Unified Interacting Quark Matter and its Astrophysical Implications
Chen Zhang
Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7, Canada
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Robert B. Mann
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

We investigate interacting quark matter (IQM),  including the perturbative QCD correction and color superconductivity, for both up-down quark matter and strange quark matter. We first derive an equation of state (EOS)  unifying all cases by a simple reparametrization and rescaling, through which we manage to maximally reduce the number of degrees of freedom. We find, in contrast to the conventional EOS  $p=1/3(\rho-4B_{\rm eff})$ for non-interacting quark matter,  that taking the extreme strongly interacting limit on the unified IQM EOS gives $p=\rho-2B_{\rm eff}$, where
$B_{\rm eff}$ is the effective bag constant. We employ the unified EOS to explore the properties of pure interacting quark stars (IQSs) composed of IQM. We describe how recent astrophysical observations, such as the pulsar-mass measurements, the NICER analysis, and the binary merger gravitational-wave events GW170817, GW190425, and GW190814, further constrain the parameter space. An upper bound for the maximum allowed mass of IQSs is found to be $M_{\rm TOV}\lesssim 3.23\, M_{\odot}$. Our analysis indicates a new possibility that the currently observed compact stars, including the recently reported GW190814's secondary component ($M=2.59^{+0.08}_{-0.09}\, M_{\odot}$), can be quark stars composed of interacting quark matter.

§ INTRODUCTION

It has been long expected that quark matter, a state consisting purely of quark and gluon degrees of freedom without confining into individual nucleons, can form at high densities or high temperatures [1, 2, 3, 4, 4]. Bodmer [5], Witten [6] and Terazawa [8] proposed that quark matter with comparable numbers of $u, \,d, \,s$ quarks, also called strange quark matter (SQM), might be the ground state of baryonic matter at the zero temperature and pressure.

However, a recent study [9] demonstrated, taking the flavor-dependent feedback of the quark gas on the QCD vacuum into account, that $u, d$ quark matter ($ud$QM) is in general more stable than SQM, and can be more stable than the ordinary nuclei at sufficiently large baryon number beyond the periodic table. This has been connected to a series of recent phenomenological explorations [10, 11, 12, 13, 15, 14, 16, 17, 18] and experimental searches [19, 20, 21].

Interacting quark matter (IQM) includes interquark effects from perturbative QCD (pQCD) and color superconductivity. pQCD corrections are due to the gluon-mediated interaction [22, 23, 24]. Color superconductivity is the superconductivity in quark matter, arising from the spin-0 Cooper-pair condensation antisymmetric in color-flavor space [25, 26, 27]. This can result in two-flavor color superconductivity, where $u$ quarks pair with $d$ quarks
[conventionally termed “2SC" (“2SC+s") without (with)  strange quarks],[Another variant of two-flavor color superconductivity is the 2SCus phase, where $u$ pairs with $s$. Ref. [28] showed that the 2SCus phase has the same free energy as the 2SC+s phase to order $m^4_s$, so we neglect the discussion of it in this paper.] or in a color-flavor locking (CFL) phase, where $u,d,s$ quarks pair with each other antisymmetrically.

Most studies of color superconductivity have assumed an effective bag constant independent of the flavor composition,  resulting in the conclusion that 2SC phases are absent in compact star physics [28]. However, improved Nambu–Jona-Lasinio models [29, 30, 31] and quark-meson models [32, 9] suggest that the effective bag constant  is very likely dependent on the flavor composition, opening up new possibilities.  A relatively small bag constant can make the two-flavor color superconductivity stable, warranting reconsideration of this possibility.

Binary mergers of compact stars produce gravitational waves, the waveforms of which encode  information about tidal deformation that is sensitive to the matter equation of state (EOS). In general, stars with stiff EOSs can easily be tidally deformed due to their large radii. The GW170817 event detected by LIGO/Virgo [34, 35] is the first confirmed merger event of compact stars. This, in conjunction with the more recent  GW190425 event [36], has inspired many investigations into EOS and the gravitational properties of  nuclear matter [37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 42], SQM [48, 49], and $ud$QM [10, 16].

More recently, a binary merger event GW190814 was reported [50], featuring a primary black hole with mass $23.2^{+1.1}_{+1.0} M_{\odot}$, and a secondary companion of $2.59^{+0.08}_{-0.09}\, M_{\odot}$, which is much larger than the upper bound   $M_{\rm TOV}\lesssim 2.3 M_{\odot}$
of the maximum mass of a non-rotating neutron star,
set by various analyses of GW170817 [51, 52, 53, 54]. Conventional non-interacting SQM and $ud$QM
have  bag constant values not sufficiently small to account for this large star mass [48, 10, 16]. It is consequently of interest to see how strongly interacting quark stars (IQSs), composed of IQM, can fit all these constraints.

We begin by first providing a unified framework for all  possible strongly interacting phases of SQM and $ud$QM by a simple reparametrization. After deriving a simple but universal EOS for IQM, we explore the properties of IQSs, investigating how recent astrophysical constraints such as   observed large pulsar masses [55, 56, 57], analysis of the NICER X-ray spectral-timing event data [58, 59], and the LIGO events [34, 35, 36, 50] constrain the IQS parameter space.

§ PROPERTIES OF IQM

We first rewrite the free energy $\Omega$ of the superconducting quark matter [28] into a general form with the pQCD correction included [23, 60, 61]:

\begin{equation}\begin{aligned}
\Omega=&-\frac{\xi_4}{4\pi^2}\mu^4+\frac{\xi_4(1-a_4)}{4\pi^2}\mu^4- \frac{ \xi_{2a} \Delta^2-\xi_{2b} m_s^2}{\pi^2}  \mu^2  \\
&-\frac{\mu_{e}^4}{12 \pi^2}+B_{\rm eff} ,
\label{omega_mu}
\end{aligned}\end{equation}

where $\mu$ and $\mu_e$ are the respective average quark and electron chemical potentials[For 2SC, $\mu=(\mu_u+2\mu_d)/3$. For 2SC+s and CFL phase, $\mu=(\mu_u+\mu_d+\mu_s)/3$.]. The first term represents the unpaired free quark gas contribution. The second term with $(1-a_4)$ represents the pQCD contribution from one-gluon exchange for gluon interaction to $O(\alpha_s^2)$ order. To phenomenologically account for  higher-order contributions, we can vary $a_4$ from $a_4=1$, corresponding to a vanishing pQCD correction, to very small values where these corrections become large [23, 60, 61]. The term with $m_s$ accounts for the correction from the finite strange quark mass if applicable, while the term with the gap parameter $\Delta$ represents the contribution from color superconductivity. The constant coefficients are

\begin{align}
(\xi_4,\xi_{2a}, \xi_{2b}) = \left\{ \begin{array} {ll}
(( \left(\frac{1}{3}\right)^{\frac{4}{3}}+ \left(\frac{2}{3}\right)^{\frac{4}{3}})^{-3},1,0) & \textrm{2SC phase}\\
(3,1,3/4) & \textrm{2SC+s phase}\\
(3,3,3/4)&   \textrm{CFL phase}
\end{array}
\nonumber
\right.
\end{align}

for the various types of quark matter.   $B_{\rm eff}$ is the effective bag constant that accounts for the non-perturbative contribution from the QCD vacuum, the size of which can be flavor-dependent [9].

From the  thermodynamic relations
p=-Ω,    n_q=-∂Ω/∂μ,   n_e=-∂Ω/∂μ_e,     ρ=Ω+n_q μ+n_e μ_e ,

we obtain relevant thermodynamic quantities such as pressure $p$,  quark and electron number densities $n_{q,e}$, and   total energy density $\rho$, in terms of the chemical potential.  To reduce the size of the parameter space, we define
λ=ξ_2a Δ^2-ξ_2b m_s^2/√(ξ_4 a_4),

characterizing the strength of the related strong interaction.  The relations (<ref>) then imply

n_q   =   ξ_4 a_4/π^2μ^3 + λ√(ξ_4 a_4) /π^2  2μ,   n_e=μ_e^3/3π^2,

ρ   =   3ξ_4 a_4/4π^2μ^4+μ_e^4/4 π^2+B_eff +  λ√(ξ_4 a_4) /π^2  μ^2,

and using Eqs. (<ref>) and (<ref>) we obtain
p=1/3(ρ-4B_eff)+ 4λ^2/9π^2(-1+sgn(λ)√(1+3π^2 (ρ-B_eff)/λ^2))

for the unified IQM EOS [We removed the electron contribution in this derivation. A numerical check approves this approximation.], where sgn$(\lambda)$ represents the sign of $\lambda$.
This general EOS expression above unifies the 2SC, “2SC+s", and CFL phases. It only has 2 independent parameters $(B_{\rm eff},\lambda)$, while all  other parameters   $(a_4, \Delta,m_s)$ and $(\xi_4,\xi_{2a}, \xi_{2b})$ are subsumed in  $\lambda$ using Eq. (<ref>)[As a necessary check of our general formula Eq. (<ref>), inserting Eq. (<ref>) into it with the CFL factor where $(\xi_4,\xi_{2a}, \xi_{2b})=(3,3,3/4)$ can reproduce the CFL result in Ref. [27, 62].].

