E_{2g}$ modes that govern
superconductivity in MgB2, numerous other phonon modes strongly couple to the
electronic states near the Fermi level yielding a much larger overall el-ph
coupling, and thus, resulting in a considerably higher $T_{c}$.

The calculated phonon spectrum of Mg2B4C2 monolayer, shown in Fig. 3(a),
contains a total of 24 phonon modes (8 atoms/cell) having the following mode
symmetry at $\Gamma$:

$\displaystyle\Gamma\textsubscript{acoustic}=\emph{A}\textsubscript{2u}\oplus\emph{E}\textsubscript{u},~{}\text{and}$
(1)
$\displaystyle\Gamma\textsubscript{optic}=4\emph{A}\textsubscript{1g}\oplus
3\emph{A}\textsubscript{2u}\oplus 3\emph{E}\textsubscript{u}\oplus
4\emph{E}\textsubscript{g}.$

Here, A1g and Eg are Raman-active modes, whereas, A2u and Eu are infrared-
active modes. In Fig. 3(c-f), we show the atomic vibration patterns for the
four phonon modes, namely, three nondegenerate A1g modes (index 14, 18, and
19) and one degenerate Eg mode (indices 16-17), which exhibit the dominant el-
ph coupling. All these A1g modes correspond to the out-of-plane vibrations of
the Mg, inner B-B, and outer B-C layers, while the Eg mode corresponds to the
in-plane stretching of the inner B-B layer.  The A1g modes primarily modulate
the el-ph coupling associated with the $\pi$ bonded $p{{}_{z}}$ orbitals
contributing to the electron and hole pockets located at the BZ boundaries.
Whereas, the doubly degenerate Eg mode couples with the $\sigma$ bonded
$p_{x,y}$ orbitals forming the hole pockets located at $\Gamma$. Here, it is
worth noting that the higher frequency Eg modes (indices 21-22) that
correspond to the in-plane stretching of the outer B-C layers do not make a
significant contribution to the overall el-ph in this system, which is as
expected since these modes modulate the occupied $\sigma_{outer}$ bands
located well-below the Fermi level at $\Gamma$ [see Fig. 2(b)]. However, these
modes may participate in the superconductivity when the system is doped with
$p$-type charge carriers Bekaert  _et al._ (2019).

Since the electronic and vibrational band structures of inner B-B and outer
B-C layers are essentially independent of each other, we predicate that the
reported properties of the studied Mg2B4C2 monolayer would be retained even
when the number of the inner B-B layers are repeated (until a critical
thickness), thus making the system thicker. This feature might greatly
simplify the eventual realization of superconductivity in Mg2B4C2.

To quantify the superconducting properties of Mg2B4C2 monolayer, we employ the
McMillian-Allen-Dynes theory derived from the isotropic Migdal-Eliashberg
formalism Migdal (1958); Eliashberg (1960); Allen and Mitrović (1983) which
relies on the calculation of the el-ph coupling matrix elements within DFT.
The calculated matrix elements correspond to the transition probabilities of
different Kohn-Sham states induced by a change in the potential due to a small
ionic displacement. Thus, these matrix elements provide the main ingredients
to calculate the el-ph coupling strength and the Eliashberg spectral function
$\alpha^{2}F(\omega)$ as a function of the phonon frequency $\omega$. Since
the physical process behind the phonon-mediated superconductivity is the
exchange of a phonon between two electrons, a strong el-ph coupling is desired
to achieve a high-$T_{c}$ in a BCS superconductor. Theoretical details of such
calculations are explained in numerous other papers Giustino (2017); Giustino
_et al._ (2007); Margine and Giustino (2013).

In Fig. 3(a), we plot the calculated phonon linewidth $\lambda({\bf{q}},n)$
for each phonon mode $n$ at each wave vector $\bf{q}$ using blue color. Note
that the plotted phonon linewidth is scaled down by a factor of two to avoid
large overlap with the neighboring phonon branches. The largest contribution
to the total el-ph coupling strength comes from three nondegenerate A1g modes
and one doubly degenerate Eg mode, as marked in Fig. 3(a). We note that the
A2u mode (index 15), marked using ‘$\times$ in Fig. 3(a), does not contribute
to the total el-ph coupling, although it appears buried in the large
$\lambda(\mathbf{q},n)$ overlap from the $E_{g}$ mode.  Notably, in addition
to the aforementioned A1g and Eg phonon modes, various other modes make
relatively smaller contributions to the overall el-ph coupling strength, as
revealed by the Eliashberg spectral function $\alpha^{2}F(\omega)$ plot shown
in the right panel of Fig. 3(a).

In addition to the el-ph coupling, the net phonon linewidth
$\lambda({\bf{q}},n)$ can have some contribution from the phonon-phonon (ph-
ph) interactions owing to the phonon anharmonicity Choi  _et al._ (2002b).
Therefore, we thoroughly investigate ph-ph interactions by computing ph-ph
linewidth using the ab-initio molecular dynamics simulations. In this
approach, we mapped the forces, obtained from the finite-temperature molecular
dynamics simulations, evaluated in a 3 $\times$ 3 $\times$ 1 supercell onto a
model Hamiltonian describing the lattice dynamics. This temperature dependent
effective potential (TDEP) technique Hellman and Abrikosov (2013); Hellman
_et al._ (2013) enabled us to calculate the third-order response from the
effective renormalized interatomic force constants. Our calculations revealed
that the ph-ph linewidths are an order of magnitude smaller than the el-ph
linewidths. The maximum value of obtained ph-ph linewidth is $\sim$2 meV,
which is much smaller compared to the el-ph linewidth values that are
typically larger than $\sim$70 meV in the studied system. This result implies
that, although the system inherits some anharmonic effects, we can safely
discard the ph-ph contributions in the study of its superconducting
properties.