We can further remove the $B_{\rm eff}$ parameter  by doing the dimensionless rescaling:
ρ̅=ρ/4 B_eff,    p̅=p/4 B_eff,

λ̅=λ^2/4B_eff= (ξ_2a Δ^2-ξ_2b m_s^2)^2/4 B_effξ_4 a_4,

so that the EOS Eq. (<ref>) reduces to the dimensionless form
p̅=1/3(ρ̅-1)+ 4/9π^2λ̅ (-1+sgn(λ)√(1+3π^2/λ̅ (ρ̅-1/4))),

In this paper, we only explore the positive $\lambda$ space.

With Eq. (<ref>), it is easy to show that $\partial \bar{p}/\partial \bar{\lambda}>0$, so a larger $\bar{\lambda}$ (i.e., smaller $B_{\rm eff}, a_4, m_s$ or larger $\Delta$) makes the EOS stiffer,  resulting in a larger star mass and radius.

As $\bar{\lambda}\to0$, Eq. (<ref>) reduces to the conventional non-interacting rescaled quark matter EOS  $\bar{p}=(\bar{\rho}-1)/3$. At the opposite limit where $\bar{\lambda}$ is extremely large, Eq. (<ref>) approaches the special form

which is equivalent to $p={\rho}-2B_{\rm eff}$ using Eq. (<ref>). We see that strong interaction effects can reduce the surface mass density of the quark star from $\rho_0= 4B_{\rm eff}$ down to $\rho_0=2B_{\rm eff}$, and increase the quark matter sound speed $c_s^2=\partial p/\partial \rho$ from $1/3$ up to $1$ (the light speed) maximally.

Since the energy per baryon number $E/A=3\mu\vert_{P=0}=3\mu\vert_{\Omega=0}$, we have
E/A=3√(2) π/(ξ_4 a_4)^1/4 B_eff^1/4/√(√(4λ̅+π^2)+2√(λ̅))

from Eq. (<ref>).
We see that a smaller bag constant or a larger $\bar{\lambda}$ yields a smaller $E/A$.  As $\bar{\lambda} \to 0, \,a_4\to1$,   we recover the results for non-interacting quark matter [9, 63].

§ PROPERTIES OF IQS$S$

To study gravitational effects of interacting quark stars, we  further rescale the mass and radius into dimensionless form [64, 65]  in geometric units where $ c=G=1$
m̅=m√(4 B_eff),   r̅=r√(4 B_eff),

so that the Tolman-Oppenheimer-Volkov (TOV) equation [67, 66]

dp(r)dr   =-[m(r)+4πr^3p(r)][ρ(r)+p(r)]r(r-2m(r)) ,

dm(r)dr   =4πρ(r)r^2,

can also be rescaled into dimensionless form (simply replace nonbarred symbols with barred ones). The rescaled TOV solution on $(\bar{M}, \bar{R}) =(M\sqrt{4B_{\rm eff}}, R\sqrt{4B_{\rm eff}})$ can thus be obtained with the rescaled EOS Eq. (<ref>) with respect to any given value of $\bar{\lambda}$.
We depict the solutions to Eq. (<ref>) in Fig. <ref> for the rescaled mass and radius for several different values of $\bar{\lambda}$. The physical $(M, R)$ with respect to any specific $B_{\rm eff}$ value can then straightforwardly be obtained directly from $(\bar{M}, \bar{R})$ using Eq. (<ref>).

$\bar{M}$-$\bar{R}$ of strong-interacting quark stars for given $\bar{\lambda}$, sampling $(0,0.1, 0.25, 0.5, 1, 5, 10)$ from the lighter black line to the darker black line, respectively.  The red line corresponds to $\bar{\lambda}\to \infty$, with the corresponding EOS Eq. (<ref>). The solid dots denote the maximum mass configurations for given $\bar{\lambda}$, for which $(\bar{M}_{\rm TOV}, \bar{R}_{\rm TOV})$ ranges from (0.05168, 0.1909) for $\bar{\lambda}=0$ to (0.1204, 0.3400) for $\bar{\lambda}\to \infty$.

From Fig. <ref>, we easily see that a larger $\bar{\lambda}$ leads to a larger $M_{\rm TOV}$ as expected, since a larger $\bar{\lambda}$ maps to a stiffer EOS. We can interpolate the $\bar{M}_{\rm TOV}(\bar{\lambda})$ numerical results with the following sigmoid-type function:
M̅_TOV(λ̅)= M̅_∞ /1+c_1  e^-λ̅^c_2+ (M̅_0-  M̅_∞/1+c_1  e^-λ̅^c_3 )e^-λ̅^c_4

where the coefficients $c_1\approx 0.8220 , c_2\approx0.4537, c_3\approx0.3313,c_4\approx0.2676$ are the best-fit values, with an error only at the $ 0.1\%$ level. And $\bar{M}_{0}=\bar{M}_{\rm TOV}(\bar{\lambda}\to 0)=0.05168$, $\bar{M}_{\infty}=\bar{M}_{\rm TOV}(\bar{\lambda}\to\infty)=0.1204$, corresponding to
M_TOV(λ̅→0)   ≈   15.17  M_⊙/ (B_eff/MeV  fm^-3)^1/2,

M_TOV(λ̅→∞)   ≈   35.35  M_⊙/(B_eff/MeV  fm^-3)^1/2
referring to Eq. (<ref>). We see that the strongly interacting limit has a maximum star mass 2.33 times larger than that of the non-interacting case.
In general, we have the function $M_{\rm TOV} (B_{\rm eff}, \bar{\lambda})$ from Eq. (<ref>) after inserting Eq. (<ref>). The largest measured pulsar mass therefore imposes a constraint on the $(B_{\rm eff}, \bar{\lambda})$ space.

A recent NICER analysis of PSR J0030+0451 [58, 59] points to a star with a mass around $1.4\,M_{\odot}$ with a radius around 13 km ($90\%$ C.L.). The inferred contour of joint probability density distribution on the $M-R$ plane can then be translated to a range for constraints on the $(B_{\rm eff}, \bar{\lambda})$ space, utilizing the derived $\bar{M}(\bar{R})$ results presented in  Fig. <ref>.

§ TIDAL DEFORMABILITY

The response of compact stars to external disturbances is characterized by the Love number $k_2$ [68, 69, 70, 71]:

k_2    = 8 C^5/5 (1-2C)^2 [ 2 + 2 C (y_R-1) -y_R ]

×{ 2 C [ 6- 3y_R + 3C (5y_R-8)]+4C^3[ 13-11 y_R
+ C (3y_R-2)+2C^2(1+y_R)]

+ 3 (1-2C)^2 [ 2 - y_R + 2C (y_R-1)] log(1-2C )}^-1 ,

where $C=M/R=C(\bar{M})$ for given $\bar{\lambda}$.  The quantity $y_R$ is $y(r)$ evaluated at the star surface, which can be obtained by solving the following equation [71]:

ry^'(r)+y(r)^2+ r^2Q(r)

+y(r)e^λ(r)[1+4πr^2(p(r)-ρ(r))]=0 ,

with the boundary condition $y(0)=2$. Here

Q(r)   =4πe^λ(r) (5ρ(r)+9p(r)+ρ(r)+p(r)/c_s^2(r) )

\begin{equation}
e^{\lambda(r)}=\left[1-{2m(r)\over r}\right]^{-1} \quad
\nu^\prime(r)=2e^{\lambda(r)}{m(r)+4\pi p(r)r^3\over r^2}
\label{eq:met}
\end{equation}

$c_s^2(r)\equiv dp/d\rho$ denotes the sound speed squared.
For stars with a finite surface density like quark stars, a matching condition [72, 73]  $y_R^{\rm ext}=y_R^{\rm int}-  4\pi R^3\rho_s/M=y_R^{\rm int}-  4\pi \bar{R}^3\bar{\rho}_s/\bar{M}$ should be used at the boundary. Solving Eq. (<ref>) with  $\rho(r)$ and $p(r)$ obtained from Eq. (<ref>), we obtain the function $k_2(C)$ for a given $\bar{\lambda}$.  The dimensionless tidal deformability $\Lambda=2k_2/(3C^5)$ as a function of ($\bar{M}$, $\bar{\lambda}$) is thus obtained accordingly.