Based on the BCS theory of superconductivity and above results, we estimate
the critical temperature $T_{c}$ using the McMillian-Allen-Dynes formula
Bardeen  _et al._ (1957); McMillan (1968); Allen and Dynes (1975):

$\displaystyle
T_{c}=\frac{\omega_{log}}{1.2}exp\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right],$
(2)

where $\omega_{log}$ is the logarithmic averaged phonon frequency, $\lambda$
is the total el-ph coupling constant, and $\mu^{*}$ is the effective screened
Coulomb repulsion constant with a typical value ranging from 0.04–0.16 (see
Table 2) Bohnen  _et al._ (2001); Choi  _et al._ (2002a, b, 2009); Dai  _et
al._ (2012).  We obtain $\lambda$ by integrating the cumulative frequency-
dependent el-ph coupling $\lambda(\omega)$ given by the following expression:

$\displaystyle\lambda(\omega)=2\int_{0}^{\omega}\frac{\alpha^{2}F(\omega)}{\omega}d\omega$
(3)

We find a fairly large value of $\lambda=1.40$, which is considerably larger
than the one reported for bulk MgB2 ($\lambda_{bulk}=0.73$ Bohnen  _et al._
(2001), and 0.61 Choi  _et al._ (2002b)). We observe that the estimated
$T_{c}$ does not vary drastically as a function of $\mu^{*}$, as shown in Fig.
3(b). This is consistent with an earlier work by Choi et al. Choi  _et al._
(2002b), which reported that the superconducting properties of MgB2 are not
very sensitive to the $\mu^{*}$ parameter within the isotropic McMillian-
Allen-Dynes formalism.  We note that for bulk MgB2, $\mu^{*}$ = 0.05 has been
used to get the correct estimate of $T_{c}$ $\sim$ 40 K Bohnen  _et al._
(2001). Therefore, using the McMillian-Allen-Dynes formula Bardeen  _et al._
(1957); McMillan (1968); Allen and Dynes (1975), we estimate the $T_{c}$ of
Mg2B4C2 monolayer to be in the range 47–48 K without any doping or strain. Our
results are consistent with a recent study Bekaert  _et al._ (2019) in which
$T_{c}=67\,K$ and $\lambda=1.46$ was predicted in a hydrogenated MgB2
monolayer by solving the fully anisotropic Eliasberg equations. We argue that
the predicted $T_{c}$ in Mg2B4C2 monolayer can be further enhanced by biaxial
strain Bekaert  _et al._ (2019); Yan  _et al._ (2020) or by p-doping Bekaert
_et al._ (2019).  In passing, we would like to mention that the predicted
$T_{c}$ could moderately vary if a fully anisotropic Migdal-Eliashberg theory
Choi  _et al._ (2002b); Margine and Giustino (2013); Zheng and Margine (2016);
Bekaert  _et al._ (2019) or SC-DFT Lüders  _et al._ (2005); Marques  _et al._
(2005); Sanna  _et al._ (2007, 2020) is employed. This is particularly
important here because the applicability of the McMillian-Allen-Dynes formula
becomes limited in the case of large el-ph coupling.

Table 2: Listing of superconducting parameters required for the prediction of
$T_{c}$ using the McMillian-Allen-Dynes formula for some reported 2D phonon-
mediated superconductors (data for bulk MgB2 is included for comparison). This
table includes data of effective Coulomb screening parameter $\mu^{*}$,
electronic DOS at the Fermi level $N$(EF) (in states/spin/Ry/cell),
logarithmic averaged phonon frequency $\omega_{log}$ (in K), total electron-
phonon coupling constant $\lambda$, and estimated $T_{c}$ (in K). Experimental
$T_{c}$ values are noted in the table.
Compounds | $\mu^{*}$ |  $N$(EF) | $\omega_{log}$ | $\lambda$ | $T_{c}$ | Ref.
---|---|---|---|---|---|---
B2C | 0.10 |  | 315 | 0.92 | 19 | Dai  _et al._ (2012)
CaC6 | 0.115 |  | 446 | 0.40 | 1.4 | Profeta  _et al._ (2012)
LiC6 | 0.115 |  | 400 | 0.61 | 8.1 | Profeta  _et al._ (2012)
LiC6 |  |  |  | 0.58 $\pm$ 0.05 | 5.9 [Exp.] | Ludbrook et al. (2015)
LiC6 | 0.12/0.14/0.16 |  |  | 0.55 | 7.6/5.9/5.1 | Zheng and Margine (2016)
2H-NbSe2 |  |  |  | 0.75 | 3.1 [Exp.] | Xi  _et al._ (2015)
2H-NbSe2 | 0.15, 0.16 |  | 134, 145 | 0.84, 0.67 | 4.5, 2.7 | Lian  _et al._ (2018); Zheng and Feng (2019)
C6CaC6 |  |  |  |  | 4.0 [Exp.] | Ichinokura  _et al._ (2016)
C6CaC6 | 0.207/0.155 |  |  |  | 6.8/8.1 | Margine  _et al._ (2016); Mazin and Balatsky (2010); Jishi  _et al._ (2011)
B2O | 0.10 | 5.4 | 250 | 0.75 | 10.3 | Yan  _et al._ (2020)
LiBC | 0.13 | 10.9 |  | 0.59 | 65 | Modak  _et al._ (2021)
bulk MgB2 | 0.05 | 9.8 | 707 | 0.73 | 40 | Bohnen  _et al._ (2001)
bulk MgB2 | 0.13 | 9.8 |  | 0.61 | 39 | Bekaert  _et al._ (2017a)
monolayer MgB2 | 0.13 | 13.1 |  | 0.68 | 20 | Bekaert  _et al._ (2017a)
monolayer H-MgB2 | 0.13 | 19.2 |  | 1.46 | 67 | Bekaert  _et al._ (2019)
Mg2B4C2 | 0.04 | 12.6 | 506 | 1.40 | 48.1 | Our work
0.10 | 47.2 | Our work
0.14 | 47.0 | Our work

In order to highlight the novelty of our results, in Table 2 we list the
theoretical superconducting parameters along with the estimated $T_{c}$ for
some reported 2D phonon-mediated superconductors. The good agreement between
the experimental data for LiC6 Profeta  _et al._ (2012); Ludbrook et al.
(2015); Zheng and Margine (2016), $2H$-NbSe2 Xi  _et al._ (2015); Heil  _et
al._ (2017); Lian  _et al._ (2018); Zheng and Feng (2019), and C6CaC6
Ichinokura  _et al._ (2016); Margine  _et al._ (2016) and the corresponding
theoretical results obtained from the McMillian-Allen-Dynes theory further
boosts our confidence in the predictive power of the employed theory.

## Summary

In summary, we present a 2D material Mg2B4C2, similar to MgB2, but with inert
surfaces obtained by the replacement of outer B-B layers by B-C layers. Our
calculations suggest that this structure is dynamically, elastically and
mechanically stable. It also features a nontrivial topological electronic band
structure together with a large el-ph coupling ($\lambda=1.40$), which is more
than twice as large as that of the bulk MgB2 and comparable to that of in a
hydrogenated monolayer MgB2 Bekaert  _et al._ (2019).  Use of the standard
McMillian-Allen-Dynes theory predicts the superconducting transition
temperature $T_{c}$ to be in the range of 47–48 K without any doping or tuning
of external parameters such as strain. To the best of our knowledge, this is
among the highest predicted intrinsic $T_{c}$ in a conventional BCS-type 2D
superconductor to date. The studied material offers the two expected
ingredients: (i) topological nontrivial electronic properties, and (ii) large
intrinsic $T_{c}$, for practical realization of nontrivial topological
superconductivity in 2D Qi and Zhang (2011). In addition to the large el-ph
coupling, the presence of sharp and well-defined flat boundaries of the
charge-carrier pockets at the Fermi surface imply the possible realization of
Kohn-like divergencies and charge-density wave ordering in this 2D system,
which calls for a dedicated study in future.

## Methods

The electronic bands structure and phonon calculations were performed using
density-functional theory (DFT) as implemented in the VASP package Kresse and
Hafner (1993, 1994); Kresse and Furthmüller (1996a, b). The phonopy Togo and
Tanaka (2015) and PyProcar Herath  _et al._ (2020) tools were used for the
post-processing of data. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation
functional Perdew  _et al._ (1996) and PAW pseudo-potentials Blöchl (1994);
Kresse and Joubert (1999) were used. The employed $k$-point grid for self-
consistent calculations was $30\times 30\times 1$, and the cutoff for the
kinetic energy of plane waves was set to 700 eV. A vacuum of thickness
$\sim$30 Å was added to avoid the periodic interactions along the $c$-axis.
Since the spin-orbit coupling (SOC) effects were found to be negligible in the
studied system, SOC was not included in the reported calculations. The elastic
and mechanical properties were analyzed using the MechElastic code Singh  _et
al._ (2018, 2021). The exfoliation energy was calculated using four different
exchange-correlation approximations: the (PBE) GGA approximation Perdew  _et
al._ (1996), the SCAN Sun  _et al._ (2015) meta-GGA, vdW-DF2 GGA functional
Lee  _et al._ (2010), and SCAN together with the rVV10 correlation functional
(SCAN+rVV10) Peng  _et al._ (2016). The topological properties of Mg2B4C2 were
studied by fitting the DFT calculated bandstructure to a real space tight-
binding Hamiltonian obtained using the maximally localized Wannier functions
(MLWFs) approach Mostofi  _et al._ (2014); Wu  _et al._ (2018). The local
density of states at (100) and (010) edges were calculated for 60 unit cells
thick nano-ribbons using the WannierTools package Wu  _et al._ (2018) with
vacuum added along the $c$-axis of the ribbon.

For the electron-phonon coupling matrix elements calculations, we used the
abinit package Gonze  _et al._ (2002); Gonze (2005); Gonze  _et al._ (2009,
2016); Romero  _et al._ (2020). We employed norm conserving pseudopotentials
(using the ONCVPSP scheme of Hamann Hamann (2013)), and a plane wave basis set
up to kinetic energies of 35 Ha. Cell parameters were optimized by using the
PBE exchange-correlation functional as in VASP calculations. We used a uniform
grid of $18\times 18\times 1$ for the ground state calculations, and a phonon
grid of $9\times 9\times 1$ for the phonon part. A total of 288 el-ph matrix
elements were calculated. Calculations of the phonon interatomic force
constants, and the el-ph coupling matrix elements performed in this work used
the second-order perturbation theory Baroni  _et al._ (2001); Gonze (1995).
The temperature dependent effective potential (TDEP) technique Hellman and
Abrikosov (2013); Hellman  _et al._ (2013) was used to study the phonon-phonon
interactions and phonon anharmonic effects.

## Acknowledgements

This work was supported in part by Fondecyt Grants No. 1191353 (F.M.), 3200697
(J.D.M.), 1180175 (V.E.), 1190361(E.M.), Center for the Development of
Nanoscience and Nanotechnology CEDENNA AFB180001, and from Conicyt PIA/Anillo
ACT192023. S.S., K.R., and D.V. acknowledge the support from ONR Grants
N00014-19-1-2073 and N00014-16-1-2951. We also thank the NSF CAREER Award
CHE-1351968 to A.N.A and DOE DE- SC0021375 project to A.H.R. This research was
partially supported by the supercomputing infrastructure of the NLHPC (ECM-02)
and XSEDE which is supported by National Science Foundation grant number
ACI-1053575. The authors also acknowledge the support from the Texas Advances
Computer Center (with the Stampede2 and Bridges supercomputers).

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Mixed vs Stable Anti-Yetter–Drinfeld Contramodules

Mixed vs Stable Anti-Yetter–Drinfeld Contramodules

Ilya SHAPIRO

I. Shapiro

Department of Mathematics and Statistics, University of Windsor,
401 Sunset Avenue, Windsor, Ontario N9B 3P4, Canada<EMAIL_ADDRESS>http://http://web2.uwindsor.ca/math/ishapiro/

Received November 09, 2020, in final form March 04, 2021; Published online
March 17, 2021

We examine the cyclic homology of the monoidal category of modules over a
finite dimensional Hopf algebra, motivated by the need to demonstrate that
there is a difference between the recently introduced mixed anti-
Yetter–Drinfeld contramodules and the usual stable anti-Yetter–Drinfeld
contramodules. Namely, we show that Sweedler’s Hopf algebra provides an
example where mixed complexes in the category of stable anti-Yetter–Drinfeld
contramodules (previously studied) are not equivalent, as differential graded
categories to the category of mixed anti-Yetter–Drinfeld contramodules
(recently introduced).

Hopf algebras; homological algebra; Taft algebras

16E35; 16T05; 18G90; 19D55

##  1 Introduction

Cyclic (co)homology was introduced independently by Boris Tsygan and Alain
Connes in the 1980s. It has since been generalized, applied to many fields,
and now reaches into many different settings. Our investigations in this paper
focus on the equivariant flavour that began with Connes–Moscovici [6] and was
generalized into Hopf-cyclic cohomology by
Hajac–Khalkhali–Rangipour–Sommerhäuser [8, 9] and Jara–Stefan [11]
(independently). Roughly speaking, the original theory defines cohomology
groups for an associative algebra that play the role of the de Rham cohomology
in the noncommutative setting. The equivariant version considers an algebra
with an action of a Hopf algebra. It turns out that just as in the de Rham
cohomology, one has coefficients in the Hopf setting; it is an interesting
fact that, unlike the de Rham setting, Hopf-cyclic cohomology requires
coefficients, i.e., there are no canonical trivial coefficients. These
coefficients are known as stable anti-Yetter–Drinfeld modules, due to their
similarity to the usual Yetter–Drinfeld modules. It turns out that the more
natural, from a conceptual point of view, version of coefficients are stable
anti-Yetter–Drinfeld contramodules [5]. It is the desire to understand the
coefficients themselves that motivated a series of papers by the author of the
present one. This paper is a natural next step.