We depict the results of $\Lambda(\bar{M},\bar{\lambda})$  in Fig. <ref>. Note that the lower end of each curve is determined by requiring the star mass not to exceed its maximum allowed mass. We can see that a larger $\bar{\lambda}$ results in a larger tidal deformability for a given $\bar{M}$, as expected since a larger $\bar{\lambda}$ maps to a stiffer EOS.

$\Lambda$-$\bar{M}$ of IQSs for $\bar{\lambda}=(0,0.1, 0.25, 0.5, 1, 5, 10)$, with a darker black color for a larger value. The red line corresponds to $\bar{\lambda}\to \infty$ utilizing the corresponding EOS (<ref>). The solid dots denote the maximum mass configurations for the given $\bar{\lambda}$, with $(\bar{M}_{\rm TOV}, \Lambda_{\rm \bar{M}_{\rm TOV}})$ ranging from (0.0517, 22.9) for $\bar{\lambda}=0$ to (0.120, 2.17) for $\bar{\lambda}\to \infty$.

Assuming the compact objects detected by the recent LIGO event are pure IQSs, we can use the LIGO constraint on $\Lambda(1.4 M_\odot)$ to narrow down the parameter space $(B_{\rm eff},\bar{\lambda})$ with the $\Lambda(\bar{M},\bar{\lambda})$ calculated above.

The average tidal deformability of a binary system is defined as
Λ̃   =    16/13
(1+12q)/(1+q)^5 Λ (M_1)+ 16/13  q^4 (12+q)/(1+q)^5Λ(M_2),

where $M_1$ and $M_2$ are the masses of the binary components, and $q=M_2/M_1=\bar{M}_2/\bar{M}_1$, with $M_2$ being the smaller mass so that $0<q\lesssim1$. For any given chirp mass  $M_{c} = (M_1 M_2)^{3/5}/(M_1+M_2)^{1/5}$, one thus has
M_2=(q^2(q+1))^{1/5} M_c \text{ and }$ and $M_1=((1+q)/q^3)^{1/5} M_c\; .$
Using the rescaled $\bar{M}=M\sqrt{4 B_{\rm eff}}$, we eventually  obtain $\tilde \Lambda=\tilde \Lambda(\bar{M}_1,\bar{M}_2,\bar{\lambda})=\tilde \Lambda(M_c,q, B_{\rm eff},\bar{\lambda})$.  LIGO constraints on $\tilde{\Lambda}$ can then be used to narrow down the parameter space $(B_{\rm eff},\bar{\lambda})$ for given $M_c$ and $q$, assuming the objects detected by the recent LIGO events are pure IQSs.

The resultant constrained parameter space is shown in Fig. <ref>. Since we are studying quark stars, we do not use assumed hadronic EOS constraints. In our results, only the stability lines have an explicit dependence on the flavor composition and the size of $a_4$.

Astrophysical constraints on the parameter space in the $(B_{\rm eff}, \bar{\lambda}^{1/4})$ plane.
Allowed regions are to the  left of each of the three solid black curves, which correspond to  maximum pulsar masses $M_{\rm TOV}\gtrsim 2.59 M_{\odot}$ (GW190814 [50]), $2.14 M_{\odot}$(J0740+6620 [57]), and $2.01 M_{\odot}$(J0348+0432 [56]), from left to right respectively. Colored dot-dash lines bound regions of the same color whose parameters satisfy various constraints ($90\%$ C.L.) :
blue for  the GW170817 constraint $\tilde{\Lambda}=300^{+420}_{-230}$ with $M_c=1.186 \, M_{\odot}$  and $q=0.73-1.00$ [34, 35], red for GW190425 constraints $ \tilde{\Lambda} \lesssim 600$ with $M_c=1.44\, M_\odot$, $q=0.8-1.0$ [36], and   cyan for constraints from the recent NICER analysis of PSR J0030+0451 [59]; overlapping regions have correspondingly different colors.
The region to the right of the dotted blue line satisfies the GW170817 constraint $\Lambda_{1.4M_\odot}\lesssim 800$.
The yellow and green curves represent the stability lines derived from Eq. (<ref>), above which the two-flavor quark matter and three-flavor quark matter can be more stable than ordinary nuclei (i.e., $E/A <930\,\rm MeV$), respectively.  Dashed curves include the pQCD correction at $a_4=0.5$ order, whereas  the solid ones do not.

As described previously, the solid black curves are determined from Eq. (<ref>) for the measured pulsar masses.  The blue- and red-colored bands (bounded by dot-dashed lines) represent the $\tilde \Lambda$ constraint of GW170817 and GW190425 translated from the previously obtained $\tilde \Lambda(M_c,q, B_{\rm eff},\bar{\lambda})$ result. For a given $M_c$, the left edge of the band is determined by the upper bound of $\tilde{\Lambda}$ with the smallest allowed $q$ value, while the right edge is determined jointly by the lower bound of $\tilde{\Lambda}$ with $q=1$ and the requirement that $\bar{M}\lesssim\bar{M}_{\rm TOV}$.

Considering the solid black curve always goes leftward for a larger mass, there is then a critical mass value $M_c\approx3.23 M_{\odot}$ beyond which the associated solid black curve no longer intersects with the blue band (GW170817) for any $\bar{\lambda}$ value. Therefore, the possibility of IQSs is only allowed for $M_{\rm TOV}\lesssim 3.23\, M_{\odot}$.

One can easily observe that for any $\bar{\lambda}$ value, the lower bound of $B_{\rm eff}$ is determined by the LIGO constraint $\bar{\Lambda}\lesssim720$ of GW170817. The upper bound of $B_{\rm eff}$ is set by the constraint of $M_{\rm TOV}$ (the solid black curves)  for small $\bar{\lambda}$, and by the NICER constraint (the right edge of the cyan band) for large $\bar{\lambda}$. More explicitly, when $M_{\rm TOV}\gtrsim 2.59 M_{\odot}$ (the left of the left solid black curve) as suggested by GW190814,  the joint constraints tell that any  $B_{\rm eff}$ is excluded for $\bar{\lambda}^{1/4} \lesssim 1.17$, while $92.7 \lesssim B_{\rm eff}/ (\rm MeV/fm^3)\lesssim 111$ for $\bar{\lambda}^{1/4} \lesssim 1.37$, and $111 \lesssim B_{\rm eff}/ (\rm MeV/fm^3)\lesssim 130$ for $\bar{\lambda}^{1/4} \gtrsim 1.37$.  For comparison, relaxing the maximum mass to $M_{\rm TOV}\gtrsim 2.14\, M_{\odot}$ (the left of the middle solid black curve), we have the constrained parameter space $49.5 \lesssim B_{\rm eff}/ (\rm MeV/fm^3) \lesssim 76.2$ for  $\bar{\lambda}^{1/4} \lesssim 0.67$, and $76.2 \lesssim B_{\rm eff}/ (\rm MeV/fm^3)\lesssim 130$ for $\bar{\lambda}^{1/4} \gtrsim 0.67$.
We see that the constrained region can be well above the two-flavor and three-flavor stability lines within reasonable uncertainties of $a_4$, so that both interacting $ud$QM and interacting SQM can be more stable than ordinary nuclei in the allowed parameter space.

§ SUMMARY

We have explored IQM, i.e., quark matter with interquark effects from pQCD corrections and color superconductivity, in a general parametrization, reducing the EOS to a dimensionless form depending on one single parameter $\bar{\lambda}$, which characterizes the relative size of strong interaction effects. A larger $\bar{\lambda}$ results in a stiffer EOS. At the large $\bar{\lambda}$ limit, the EOS becomes $\bar{p}=\bar{\rho}-1/2$, or equivalently $p=\rho-2B_{\rm eff}.$

Using this we studied the properties of IQSs (compact stars composed of IQM) and found that a larger $\bar{\lambda}$ results in a larger (rescaled) maximum mass, with an upper bound $M_{\rm TOV}(\bar{\lambda}\to \infty)=35.35\, M_{\odot} (B_{\rm eff}/\rm MeV\, fm^{-3})^{-1/2}$. We also obtained an explicit expression of $\bar{M}_{\rm TOV}(\bar{\lambda})$, and computed the deformability of IQSs.

Assuming the related compact objects are IQSs, we translated recent astrophysical observations into various constraints on the  $(B_{\rm eff}, \bar{\lambda}^{1/4})$ parameter space, and obtained an upper bound for the maximum allowed mass for IQSs: $M_{\rm TOV}\lesssim 3.23 M_{\odot}$. Remarkably, the parameter space is confined to a window of moderate $B_{\rm eff}$ and large $\bar{\lambda}$ above the two-flavor stability line, enclosed by the constraints from the upper bound of $\tilde \Lambda$ of GW170817, the recent NICER analysis, and the mass of the most massive compact star identified. Compact stars identified using the recent astrophysical observations can thus be consistently interpreted as quark stars composed of interacting $ud$QM or  interacting SQM. This study paves the way for future astrophysical observations to confirm this possibility, further constrain it,  or rule it out entirely.

We thank Bob Holdom and Jing Ren for helpful discussions. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada.

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Transposed Poisson structures on the Lie algebra of
upper triangular matrices 111 The work was supported by FCT
UIDB/MAT/00212/2020, UIDP/MAT/00212/2020, 2022.02474.PTDC and by CMUP, member
of LASI, which is financed by national funds through FCT — Fundação para a
Ciência e a Tecnologia, I.P., under the project with reference
UIDB/00144/2020. 222Corresponding author<EMAIL_ADDRESS>

Ivan Kaygorodov333CMA-UBI, Universidade da Beira Interior, Covilhã, Portugal;
<EMAIL_ADDRESS>& Mykola Khrypchenko444 Departamento de Matemática,
Universidade Federal de Santa Catarina, Brazil; and CMUP, Departamento de
Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre
s/n, 4169–007 Porto, Portugal<EMAIL_ADDRESS>

Abstract: We describe transposed Poisson structures on the upper triangular
matrix Lie algebra $T_{n}(F)$, $n>1$, over a field $F$ of characteristic zero.
We prove that, for $n>2$, any such structure is either of Poisson type or the
orthogonal sum of a fixed non-Poisson structure with a structure of Poisson
type, and for $n=2$, there is one more class of transposed Poisson structures
on $T_{n}(F)$. We also show that, up to isomorphism, the full matrix Lie
algebra $M_{n}(F)$ admits only one non-trivial transposed Poisson structure,
and it is of Poisson type.

Keywords: Transposed Poisson algebra, upper triangular matrix Lie algebra,
full matrix Lie algebra, $\delta$-derivation.

MSC2020: primary 17A30, 15B30; secondary 17B40, 17B63.

## Introduction

Since their origin in the 1970s in Poisson geometry, Poisson algebras have
appeared in several areas of mathematics and physics, such as algebraic
geometry, operads, quantization theory, quantum groups, and classical and
quantum mechanics. One of the natural tasks in the theory of Poisson algebras
is the description of all such algebras with fixed Lie or associative part
[14, 6, 7].

Recently, Bai, Bai, Guo, and Wu [1] have introduced a dual notion of the
Poisson algebra, called a transposed Poisson algebra, by exchanging the roles
of the two multiplications in the Leibniz rule defining a Poisson algebra. A
transposed Poisson algebra defined this way not only shares some properties of
a Poisson algebra, such as the closedness under tensor products and the Koszul
self-duality as an operad, but also admits a rich class of identities. It is
important to note that a transposed Poisson algebra naturally arises from a
Novikov-Poisson algebra by taking the commutator Lie algebra of its Novikov
part.

Thanks to [3], any unital transposed Poisson algebra is a particular case of a
“contact bracket” algebra and a quasi-Poisson algebra. In a recent paper by
Ferreira, Kaygorodov, and Lopatkin a relation between
$\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras has
been established [4]. These ideas were used to describe all transposed Poisson
structures on Witt and Virasoro algebras in [4]; on twisted Heisenberg-
Virasoro, Schrödinger-Virasoro and extended Schrödinger-Virasoro algebras in
[15]; on oscillator Lie algebras in [3]; on Schrodinger algebra in
$(n+1)$-dimensional space-time in [13]; on Witt type Lie algebras in [9]; on
generalized Witt algebras in [10] and Block Lie algebras in [8, 10]. Any
complex finite-dimensional solvable Lie algebra was proved to admit a non-
trivial transposed Poisson structure [11]. The algebraic and geometric
classification of $3$-dimensional transposed Poisson algebras was given in
[2]. For the list of actual open questions on transposed Poisson algebras, see
[3].

In this paper we describe transposed Poisson structures on the upper
triangular matrix Lie algebra $T_{n}(F)$ over a field $F$ of characteristic
zero. To this end, we first characterize $\frac{1}{2}$-derivations of
$T_{n}(F)$ in Proposition 10. Then the case $n>2$ is solved in Theorem 11, and
the case $n=2$ is treated separately in Theorem 12. Namely, we prove in
Theorem 11 that any transposed Poisson structure on $T_{n}(F)$, $n>2$, is
either of Poisson type or the orthogonal sum of the fixed non-Poisson
structure

$\displaystyle e_{11}\cdot e_{11}=-e_{11}\cdot e_{nn}=e_{nn}\cdot
e_{nn}=e_{1n}$

with a structure of Poisson type. If $n=2$, then there appears one more
separate class of transposed Poisson structures on $T_{n}(F)$ consisting of
the (non-orthogonal) sums of a structure of the family

$\displaystyle e_{11}\cdot e_{11}=ce_{11},e_{11}\cdot e_{12}=-e_{12}\cdot
e_{22}=ce_{12},e_{11}\cdot e_{22}=-e_{22}\cdot e_{22}=ce_{22},\ c\neq 0,$

with a structure of Poisson type. As a complementary result, we prove in
Theorem 14 that the full matrix Lie algebra $M_{n}(F)$ admits only one non-
trivial transposed Poisson structure $e_{ii}\cdot e_{jj}=\delta$, $1\leq
i,j\leq n$, and it is of Poisson type. In fact, is isomorphic to the extension
by zero of the product $\delta\cdot\delta=\delta$ as observed in Remark 15.

##  1\. Definitions and preliminaries

All the algebras below will be over a field $F$ of characteristic zero and all
the linear maps will be $F$-linear, unless otherwise stated. The notation
$\langle S\rangle$ means the $F$-subspace generated by $S$.

######  Definition 1.

Let ${\mathfrak{L}}$ be a vector space equipped with two nonzero bilinear
operations $\cdot$ and $[\cdot,\cdot].$ The triple
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ is called a transposed Poisson algebra
if $({\mathfrak{L}},\cdot)$ is a commutative associative algebra and
$({\mathfrak{L}},[\cdot,\cdot])$ is a Lie algebra that satisfies the following
compatibility condition

$\displaystyle 2z\cdot[x,y]=[z\cdot x,y]+[x,z\cdot y].$ (1)

Transposed Poisson algebras were first introduced in a paper by Bai, Bai, Guo,
and Wu [1].

######  Definition 2.

Let $({\mathfrak{L}},[\cdot,\cdot])$ be a Lie algebra. A transposed Poisson
algebra structure on $({\mathfrak{L}},[\cdot,\cdot])$ is a commutative
associative multiplication $\cdot$ on $\mathfrak{L}$ which makes
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ a transposed Poisson algebra.

######  Definition 3.

Let $({\mathfrak{L}},[\cdot,\cdot])$ be an algebra and
$\varphi:\mathfrak{L}\to\mathfrak{L}$ a linear map. Then $\varphi$ is a
$\frac{1}{2}$-derivation if it satisfies

$\displaystyle\varphi\big{(}[x,y]\big{)}=\frac{1}{2}\big{(}[\varphi(x),y]+[x,\varphi(y)]\big{)}.$
(2)

Observe that $\frac{1}{2}$-derivations are a particular case of
$\delta$-derivations introduced by Filippov in [5]. The space of all
$\frac{1}{2}$-derivations of an algebra $\mathfrak{L}$ will be denoted by
$\Delta(\mathfrak{L}).$ It is easy to see from 2 that
$[\mathfrak{L},\mathfrak{L}]$ and $\operatorname{Ann}(\mathfrak{L})$ are
invariant under any $\frac{1}{2}$-derivation of $\mathfrak{L}$.

Definitions 1 and 3 immediately imply the following key Lemma.

######  Lemma 4.

Let $({\mathfrak{L}},[\cdot,\cdot])$ be a Lie algebra and $\cdot$ a new binary
(bilinear) operation on ${\mathfrak{L}}$. Then
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ is a transposed Poisson algebra if and
only if $\cdot$ is commutative and associative and for every
$z\in{\mathfrak{L}}$ the multiplication by $z$ in $({\mathfrak{L}},\cdot)$ is
a $\frac{1}{2}$-derivation of $({\mathfrak{L}},[\cdot,\cdot]).$

The basic example of a $\frac{1}{2}$-derivation is the multiplication by a
field element. Such $\frac{1}{2}$-derivations will be called trivial.