This paper is a descendant of [20], where it is shown that the classic stable
anti-Yetter–Drinfeld contramodules are simply objects in the naive cyclic
homology category of ${{}_{H}\mathcal{M}}$, the monoidal category of modules
over the Hopf algebra $H$. It is furthermore conjectured there, that the new
coefficients introduced (mixed anti-Yetter–Drinfeld contramodules) are
obtained via the true cyclic homology category; this makes exact the analogy
between the de Rham coefficients in the geometric setting and the Hopf-cyclic
coefficients. Namely, while the latter are obtained from the cyclic homology
of ${{}_{H}\mathcal{M}}$, the former are shown in [4] to arise from the cyclic
homology of quasi-coherent sheaves on the space $X$. More precisely, in [20],
a category of mixed anti-Yetter–Drinfeld contramodules is defined by analogy
with the derived algebraic geometry case of [4]. This new generalization is
conceptual, and furthermore allows the expression of the Hopf-cyclic
cohomology of an algebra $A$ with coefficients in $M$ as an ${\rm Ext}$ (in
this category) between $\mathop{\rm ch}(A)$, the Chern character object
associated to $A$, and $M$ itself. Even if one takes $M$ to be a stable anti-
Yetter–Drinfeld contramodule, the object $\mathop{\rm ch}(A)$ is truly a mixed
anti-Yetter–Drinfeld contramodule. It is conjectured that mixed anti-
Yetter–Drinfeld contramodules are the objects in the cyclic homology category
of ${{}_{H}\mathcal{M}}$.

The comparison in [20] between anti-Yetter–Drinfeld contramodules and the
cyclic homology category of ${{}_{H}\mathcal{M}}$ involves a monad on
${{}_{H}\mathcal{M}}$ with a central element $\sigma$. When we talk about the
$S^{1}$-action we mean the action of this central element on the category of
modules over the monad. It is this description that allows us here to reduce
the investigations into the differences between the previously studied and the
new Hopf-cyclic cohomology to the analysis of categories of modules over two
differential graded algebras (DGAs). Namely, in the notation of the paper, we
have an algebra $\widehat{D}(H)$ whose modules are the anti-Yetter–Drinfeld
contramodules, we have a DGA $\widehat{D}(H)[\theta]$ with $d\theta=\sigma-1$
that yields the new mixed anti-Yetter–Drinfeld contramodules, and we have a
DGA $\widehat{D}(H)/(\sigma-1)[\theta]$ with $d\theta=0$ that yields the
previously studied setting, i.e., the mixed complexes in stable anti-
Yetter–Drinfeld contramodules. Thus, it suffices for our purposes to compare
the DG categories of modules over these two DGAs. We concentrate on finite
dimensional Hopf algebras $H$. We show that if the square of the antipode is
trivial, i.e., $S^{2}={\rm Id}$ then the DG categories coincide (Proposition
2.5):

######  Result 1.1.

Let $H$ be a finite dimensional Hopf algebra such that the square of the
antipode is equal to the identity, i.e., $S^{2}={\rm Id}$. Then the categories
of mixed complexes in stable $aYD$-contramodules and mixed $aYD$-contramodules
are $DG$-equivalent.

On the other hand, if we consider Sweedler’s Hopf algebra $T_{2}(-1)$ (the
smallest case of $S^{2}\neq{\rm Id}$) then they do not (Proposition 3.6):

######  Result 1.2.

Let $H=T_{2}(-1)$, then the mixed complexes in the category of stable anti-
Yetter–Drinfeld contramodules are not DG-equivalent to the category of mixed
anti-Yetter–Drinfeld contramodules.

Conventions. All algebras $A$ in monoidal categories are assumed to be unital
associative. Our $H$ is a Hopf algebra over some fixed algebraically closed
field $k$, of characteristic $0$, and Vec denotes the category of $k$-vector
spaces. For the purposes of this paper we are only interested in finite
dimensional Hopf algebras. We use the following version of Sweedler’s
notation: For $h\in H$ we denote the coproduct $\Delta(h)\in H\otimes H$ by
$h^{1}\otimes h^{2}$. The letter $S$ denotes the antipode of $H$. The number
$p$ is prime. Finally, DG stands for differential graded.

##  2 Twisted Drinfeld double

Let $H$ be a Hopf algebra. From [20] we see that the study of the Hochschild
and cyclic homologies of ${{}_{H}\mathcal{M}}$, the monoidal category of
$H$-modules, reduces to the study of modules over a certain monad on
${{}_{H}\mathcal{M}}$. Recall that the consideration of Hochschild and cyclic
homologies of monoidal categories is motivated by their recently discovered
role [20] in the understanding of Hopf-cyclic theory coefficients.

Briefly, we have the monad (see [20] for more details):

$\displaystyle\mathop{\rm Hom}\nolimits_{k}(H,-)\colon\
{{}_{H}\mathcal{M}}\to{{}_{H}\mathcal{M}}$ (2.1)

with the $H$-module structure on $\mathop{\rm Hom}\nolimits_{k}(H,V)$:

$\displaystyle
x\cdot\varphi=x^{2}\varphi\big{(}S\big{(}x^{3}\big{)}(-)x^{1}\big{)},$ (2.2)

for $x\in H$ and $\varphi\in\mathop{\rm Hom}\nolimits_{k}(H,V)$. The unit
$1_{V}\colon{\rm Id}(V)\to\mathop{\rm Hom}\nolimits_{k}(H,V)$ is

$\displaystyle 1_{V}(v)(h)=\epsilon(h)v$ (2.3)

and a crucial central element $\sigma_{V}\colon{\rm Id}(V)\to\mathop{\rm
Hom}\nolimits_{k}(H,V)$ is

$\displaystyle\sigma_{V}(v)(h)=hv.$ (2.4)

The anti-Yetter–Drinfeld contramodules then coincide with modules over this
monad (shown in [20]), while the stable ones consist of those for which the
action of $\sigma$ agrees with that of $1$, and the mixed ones introduced in
[20] are the homotopic version of this on the nose requirement.