######  Theorem 5.

Let ${\mathfrak{L}}$ be a Lie algebra without non-trivial
$\frac{1}{2}$-derivations. Then all transposed Poisson algebra structures on
${\mathfrak{L}}$ are trivial.

Another well-known class of $\frac{1}{2}$-derivations of
$(\mathfrak{L},[\cdot,\cdot])$ is formed by linear maps
$\mathfrak{L}\to\operatorname{Ann}(\mathfrak{L})$ annihilating
$[\mathfrak{L},\mathfrak{L}]$. If $(\mathfrak{L},[\cdot,\cdot])$ is a Lie
algebra, such $\frac{1}{2}$-derivations of $\mathfrak{L}$ correspond to the
following transposed Poisson structures on $\mathfrak{L}$. Denote by
$Z(\mathfrak{L})$ the center of $\mathfrak{L}$ and fix a complement $V$ of
$[\mathfrak{L},\mathfrak{L}]$ in $\mathfrak{L}$. Then any commutative
associative product $*:V\times V\to Z(\mathfrak{L})$ defines a transposed
Poisson algebra structure $\cdot$ on $\mathfrak{L}$ by means of

$\displaystyle(a_{1}+a_{2})\cdot(b_{1}+b_{2})=a_{1}*b_{1},$ (3)

where $a_{1},b_{1}\in V$ and $a_{2},b_{2}\in[\mathfrak{L},\mathfrak{L}]$.
Indeed, the right-hand side of 1 is zero, because $z\cdot x,z\cdot y\in
Z(\mathfrak{L})$, and the left-hand side of 1 is zero by 3, because
$[x,y]\in[\mathfrak{L},\mathfrak{L}]$. We say that $\cdot$ is the extension by
zero of $*$. Observe that $\cdot$ is, at the same time, a usual Poisson
structure on $(\mathfrak{L},[\cdot,\cdot])$. Thus, such transposed Poisson
structures are said to be of Poisson type.

Given two transposed Poisson structures $\cdot_{1}$ and $\cdot_{2}$ on
$(\mathfrak{L},[\cdot,\cdot])$, their sum $*$ defined by

$\displaystyle a*b=a\cdot_{1}b+a\cdot_{2}b$

is clearly commutative and satisfies 1. In general, $*$ may be non-
associative, but it is associative, if

$\displaystyle\mathfrak{L}\cdot_{1}\mathfrak{L}\subseteq\operatorname{Ann}(\mathfrak{L},\cdot_{2})\text{
and
}\mathfrak{L}\cdot_{2}\mathfrak{L}\subseteq\operatorname{Ann}(\mathfrak{L},\cdot_{1}).$

In this case we say that $\cdot_{1}$ and $\cdot_{2}$ are orthogonal, and $*$
is the orthogonal sum of $\cdot_{1}$ and $\cdot_{2}$.

Let $\cdot$ be a transposed Poisson algebra structure on a Lie algebra
$({\mathfrak{L}},[\cdot,\cdot])$. Then any automorphism $\phi$ of
$({\mathfrak{L}},[\cdot,\cdot])$ induces the transposed Poisson algebra
structure $*$ on $({\mathfrak{L}},[\cdot,\cdot])$ given by

$\displaystyle x*y=\phi\big{(}\phi^{-1}(x)\cdot\phi^{-1}(x)\big{)},\ \
x,y\in{\mathfrak{L}}.$

Clearly, $\phi$ is an isomorphism of transposed Poisson algebras
$({\mathfrak{L}},\cdot,[\cdot,\cdot])$ and $({\mathfrak{L}},*,[\cdot,\cdot])$.

##  2\. Transposed Poisson structures on the upper triangular matrix Lie
algebra

###  2.1. Upper triangular matrix algebra

Let $n$ be a positive integer. Denote by $T_{n}(F)$ the algebra of upper
triangular $n\times n$ matrices over $F$. The usual matrix product on
$T_{n}(F)$ will be denoted by $\cdot$ or just by the concatenation, and the
commutator product by $[a,b]=ab-ba$. Following the terminology of incidence
algebras, we denote the identity matrix by $\delta$ and, given $a\in T_{n}(F)$
and $1\leq i,j\leq n$, we write $a(i,j)$ for the $(i,j)$-entry of $a$. The
algebra $T_{n}(F)$ has the natural basis formed by the matrix units $e_{ij}$,
$1\leq i\leq j\leq n$, where $e_{ij}(k,l)=\delta(i,k)\delta(j,l)$. It is well-
known that

$\displaystyle Z(T_{n}(F))=\langle\delta\rangle\text{ and
}[T_{n}(F),T_{n}(F)]=\langle e_{ij}\mid 1\leq i<j\leq n\rangle\text{ for all
}n\geq 1.$

Moreover,

$\displaystyle Z([T_{n}(F),T_{n}(F)])=\langle e_{1n}\rangle\text{ for all
}n>1.$ (4)

Clearly, for all $a\in T_{n}(F)$ and $1\leq i\leq j\leq n$ we have

$\displaystyle e_{ij}a=\sum_{j\leq k}a(j,k)e_{ik},\ \ ae_{ij}=\sum_{l\leq
i}a(l,i)e_{lj},$

so

$\displaystyle e_{ij}ae_{kl}=\begin{cases}a(j,k)e_{il},&j\leq k,\\\
0,&j>k.\end{cases}$

These equalities will be used numerous times throughout the text without any
reference.

###  2.2. $\frac{1}{2}$-derivations of the upper triangular matrix Lie algebra

We denote by $\Delta(T_{n}(F))$ the space of $\frac{1}{2}$-derivations of the
Lie algebra $(T_{n}(F),[\cdot,\cdot])$.

######  Lemma 6.

Let $\varphi\in\Delta(T_{n}(F))$. Then for all $1\leq i<j\leq n$ we have

  1. (i)

$\varphi(e_{ij})e_{ii}=0;$

  2. (ii)

$e_{ij}\varphi(e_{ii})=\varphi(e_{ii})(j,j)e_{ij};$

  3. (iii)

$\varphi(e_{ii})(i,j)=-\varphi(e_{jj})(i,j);$

  4. (iv)

$\varphi(e_{ii})e_{ij}=\varphi(e_{ii})(i,i)e_{ij}$.

###### Proof.

(i). Since $e_{ij}=[e_{ij},e_{jj}]$, then

$\displaystyle
2\varphi(e_{ij})=[\varphi(e_{ij}),e_{jj}]+[e_{ij},\varphi(e_{jj})]=\varphi(e_{ij})e_{jj}-e_{jj}\varphi(e_{ij})+e_{ij}\varphi(e_{jj})-\varphi(e_{jj})e_{ij}.$
(5)

Multiplying 5 by $e_{ii}$ on the right (under $\cdot$), we obtain the desired
equality.

(ii). Apply $\varphi$ to $[e_{ii},e_{jj}]=0$:

$\displaystyle
0=\varphi([e_{ii},e_{jj}])=\varphi(e_{ii})e_{jj}-e_{jj}\varphi(e_{ii})+e_{ii}\varphi(e_{jj})-\varphi(e_{jj})e_{ii}.$
(6)

If there exists $k>j$, then multiplying 6 by $e_{jj}$ on the left and by
$e_{kk}$ on the right, we obtain

$\displaystyle\varphi(e_{ii})(j,k)=0\text{ for }j<k.$ (7)

It follows that

$\displaystyle e_{ij}\varphi(e_{ii})=\sum_{j\leq
k}\varphi(e_{ii})(j,k)e_{ik}=\varphi(e_{ii})(j,j)e_{ij},$

as needed.

(iii). This follows by multiplying 6 by $e_{ii}$ on the left and by $e_{jj}$
on the right.

(iv). Applying $\varphi$ to $e_{ij}=[e_{ii},e_{ij}]$, we have

$\displaystyle
2\varphi(e_{ij})=[\varphi(e_{ii}),e_{ij}]+[e_{ii},\varphi(e_{ij})]=\varphi(e_{ii})e_{ij}-e_{ij}\varphi(e_{ii})+e_{ii}\varphi(e_{ij})-\varphi(e_{ij})e_{ii}.$
(8)

If there exists $l<i$, then the multiplication of 8 by $e_{ll}$ on the left
and by $e_{jj}$ on the right gives

$\displaystyle 2\varphi(e_{ij})(l,j)=\varphi(e_{ii})(l,i).$ (9)

On the other hand, $[e_{ll},e_{ij}]=0$, so

$\displaystyle
0=\varphi(e_{ll})e_{ij}-e_{ij}\varphi(e_{ll})+e_{ll}\varphi(e_{ij})-\varphi(e_{ij})e_{ll}.$

Multiplying this by $e_{ll}$ on the left and by $e_{jj}$ on the right, we come
to

$\displaystyle\varphi(e_{ij})(l,j)=-\varphi(e_{ll})(l,i).$ (10)

The latter is $\varphi(e_{ii})(l,i)$ by item (iii). Thus, 9 and 10 result in

$\displaystyle\varphi(e_{ii})(l,i)=0\text{ for }l<i.$ (11)

Consequently,

$\displaystyle\varphi(e_{ii})e_{ij}=\sum_{l\leq
i}\varphi(e_{ii})(l,i)e_{ik}=\varphi(e_{ii})(i,i)e_{ij},$

as desired. ∎

######  Lemma 7.

Let $\varphi\in\Delta(T_{n}(F))$. Then for all $1\leq i<j\leq n$ we have

$\displaystyle\varphi(e_{ij})=(\varphi(e_{ii})(i,i)-\varphi(e_{ii})(j,j))e_{ij}=(\varphi(e_{jj})(j,j)-\varphi(e_{jj})(i,i))e_{ij}.$
(12)

###### Proof.

By 8 and (i), (ii) and (iv) of Lemma 6 we have

$\displaystyle
2\varphi(e_{ij})=(\varphi(e_{ii})(i,i)-\varphi(e_{ii})(j,j))e_{ij}+e_{ii}\varphi(e_{ij}).$
(13)

Multiplying 13 by $e_{ii}$ on the left, we get

$\displaystyle
2e_{ii}\varphi(e_{ij})=(\varphi(e_{ii})(i,i)-\varphi(e_{ii})(j,j))e_{ij}+e_{ii}\varphi(e_{ij}).$

whence

$\displaystyle
e_{ii}\varphi(e_{ij})=(\varphi(e_{ii})(i,i)-\varphi(e_{ii})(j,j))e_{ij}.$

Substituting this into 13 and dividing by $2$ (recall that
$\mathrm{char}(F)=0$), we prove the first equality of 12. The second one is
obtained from 5 using $\varphi(e_{ij})\in\langle e_{ij}\rangle$ (which holds
by the first equality of 12). ∎

######  Lemma 8.

Let $\varphi\in\Delta(T_{n}(F))$.

  1. (i)

For all $1\leq i\leq n$ and $1\leq j\leq k\leq n$ with $i\not\in\\{j,k\\}$ we
have $\varphi(e_{ii})(j,j)=\varphi(e_{ii})(k,k)$.

  2. (ii)

For all $1\leq i<j\leq n$ we have $\varphi(e_{ij})(i,j)=\varphi(e_{1n})(1,n)$.

  3. (iii)

For all $1\leq i\leq n$ and $1\leq j<k\leq n$ we have
$\varphi(e_{ii})(j,k)=0$, unless $i\in\\{1,n\\}$ and $(j,k)=(1,n)$.

###### Proof.

(i). Applying $\varphi$ to $[e_{ii},e_{jk}]=0$, we get

$\displaystyle
0=\varphi(e_{ii})e_{jk}-e_{jk}\varphi(e_{ii})+e_{ii}\varphi(e_{jk})-\varphi(e_{jk})e_{ii}.$

Then the desired equality is obtained by the left multiplication by $e_{jj}$
and right multiplication by $e_{kk}$.

(ii). Since $i<j$, then $i\neq n$, so $i\not\in\\{j,n\\}$ and
$n\not\in\\{1,i\\}$. We use item (i) and 12:

$\displaystyle\varphi(e_{ij})(i,j)$
$\displaystyle=\varphi(e_{ii})(i,i)-\varphi(e_{ii})(j,j)=\varphi(e_{ii})(i,i)-\varphi(e_{ii})(n,n)$
$\displaystyle=\varphi(e_{nn})(n,n)-\varphi(e_{nn})(i,i)=\varphi(e_{nn})(n,n)-\varphi(e_{nn})(1,1)=\varphi(e_{1n})(1,n).$

(iii). We have already seen in 7 that $\varphi(e_{ii})(j,k)=0$ for $j<k$ and
$i\not\in\\{j,k\\}$ (although it is required that $i<j$ in the statement of
Lemma 6, the proof of 7 uses only $j<k$ and $i\not\in\\{j,k\\}$). Moreover,
$\varphi(e_{ii})(l,i)=0$ for all $l<i$ by 11 under the assumption that there
exists $j>i$ (i.e., $i<n$). Similarly, 5 and 7 imply that
$\varphi(e_{jj})(j,k)=2\varphi(e_{ij})(j,k)=0$ for all $i<j<k$, so
$\varphi(e_{ii})(i,k)=0$ for all $1<i<k$. It remains to prove that

$\displaystyle\varphi(e_{11})(1,k)=\varphi(e_{nn})(j,n)=0\text{ for }1<j,k<n.$

But Lemma 6(iii) shows that $\varphi(e_{11})(1,k)=-\varphi(e_{kk})(1,k)$,
which is proved to be $0$ for $1<k<n$. Similarly,
$\varphi(e_{nn})(j,n)=-\varphi(e_{jj})(j,n)=0$ for $1<j<n$. ∎

######  Lemma 9.

Let $n>1$. Then the linear map $\alpha:T_{n}(F)\to T_{n}(F)$ given by

$\displaystyle\alpha(e_{ij})=\begin{cases}e_{1n},&(i,j)=(1,1),\\\
-e_{1n},&(i,j)=(n,n),\\\ 0,&(i,j)\not\in\\{(1,1),(n,n)\\},\end{cases}$

is a $\frac{1}{2}$-derivation of $T_{n}(F)$.

###### Proof.

We are going to prove that $\varphi=\alpha$ satisfies 2 for $x=e_{ij}$ and
$y=e_{kl}$. Since $\alpha$ annihilates the commutators of $T_{n}(F)$, the
left-hand side of 2 is always zero. In view of the anti-commutativity of
$[\cdot,\cdot]$, we have to deal only with the following $2$ cases.

Case 1. $(i,j)=(1,1)$. Then

$\displaystyle[\alpha(e_{ij}),e_{kl}]+[e_{ij},\alpha(e_{kl})]=[e_{1n},e_{kl}]+[e_{11},\alpha(e_{kl})].$
(14)

Both summands on the right-hand side of 14 are zero, unless
$(k,l)\in\\{(1,1),(n,n)\\}$. If $(k,l)=(1,1)$, then the right-hand side of 14
is zero due to the anti-commutativity of $[\cdot,\cdot]$. And if
$(k,l)=(n,n)$, then $[e_{1n},e_{kl}]+[e_{11},\alpha(e_{kl})]=e_{1n}-e_{1n}=0$.

Case 2. $(i,j)=(n,n)$. Then

$\displaystyle[\alpha(e_{ij}),e_{kl}]+[e_{ij},\alpha(e_{kl})]=-[e_{1n},e_{kl}]+[e_{nn},\alpha(e_{kl})].$

We again have only $2$ non-trivial subcases $(k,l)=(1,1)$ and $(k,l)=(n,n)$
that are similar to the corresponding subcases of Case 1. ∎

We also introduce the linear maps $\beta_{i}:T_{n}(F)\to T_{n}(F)$, $1\leq
i\leq n$, by

$\displaystyle\beta_{i}(e_{jk})=\begin{cases}\delta,&(j,k)=(i,i),\\\
0,&(j,k)\neq(i,i).\end{cases}$

Obviously, $\beta_{i}$, $1\leq i\leq n$, constitute a basis of the space of
linear maps $T_{n}(F)\to Z(T_{n}(F))$ annihilating $[T_{n}(F),T_{n}(F)]$. In
particular, $\beta_{i}\in\Delta(T_{n}(F))$ for all $1\leq i\leq n$.

######  Proposition 10.

Let $n>1$. Then
$\Delta(T_{n}(F))=\langle\mathrm{id},\alpha\rangle\oplus\langle\beta_{i}\mid
1\leq i\leq n\rangle$.