Recall the mixed complexes of [12]. These are complexes of vector spaces
$(V^{\bullet},d)$ with a homotopy $h$ such that $dh+hd=0$. We can replace
vector spaces with $R$-modules for some ring $R$. Note that as observed in
[12], the DG-category of mixed complexes in $R$-modules is isomorphic to the
DG-category of DG-modules over $R[\theta]$, where $\theta$ is a freely and
centrally adjoined degree $-1$ graded commutative variable (naturally $R$
itself is placed in degree $0$, so that $R[\theta]=R\to R$ as a complex) and
$d=0$ on $R[\theta]$. The action of $\theta$ gives the homotopy $h$.

We can generalize the considerations of [12] so as to apply to our particular
situation. Namely, let $z\in Z(R)$, i.e., $zr=rz$ for all $r\in R$. Define a
DG-algebra $R[\theta]$ by placing $R$ in degree $0$ and $\theta$ in degree
$-1$. Let $\theta$ commute with $R$ and itself, so in particular
$\theta^{2}=0$. So far it is as above. Now define the differential to be $0$
on $R$ and $d\theta=z$. This is well defined and unique by the Leibniz rule.
We observe that the category of DG-modules over $R[\theta]$ consists of
complexes of $R$-modules equipped with a homotopy $h$ such that $dh+hd=z$.

Now recall from [20]:

######  Definition 2.1.

We say that $(M^{\bullet},d,h)$ is a mixed anti-Yetter–Drinfeld contramodule
if $(M^{\bullet},d)$ is a complex of contramodules, i.e., modules over the
monad (2.1), and $h$ is a homotopy annihilating $\sigma-1$. More precisely,

$\displaystyle dh+hd=\sigma-1.$ (2.5)

In this section we will define an explicit DG-algebra that will yield the
mixed anti-Yetter–Drinfeld (aYD) contramodules (for $H$ finite dimensional) as
its DG-modules. The construction of the twisted convolution algebra below is
analogous to the classical Drinfeld double $D(H)$ and its anti-version
$D_{a}(H)$ [9] (we review these in Appendix A, where we expand upon this
comparison).

######  Definition 2.2.

Let $H$ be a Hopf algebra with an invertible antipode $S$, define a twisted
double $\widehat{D}(H)$ as follows. The multiplication on
$\widehat{D}(H):={\rm End}(H)$ is

$\displaystyle(f\star
g)(h)=f\big{(}h^{1}\big{)}^{2}g\big{(}S\big{(}f\big{(}h^{1}\big{)}^{3}\big{)}h^{2}f\big{(}h^{1}\big{)}^{1}\big{)},$
(2.6)

thus the multiplicative identity, which we denote by $1$ is $\epsilon(-)1$,
and the central element $\sigma(h)=h$ is invertible with inverse $S^{-1}$.

Definition 2.2 is extracted from the monad (2.1) with the sole purpose
consisting of making the following lemma a tautology.

######  Lemma 2.3.

Let $H$ be a finite dimensional Hopf algebra.

  * •

The category of anti-Yetter–Drinfeld contramodules over $H$ is isomorphic to
$\widehat{D}(H)$-modules.

  * •

The category of stable anti-Yetter–Drinfeld contramodules is isomorphic to
modules over

$\displaystyle A:=\widehat{D}(H)/(\sigma-1).$ (2.7)

  * •

The DG-category of mixed anti-Yetter–Drinfeld contramodules is isomorphic to
DG-modules over the DG algebra

$\displaystyle B:=\widehat{D}(H)[\theta],$ (2.8)

where $\theta$ is a freely adjoined degree $-1$ graded commutative variable
and $d\theta=\sigma-1$, with $d|_{\widehat{D}(H)}=0$.

###### Proof.

In the finite dimensional case, as vector spaces, $\widehat{D}(H)={\rm
End}(H)\simeq H^{*}\otimes H$. Furthermore, as an algebra, $\widehat{D}(H)$ is
the quotient of the free product algebra, generated by $H^{*}$ and $H$, by the
relation:

$\displaystyle h\chi=\chi\big{(}S\big{(}h^{3}\big{)}(-)h^{1}\big{)}h^{2},$
(2.9)

where $h\in H$, $\chi\in H^{*}$. Thus, modules over the algebra are both
$H$-modules and $H$-contramodules (same as $H^{*}$-modules for $H$ finite
dimensional). The two actions satisfy the requisite compatibility condition
for contramodules, as specified in [5], and ensured by (2.9). Clearly, modules
over $A$ consist of the full subcategory of objects on which $\sigma$ acts by
identity, these are exactly the stable contramodules. Finally, a DG-module
over $B$ is just a complex of $\widehat{D}(H)$-modules with a homotopy given
by the action of $\theta$. The condition $d\theta=\sigma-1$ ensures that
$dh+hd=\sigma-1$ on $M^{\bullet}$. ∎

Our main goal is to compare the category of mixed aYD contramodules to the
category of mixed complexes of stable aYD contramodules. By the preceding
lemma this means determining when, and more interestingly when not, the
category of DG-modules over $B$ is DG-equivalent to $A[\theta]$-modules (with
$\theta$ of degree $-1$ and $d=0$). The study of Hopf-cyclic cohomology has
thus far only concerned itself with the latter.

The following simple lemma takes care of a lot of cases.

######  Lemma 2.4.

Let $H$ be a finite dimensional Hopf algebra and suppose that the action of
$\sigma-1$ on $\widehat{D}(H)$ is diagonalizable. Then the categories of mixed
complexes in stable $aYD$-contramodules and mixed $aYD$-contramodules are
$DG$-equivalent.

###### Proof.

Since the action of the central element $\sigma-1$ on $\widehat{D}(H)$ is
diagonalizable we may decompose $\widehat{D}(H)$ as a product of algebras
$\widehat{D}(H)_{0}\oplus\widehat{D}(H)_{+}$ with $\widehat{D}(H)_{0}$ being
the $0$-eigenspace and $\widehat{D}(H)_{+}$ all the other eigenspaces. We have
an inclusion of DGAs: $\widehat{D}(H)_{0}[\theta]\to B$ that induces an
isomorphism on cohomology. Namely, as complexes $B$ is
$\widehat{D}(H)\stackrel{{\scriptstyle\sigma-1}}{{\to}}\widehat{D}(H)$ whereas
$\widehat{D}(H)_{0}[\theta]$ is $\widehat{D}(H)_{0}\stackrel{{\scriptstyle
0}}{{\to}}\widehat{D}(H)_{0}$ and $\sigma-1$ is invertible on
$\widehat{D}(H)_{+}$. Note that $\widehat{D}(H)_{0}\simeq A$ and we are done.
∎

######  Proposition 2.5.

Let $H$ be a finite dimensional Hopf algebra such that the square of the
antipode is identity, i.e., $S^{2}={\rm Id}$. Then the categories of mixed
complexes in stable $aYD$-contramodules and mixed $aYD$-contramodules are
$DG$-equivalent.