###### Proof.

In view of Lemma 9 we only need to prove that each
$\varphi\in\Delta(T_{n}(F))$ is a linear combination of $\mathrm{id}$,
$\alpha$ and $\beta_{i}$, $1\leq i\leq n$. Setting $a=\varphi(e_{1n})(1,n)$,
we have

$\displaystyle\varphi(e_{ij})=ae_{ij}\text{ for all }1\leq i<j\leq n$ (15)

by Lemma 7 and Lemma 8(ii). Then

$\displaystyle\varphi(e_{ii})(i,i)-\varphi(e_{ii})(j,j)=a=\varphi(e_{ii})(i,i)-\varphi(e_{ii})(k,k)$

for all $k<i<j$ by 12. Hence, defining $b_{i}=\varphi(e_{ii})(j,j)$ for some
$j\neq i$, we have

$\displaystyle\varphi(e_{ii})(j,j)=\begin{cases}a+b_{i},&j=i,\\\ b_{i},&j\neq
i.\end{cases}$ (16)

Finally, let $c=\varphi(e_{11})(1,n)$. By Lemma 6(iii) and Lemma 8(iii) for
all $1\leq i\leq n$ and $1\leq j<k\leq n$ we have

$\displaystyle\varphi(e_{ii})(j,k)=\begin{cases}c,&(i,j,k)=(1,1,n),\\\
-c,&(i,j,k)=(n,1,n),\\\ 0,&(i,j,k)\not\in\\{(1,1,n),(n,1,n)\\}.\end{cases}$
(17)

It follows from 15, 16 and 17 that
$\varphi=a\cdot\mathrm{id}+c\alpha+\sum_{i=1}^{n}b_{i}\beta_{i}$. ∎

######  Theorem 11.

Let $\mathrm{char}(F)=0$ and $n>2$. Then any transposed Poisson algebra
structure on $T_{n}(F)$ is of one of the following two non-isomorphic forms:

  1. (i)

transposed Poisson algebra structure of Poisson type;

  2. (ii)

the orthogonal sum of the transposed Poisson algebra structure

$\displaystyle e_{11}\cdot e_{11}=-e_{11}\cdot e_{nn}=e_{nn}\cdot
e_{nn}=e_{1n}$ (18)

with a transposed Poisson algebra structure of Poisson type.

###### Proof.

Let $\cdot$ be a transposed Poisson algebra structure on $T_{n}(F)$. By
Propositions 10 and 4 for all $1\leq i\leq j\leq n$ there exist
$x_{ij},y_{ij}\in F$ and $\\{z_{ij}^{k}\\}_{k=1}^{n}\subseteq F$, such that

$\displaystyle e_{ij}\cdot
e_{kl}=x_{ij}e_{kl}+y_{ij}\alpha(e_{kl})+\sum_{s=1}^{n}z_{ij}^{s}\beta_{s}(e_{kl})=\begin{cases}x_{ij}e_{kl},&k<l,\\\
x_{ij}e_{kk}+z_{ij}^{k}\delta,&k=l\not\in\\{1,n\\},\\\
x_{ij}e_{11}+y_{ij}e_{1n}+z_{ij}^{1}\delta,&k=l=1,\\\
x_{ij}e_{nn}-y_{ij}e_{1n}+z_{ij}^{n}\delta,&k=l=n.\end{cases}$

Let $1\leq i<j\leq n$ and $1<k<n$. Then $e_{kk}\cdot e_{ij}=x_{kk}e_{ij}$,
while $e_{ij}\cdot e_{kk}=x_{ij}e_{kk}+z_{ij}^{k}\delta$, whence

$\displaystyle x_{kk}=0\text{ and }x_{ij}=z_{ij}^{k}=0\text{ for }1\leq
i<j\leq n\text{ and }1<k<n.$ (19)

Now take $1\leq i<j\leq n$ with $(i,j)\neq(1,n)$. Then $e_{11}\cdot
e_{ij}=x_{11}e_{ij}$, while $e_{ij}\cdot
e_{11}=y_{ij}e_{1n}+z_{ij}^{1}\delta$. Similarly, $e_{nn}\cdot
e_{ij}=x_{nn}e_{ij}$, while $e_{ij}\cdot
e_{nn}=-y_{ij}e_{1n}+z_{ij}^{n}\delta$. Hence,

$\displaystyle x_{11}=x_{nn}=0\text{ and }y_{ij}=z_{ij}^{1}=z_{ij}^{n}=0\text{
for }1\leq i<j\leq n\text{ with }(i,j)\neq(1,n).$ (20)

Considering the products $e_{11}\cdot e_{1n}=0$, $e_{1n}\cdot
e_{11}=y_{1n}e_{1n}+z_{1n}^{1}\delta$, $e_{nn}\cdot e_{1n}=0$ and $e_{1n}\cdot
e_{nn}=-y_{1n}e_{1n}+z_{1n}^{n}\delta$, we have

$\displaystyle y_{1n}=z_{1n}^{1}=z_{1n}^{n}=0.$ (21)

Now, for any $1<i<n$ it follows from $e_{11}\cdot e_{ii}=z_{11}^{i}\delta$,
$e_{ii}\cdot e_{11}=y_{ii}e_{1n}+z_{ii}^{1}\delta$, $e_{nn}\cdot
e_{ii}=z_{nn}^{i}\delta$ and $e_{ii}\cdot
e_{nn}=-y_{ii}e_{1n}+z_{ii}^{n}\delta$ that

$\displaystyle y_{ii}=0,\ z_{ii}^{1}=z_{11}^{i}\text{ and
}z_{ii}^{n}=z_{nn}^{i}\text{ for }1<i<n.$ (22)

Similarly, for $1<i,j<n$, it follows from $e_{ii}\cdot
e_{jj}=z_{ii}^{j}\delta$ and $e_{jj}\cdot e_{ii}=z_{jj}^{i}\delta$ that

$\displaystyle z_{ii}^{j}=z_{jj}^{i}\text{ for }1<i,j<n.$ (23)

Finally, $e_{11}\cdot e_{nn}=-y_{11}e_{1n}+z_{11}^{n}\delta$ and $e_{nn}\cdot
e_{11}=y_{nn}e_{1n}+z_{nn}^{1}\delta$ yield

$\displaystyle y_{nn}=-y_{11}\text{ and }z_{11}^{n}=z_{nn}^{1}.$ (24)

Combining 19, 20, 21, 22, 23 and 24 and denoting $a_{ij}:=z_{ii}^{j}$,
$b=y_{11}$, we see that the only (possibly) non-zero products $e_{ij}\cdot
e_{kl}$ are

$\displaystyle e_{ii}\cdot e_{jj}$ $\displaystyle=a_{ij}\delta,\ 1<i,j<n,$
(25) $\displaystyle e_{11}\cdot e_{11}$ $\displaystyle=be_{1n}+a_{11}\delta,\
e_{nn}\cdot e_{nn}=be_{1n}+a_{nn}\delta,\ e_{11}\cdot e_{nn}=e_{nn}\cdot
e_{11}=-be_{1n}+a_{1n}\delta,$ (26)

where $a_{ij}=a_{ji}$ for all $1\leq i,j\leq n$.

If $b=0$, then $\cdot$ is of Poisson type, so we are in the case (i).
Otherwise, choosing $\phi$ to be the conjugation by
$(b{}^{-1}-1)e_{11}+\delta$, we have

$\displaystyle\phi(e_{ij})=\begin{cases}e_{ij},&1<i\leq j\leq n\text{ or
}i=j=1,\\\ b{}^{-1}e_{ij},&1=i<j\leq n.´\end{cases}$

It follows that, applying $\phi$, we can replace $b\neq 0$ by $b=1$ in 26, and
we come to the form (ii).

To prove that the structures (i) and (ii) are not isomorphic, observe from
[12] that for any automorphism $\phi$ of $(T_{n}(F),[\cdot,\cdot])$

either $\displaystyle\phi(e_{ii})\in e_{ii}+(\langle\delta\rangle\oplus\langle
e_{ij}\mid i<j\rangle)\text{ for all }1\leq i\leq n,$ (27) or
$\displaystyle\phi(e_{ii})\in-
e_{n-i+1,n-i+1}+(\langle\delta\rangle\oplus\langle e_{ij}\mid
i<j\rangle)\text{ for all }1\leq i\leq n.$ (28)

Consider the case 27. Since
$e_{ii}\cdot\delta\in\langle\delta\rangle=Z(T_{n}(F),[\cdot,\cdot])$ and
$e_{ii}\cdot e_{jk}=0$ for all $1\leq i\leq n$ and $1\leq j<k\leq n$ by 25 and
26, any such $\phi$ leads to the product

$\displaystyle
e_{ii}*e_{jj}=\phi(\phi{}^{-1}(e_{ii})\cdot\phi{}^{-1}(e_{jj}))\in\phi(e_{ii}\cdot
e_{jj}+\langle\delta\rangle)=\phi(e_{ii}\cdot e_{jj})+\langle\delta\rangle,$

where $\phi(e_{ii}\cdot e_{jj})$ belongs either to $\langle\delta\rangle$
(whenever $e_{ii}\cdot e_{jj}\in\langle\delta\rangle$) or to $\pm
b\phi(e_{1n})+\langle\delta\rangle$ (whenever $e_{ii}\cdot e_{jj}\in\pm
be_{1n}+\langle\delta\rangle$). But $\phi(e_{1n})$ is a non-zero multiple of
$e_{1n}$ by 4, so