###### Proof.

We need characteristic $0$ here. Since $S^{2}={\rm Id}$, so $H$ is semi-simple
[14, 15], so $D(H)$ (its Drinfeld double) is semi-simple [17]. By Lemma A.3,
we know that it follows that $D_{a}(H)$ is semi-simple and thus by [19] so is
$\widehat{D}(H)$. Thus, by Schur’s lemma, the action of the central element
$\sigma-1$ is diagonalizable and we are done by Lemma 2.4. ∎

In light of the above we need to consider an example of $H$ with
$S^{2}\neq{\rm Id}$. It turns out that the smallest, dimension wise, such
example suffices.

##  3 Taft Hopf algebras

We fix a prime $p$ and a primitive $p$th root of unity $\xi\in k$ in the
following. The Taft Hopf algebra $T_{p}(\xi)$ [21] is generated as a
$k$-algebra by $g$ and $x$ with the relations

$\displaystyle g^{p}=1,\qquad x^{p}=0,$ (3.1) $\displaystyle gx=\xi xg.$ (3.2)

It is sometimes called the quantum ${\mathfrak{sl}}_{2}$ Borel algebra. It is
$p^{2}$ dimensional over $k$. Furthermore, the coalgebra structure is

$\displaystyle\Delta(g)=g\otimes g,\qquad\Delta(x)=x\otimes 1+g\otimes x$
(3.3)

with $\epsilon(g)=1$, $\epsilon(x)=0$, and thus $S(g)=g^{-1}$, while
$S(x)=-g^{-1}x$. Note that

$\displaystyle S^{2}(x)=\xi^{-1}x\neq x,$ (3.4)

making $T_{2}(-1)$ the smallest Hopf algebra with $S^{2}\neq{\rm Id}$. The
Taft algebra $T_{2}(-1)$ is somewhat different from the other $T_{p}(\xi)$ and
has its own name: Sweedler’s Hopf algebra.

###  3.1 The identification with the dual

The Taft algebra is isomorphic to its dual. We need some explicit formulas
establishing the isomorphism $T_{p}(\xi)\simeq T_{p}(\xi)^{*}$ and its
inverse. Suppose that $\omega$ is a $p$th root of unity, let

$\displaystyle(n)_{\omega}=1+\cdots+\omega^{n-1}$ (3.5)

and

$\displaystyle(n)_{\omega}!=(n)_{\omega}\cdots(1)_{\omega}.$ (3.6)

The verification of the following is left to the reader; key details can be
found in [18]. The lemma itself can be obtained from [16].

######  Lemma 3.1.

As Hopf algebras

$\displaystyle T_{p}(\xi)^{*}\simeq T_{p}(\xi).$ (3.7)

###### Proof.

Consider a basis of $T_{p}(\xi)$: $\big{\\{}g^{i}x^{j}\big{\\}}_{i,j=0}^{p-1}$
so that $\big{\\{}\big{(}g^{i}x^{j}\big{)}^{*}\big{\\}}$ denotes the dual
basis of $T_{p}(\xi)^{*}$. Then the isomorphism of Hopf algebras and its
inverse are given by

$\displaystyle
g^{i}x^{j}\mapsto(j)_{\xi^{-1}}!\sum_{l}\xi^{i(j+l)}\big{(}g^{l}x^{j}\big{)}^{*}$
(3.8)

and

$\displaystyle(g^{i}x^{j})^{*}\mapsto\dfrac{1}{p(j)_{\xi^{-1}}!}\sum_{l}\xi^{-l(i+j)}g^{l}x^{j}.$
(3.9)

∎

######  Corollary 3.2.

The twisted double $\widehat{D}(T_{p}(\xi))$ is a quotient of $k\left\langle
x,x^{\prime},g,g^{\prime}\right\rangle.$

  * •

The relations are

$\displaystyle x^{p}=x^{\prime p}=g^{p}-1=g^{\prime p}-1=0,$ $\displaystyle
gg^{\prime}=g^{\prime}g,\qquad gx=\xi xg,\qquad g^{\prime}x^{\prime}=\xi
x^{\prime}g^{\prime},\qquad gx^{\prime}=\xi^{-1}x^{\prime}g,\qquad
g^{\prime}x=\xi^{-1}xg^{\prime},$ $\displaystyle
xx^{\prime}-\xi^{-1}x^{\prime}x=1-\xi^{-1}g^{\prime-1}g.$

  * •

The actions of $g^{\prime}$ and $g$ on a $\widehat{D}(T_{p}(\xi))$-module $V$,
yields a $(\mathbb{Z}/p)^{2}$-grading on $V$ by their eigenspaces, i.e.,
$g^{\prime}$, $g$ act on $V_{ij}$ by $\xi^{i}$, $\xi^{j}$, respectively. Thus
$x$ and $x^{\prime}$ have degrees $(-1,1)$ and $(1,-1)$, respectively.

  * •

The $S^{1}$-action of $\sigma$ on $V_{ij}$ is

$\displaystyle\sum_{l=0}^{p-1}\dfrac{\xi^{(i-l)(j+l)}}{(l)_{\xi^{-1}}!}x^{\prime
l}x^{l}.$ (3.10)

###### Proof.

We use the identification of vector spaces
$\widehat{D}(T_{p}(\xi))=(T_{p}(\xi))^{*}\otimes T_{p}(\xi)$ as in Lemma 2.3
followed by $(T_{p}(\xi))^{*}\otimes T_{p}(\xi)\simeq T_{p}(\xi)\otimes
T_{p}(\xi)$ from Lemma 3.1. We let

$\displaystyle x^{\prime}=x\otimes 1,\qquad x=1\otimes x,\qquad
g^{\prime}=g\otimes 1,\qquad g=1\otimes g$ (3.11)

in the latter. To derive the rest of the relations we apply (2.9). The action
of $\sigma=\sum_{ij}\big{(}g^{i}x^{j}\big{)}^{*}\otimes g^{i}x^{j}$ is
computed on the graded components directly. ∎

Observe that it follows from Corollary 3.2 that
$gg^{\prime}\in\widehat{D}(T_{p}(\xi))$ is central. Since we see that
$(gg^{\prime})^{p}=1$ so its action on $\widehat{D}(T_{p}(\xi))$ is
diagonalizable with eigenvalues $\xi^{s}$, $s\in\mathbb{Z}/p$. Thus as an
algebra

$\displaystyle\widehat{D}(T_{p}(\xi))=\bigoplus_{s}\widehat{D}(T_{p}(\xi))/(gg^{\prime}-\xi^{s})$
(3.12)

so that it suffices to understand
$\widehat{D}(T_{p}(\xi))/(gg^{\prime}-\xi^{s})$. There are two cases: $p=2$
and $p>2$. We will begin by briefly discussing the latter (though without
addressing the $S^{1}$-action), and then concentrate our attention on the
former (with examining the $S^{1}$-action) to achieve the goal set out in the
abstract.

###  3.2 The case of $\boldsymbol{p>2}$

Let $p>2$, then there exists a primitive $p$th root of unity $q\in k$ such
that

$\displaystyle q^{2}=\xi^{-1}.$ (3.13)

We then have

$\displaystyle\widehat{D}(T_{p}(\xi))/(gg^{\prime}-\xi^{s})\simeq
u_{q}({\mathfrak{sl}}_{2}).$ (3.14)

More precisely, let

$\displaystyle E=\frac{q^{s+1}}{q-q^{-1}}x^{\prime},\qquad
F=xg^{\prime},\qquad\text{and}\qquad K=q^{s+1}g,$ (3.15)

so that $\widehat{D}(T_{p}(\xi))/(gg^{\prime}-\xi^{s})$ is generated by $E$,
$F$, $K$ subject to

$\displaystyle E^{p}=F^{p}=K^{p}-1=0,$
$\displaystyle[E,F]=\frac{K-K^{-1}}{q-q^{-1}},\qquad
KEK^{-1}=q^{2}E,\qquad\text{and}\qquad KFK^{-1}=q^{-2}F.$

This shows that:

$\displaystyle\widehat{D}(T_{p}(\xi))\simeq
u_{q}({\mathfrak{sl}}_{2})\otimes\mathcal{O}_{\mathbb{Z}/p}\simeq
u_{q}({\mathfrak{sl}}_{2})\otimes k\mathbb{Z}/p.$ (3.16)

As we will see below the case of $p=2$ is very different, in particular as $s$
varies, the algebra will change significantly whereas here it does not (3.14).

######  Remark 3.3.

See [13], where the Taft algebra is called the quantum ${\mathfrak{sl}}_{2}$
Borel algebra, and its Drinfeld double is computed. The result obtained is
identical to ours in (3.16), though we compute the twisted double. This is not
surprising as we see from Appendix A that our analysis of $\widehat{D}(H)$ can
be interpreted as that of $D(H)$, with $\sigma$ being a new ingredient. Note
that more generally, a comparison between Drinfeld doubles of Nichols algebras
and quantized universal enveloping algebras can be found in [2].

###  3.3 The case of $\boldsymbol{p=2}$

We need to describe the algebra $\widehat{D}(T_{2}(-1))$ in greater detail,
paying particular attention to the element $\sigma$.

By (3.12) the category of DG-modules over $\widehat{D}(T_{2}(-1))[\theta]$ is
a product of categories, $\mathcal{C}_{0}\times\mathcal{C}_{1}$, corresponding
to the cases $s=0$ and $s=1$. We will deal with both separately. More
precisely, for a $\widehat{D}(T_{2}(-1))[\theta]$-module $V$, we decompose

$\displaystyle V=(V_{00}\oplus V_{11})\oplus(V_{01}\oplus V_{10}).$ (3.17)

Observe that by the Corollary 3.2 we have that

$\displaystyle xx^{\prime}+x^{\prime}x=1+(-1)^{s},$ (3.18)

so that

$\displaystyle(x^{\prime}x)^{2}=(1+(-1)^{s})x^{\prime}x.$ (3.19)

We see from (3.19) that the minimal polynomial of $x^{\prime}x$ depends only
on $s$; this is exclusive to $p=2$ and makes this case tractable. Note that by
(3.10):

$\displaystyle\sigma|_{V_{00}}=1-x^{\prime}x,\qquad\sigma|_{V_{11}}=-1+x^{\prime}x,\qquad\text{and}\qquad\sigma|_{V_{01}}=\sigma|_{V_{10}}=1+x^{\prime}x.$
(3.20)

Let

$\displaystyle D_{s}=\widehat{D}(T_{2}(-1))/(gg^{\prime}-(-1)^{s}).$ (3.21)

We begin with $s=0$: The category of $D_{0}$-modules consists of
$\mathbb{Z}/2$-graded vector spaces ($V=V_{00}\oplus V_{11}$) equipped with
degree changing operators $x$ and $x^{\prime}$ subject to the relation
$xx^{\prime}+x^{\prime}x=2$. The action of $\sigma-1$ on $V_{00}$ is
$-x^{\prime}x$ and on $V_{11}$ it is $x^{\prime}x-2$.

######  Lemma 3.4.

The category $\mathcal{C}_{0}$ consists of the mixed complexes of [12].

###### Proof.

Note that $D_{0}/(\sigma-1)$-modules are just vector spaces. Indeed, let
$y=x^{\prime}/2$. For improved clarity, denote by $x_{i}$ and $y_{i}$ the
action of $x$ and $y$, respectively, that originates at $V_{ii}$. After
modding out by $\sigma-1$ we have by (3.20) that $y_{1}x_{0}=0\implies
x_{1}y_{0}=1$ and $y_{0}x_{1}=1$. So

$\displaystyle y_{0}:V_{00}\simeq V_{11}:x_{1}.$ (3.22)

Furthermore, $x^{2}=y^{2}=0$ so that both $x_{0}$ and $y_{1}$ are the $0$
maps. Thus

$\displaystyle V=V_{00}\oplus V_{11}\mapsto V_{00}$ (3.23)

establishes an equivalence of categories between $D_{0}/(\sigma-1)$-modules
and Vec.

The action of $\sigma-1$ on $D_{0}$ is diagonalizable by (3.19) and so by the
proof of Lemma 2.4 the algebras $D_{0}[\theta]$ ($d\theta=\sigma-1$) and
$D_{0}/(\sigma-1)[\theta]$ ($d\theta=0$) are quasi-isomorphic. Thus,
$\mathcal{C}_{0}$, being DG-equivalent to $D_{0}/(\sigma-1)[\theta]$-modules,
is by the above discussion, equivalent to $k[\theta]$-modules. These are just
the mixed complexes of [12]. ∎

######  Remark 3.5.

Note that not only does $\mathcal{C}_{0}$ consist of the usual mixed complexes
but it also does not provide any evidence of the need for the mixed
$aYD$-contramodules (see the proof above).

Moving on to $s=1$ we find that things change for the better. Recall that
$D_{1}$ is generated by

$\displaystyle x,\ x^{\prime},\ g$ (3.24)

subject to

$\displaystyle x^{2}=x^{\prime 2}=g^{2}-1=0,\qquad
xx^{\prime}=-x^{\prime}x,\qquad gx=-xg,\qquad gx^{\prime}=-x^{\prime}g.$
(3.25)

Furthermore, $\sigma-1$ acts as $x^{\prime}x$ by (3.20).

######  Proposition 3.6.

Let $H=T_{2}(-1)$, then the mixed complexes in the category of stable anti-
Yetter–Drinfeld contramodules are not DG-equivalent to the category of mixed
anti-Yetter–Drinfeld contramodules.

###### Proof.

By the preceding discussion, i.e., the decomposition of the category into a
product, it suffices to show that the categories $D_{1}[\theta]\text{-mod}$
(where $d\theta=x^{\prime}x$) and $D_{1}/(x^{\prime}x)[\theta]\text{-mod}$
(where $d\theta=0$) are not DG-equivalent.

Recall that the Hochschild cohomology $HH^{i}(C^{\bullet})$ of a DG algebra
$C^{\bullet}$ is an invariant of its DG category of modules [22]. We will
compute $HH^{-1}$. In our simple case of a DG algebra
$C=C^{-1}\stackrel{{\scriptstyle d}}{{\to}}C^{0}$ concentrated in two degrees,
we have

$\displaystyle HH^{-1}(C)={\rm
ker}\big{(}C^{-1}\stackrel{{\scriptstyle\alpha}}{{\to}}C^{0}\oplus{\rm
Hom}\big{(}C^{0},C^{-1}\big{)}\big{)},$ (3.26)

where $\alpha(x)=(dx,[x,-])$.

The key observation here is that the center of $D_{1}$ is spanned by $1$,
$xx^{\prime}$, $xx^{\prime}g$, while that of $D_{1}/(x^{\prime}x)$ is spanned
by $1$. Thus, in the first case we get that $HH^{-1}$ is spanned by
$xx^{\prime}$ and $xx^{\prime}g$. In the second case it is spanned by $1$. ∎

##  Appendix A Appendix

Our purpose in this section is to compare $\widehat{D}(H)$ of Definition 2.2
to the more familiar Drinfeld double $D(H)$ in the case of a finite
dimensional Hopf algebra $H$. Though not original, see [7] for Definition A.1
and [9] for Definition A.2, since our conventions differ from the usual ones
we spell out the definitions again below (for $H$ finite dimensional):

######  Definition A.1.

The algebra $D(H)$ is generated by $H$ and $H^{*}$ subject to the relations:

$\displaystyle\chi
h=h^{2}\chi\big{(}h^{3}(-)S^{-1}\big{(}h^{1}\big{)}\big{)},$ (A.1)

for $\chi\in H^{*}$ and $h\in H$. Thus $D(H)=H\otimes H^{*}$ as vector spaces.

######  Definition A.2.

The algebra $D_{a}(H)$ is generated by $H$ and $H^{*}$ subject to the
relations:

$\displaystyle\chi h=h^{2}\chi\big{(}h^{3}(-)S\big{(}h^{1}\big{)}\big{)},$
(A.2)

for $\chi\in H^{*}$ and $h\in H$. Thus $D_{a}(H)=H\otimes H^{*}$ as vector
spaces. The central element is $\sigma=e_{i}\otimes e^{i}$, where $e_{i}$ is
any basis of $H$ and $e^{i}$ is its dual basis of $H^{*}$.

Note that modules over $D_{a}(H)$ as specified in Definition A.2 can be
identified with what is usually called left-right anti-Yetter–Drinfeld
modules, i.e., left modules and right comodules [9]. Recall that if $H$ is
finite dimensional then we have an $S^{1}$-equivariant equivalence between
$\widehat{D}(H)$-modules and $D_{a}(H)$-modules [19]. We will thus focus on
the comparison between $D_{a}(H)$ and $D(H)$. It is known that in general they
give very different categories of modules [10]. It is immediate that if
$S^{2}={\rm Id}$ then the algebras in fact coincide, since the only difference
between them is $S$ in Definition A.2 and $S^{-1}$ in Definition A.1. Below we
extend that observation slightly so as to cover our case of Taft algebras
where we do not have $S^{2}={\rm Id}$, but instead we get

$\displaystyle S^{2}(h)=uhu^{-1}$ (A.3)

for some group-like element $u\in H$, i.e., $\Delta(u)=u\otimes u$. For
$T_{p}(\xi)$ we have $S^{2}(a)=g^{-1}ag$ with $\Delta g^{-1}=g^{-1}\otimes
g^{-1}$, so that $u=g^{-1}$.

Recall that Hopf algebras possessing such an element $u$ as above are called
pivotal, see [1, 3] for example. Their categories of representations are thus
pivotal as well, i.e., equipped with a monoidal isomorphism from the identity
functor to the double dual functor. This natural transformation is given by
the action of $u\in H$; it is monoidal since $u$ is group-like and mapping to
the double dual since $S^{2}(h)=uhu^{-1}$.

The following lemma is a straightforward computation but can be obtained from
[10]:

######  Lemma A.3.

Let $H$ be a finite dimensional Hopf algebra and suppose that there exists a
$u\in H$ with $\Delta(u)=u\otimes u$ such that $S^{2}(h)=uhu^{-1}$ for all
$h\in H$. Then

$\displaystyle D(H)$ $\displaystyle\to D_{a}(H),$ $\displaystyle h\otimes\chi$
$\displaystyle\mapsto h\otimes\chi((-)u)$

is an isomorphism of algebras.

Thus for Taft algebras, Drinfeld doubles can play the role of
$\widehat{D}(H)$, as long as we are careful to remember about the crucial
central element $\sigma$.

### Acknowledgements

This research was supported in part by the NSERC Discovery Grant number
406709. The author wishes to thank the referees for their many helpful
suggestions.

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