$\displaystyle e_{ii}*e_{jj}\in\langle\delta\rangle\Leftrightarrow e_{ii}\cdot
e_{jj}\in\langle\delta\rangle.$ (29)

for all $1\leq i,j\leq n$. This shows that any structure of type (i) can be
isomorphic only to a structure of type (i). As to the case 28, we similarly
have

$\displaystyle
e_{ii}*e_{jj}=\phi(\phi{}^{-1}(e_{ii})\cdot\phi{}^{-1}(e_{jj}))\in\phi(e_{n-i+1,n-i+1}\cdot
e_{n-j+1,n-j+1}+\langle\delta\rangle).$

However

$\displaystyle e_{n-i+1,n-i+1}\cdot e_{n-j+1,n-j+1}\in\langle\delta\rangle$
$\displaystyle\Leftrightarrow(i,j)\not\in\\{(1,1),(1,n),(n,1),(n,n)\\}\Leftrightarrow
e_{ii}\cdot e_{jj}\in\langle\delta\rangle.$

Thus, we again conclude that 29 holds for all $1\leq i,j\leq n$ showing that
the structures (i) and (ii) cannot be isomorphic.

Conversely, it is directly verified that 18 is a transposed Poisson algebra
structure on $T_{n}(F)$, and it is orthogonal to any transposed Poisson
algebra structure of Poisson type on $T_{n}(F)$. ∎

The following Theorem 12 describes transposed Poisson structures on
$T_{2}(F)$, and its proof is similar to that of Theorem 11 with the only
difference that as a result we will have $3$ types of the structures, and
inside each of the $3$ types the structures can be fully classified up to
isomorphism. On the other hand, $T_{2}(F)$ is isomorphic to the
$3$-dimensional Lie algebra $\mathfrak{g}_{2}^{0}$ from [2] whose transposed
Poisson structures were fully described in [2, Proposition 14]. By this
reason, we have chosen to omit the proof of Theorem 12 and indicate on the
left of each of the found structures its isomorphic version from [2].

######  Theorem 12.

Let $\mathrm{char}(F)=0$. Then any transposed Poisson algebra structure on
$T_{2}(F)$ is isomorphic to exactly one of the following structures:

  1. (i)

transposed Poisson algebra structure of Poisson type:

    1. (a)

$\mathrm{T}_{09}^{0,0}:$ the trivial one;

    2. (b)

$\mathrm{T}_{17}^{0}:\ e_{11}\cdot e_{11}=\delta$;

    3. (c)

$\mathrm{T}_{10}^{0}:\ e_{11}\cdot e_{11}=-e_{11}\cdot e_{22}=e_{22}\cdot
e_{22}=\delta$;

  2. (ii)

the orthogonal sum of the transposed Poisson algebra structure

$\displaystyle e_{11}\cdot e_{11}=-e_{11}\cdot e_{22}=e_{22}\cdot
e_{22}=e_{12}$

with a transposed Poisson algebra structure of Poisson type, which results in
the structures:

    1. (a)

$\mathrm{T}_{16}:\ e_{11}\cdot e_{11}=-e_{11}\cdot e_{22}=e_{22}\cdot
e_{22}=e_{12}$;

    2. (b)

$\mathrm{T}_{18}:\ e_{11}\cdot e_{11}=e_{12}+\delta$, $-e_{11}\cdot
e_{22}=e_{22}\cdot e_{22}=e_{12}$;

    3. (c)

$\mathrm{T}_{11}^{0}:\ e_{11}\cdot e_{11}=-e_{11}\cdot e_{22}=e_{22}\cdot
e_{22}=e_{12}+\delta$;

  3. (iii)

the (non-orthogonal) sum of a transposed Poisson algebra structure of the
family

$\displaystyle e_{11}\cdot e_{11}=ce_{11},e_{11}\cdot e_{12}=-e_{12}\cdot
e_{22}=ce_{12},e_{11}\cdot e_{22}=-e_{22}\cdot e_{22}=ce_{22},\ c\neq 0,$

with a transposed Poisson algebra structure of Poisson type, which results in
the structures:

    1. (a)

$\mathrm{T}_{17}^{-c}:\ e_{11}\cdot e_{11}=ce_{11},\ e_{11}\cdot
e_{12}=-e_{12}\cdot e_{22}=ce_{12},\ e_{11}\cdot e_{22}=-e_{22}\cdot
e_{22}=ce_{22}$;

    2. (b)

$\mathrm{T}_{09}^{0,-c}:\ e_{11}\cdot e_{11}=ce_{11},\ e_{11}\cdot
e_{12}=-e_{12}\cdot e_{22}=ce_{12},\ e_{11}\cdot e_{22}=ce_{22},\ e_{22}\cdot
e_{22}=-c(e_{22}+\delta)$;

    3. (c)

$\mathrm{T}_{19}^{-c}:\ e_{11}\cdot e_{11}=ce_{11},\ e_{11}\cdot
e_{12}=-e_{12}\cdot e_{22}=ce_{12},\ -e_{11}\cdot e_{22}=e_{22}\cdot
e_{22}=ce_{11}$.

##  3\. Transposed Poisson structures on the full matrix Lie algebra

In this short section we describe transposed Poisson structures on the full
matrix algebra $M_{n}(F)$. As above, we denote by $e_{ij}$, $1\leq i,j\leq n$,
the matrix units and by $\delta$ the identity matrix. Recall that

$\displaystyle Z(M_{n}(F))$ $\displaystyle=\langle\delta\rangle,$
$\displaystyle[M_{n}(F),M_{n}(F)]$ $\displaystyle=\mathfrak{sl}_{n}(F)=\\{a\in
M_{n}(F)\mid\mathrm{tr}(a)=0\\}$ $\displaystyle=\langle e_{ij}\mid 1\leq i\neq
j\leq n\rangle\oplus\langle e_{11}-e_{ii}\mid 1<i\leq n\rangle.$

Denote by $\Delta(M_{n}(F))$ the space of $\frac{1}{2}$-derivations of the Lie
algebra $(M_{n}(F),[\cdot,\cdot])$. The linear map $\gamma:M_{n}(F)\to
M_{n}(F)$ given by

$\displaystyle\gamma(e_{ij})=\begin{cases}\delta,&i=j,\\\ 0,&i\neq
j,\end{cases}$

belongs to $\Delta(M_{n}(F))$ as a linear map $M_{n}(F)\to Z(M_{n}(F))$
annihilating $[M_{n}(F),M_{n}(F)]$.

######  Proposition 13.

Let $n>1$. Then $\Delta(M_{n}(F))=\langle\mathrm{id},\gamma\rangle$.

###### Proof.

It is easy to see that, as a Lie algebra, $M_{n}(F)$ is the direct sum
$[M_{n}(F),M_{n}(F)]\oplus Z(M_{n}(F))$ Namely, every $a\in M_{n}(F)$
decomposes uniquely as

$\displaystyle
a=\left(a-\frac{\mathrm{tr}(a)}{n}\delta\right)+\frac{\mathrm{tr}(a)}{n}\delta.$

Since $\frac{1}{2}$-derivations of a Lie algebra $\mathfrak{L}$ leave
$[\mathfrak{L},\mathfrak{L}]$ and $Z(\mathfrak{L})$ invariant, we have

$\displaystyle\Delta(M_{n}(F))=\Delta([M_{n}(F),M_{n}(F)])\oplus\Delta(Z(M_{n}(F))).$

By [5, Corollary 3] any $\varphi\in\Delta([M_{n}(F),M_{n}(F)])$ is trivial.
Since $\dim(Z(M_{n}(F)))=1$, any $\varphi\in\Delta(Z(M_{n}(F)))$ is trivial as
well. Hence, given $\varphi\in\Delta(M_{n}(F))$, there exist $k_{1},k_{2}\in
F$ such that
$\varphi=k_{1}\cdot\mathrm{id}_{[M_{n}(F),M_{n}(F)]}+k_{2}\cdot\mathrm{id}_{Z(M_{n}(F))}$.
Clearly,

$\displaystyle\varphi(e_{ij})$ $\displaystyle=k_{1}e_{ij},\text{ if }i\neq j,$
$\displaystyle\varphi(e_{ii})$
$\displaystyle=k_{1}\left(e_{ii}-\frac{1}{n}\delta\right)+\frac{k_{2}}{n}\delta=k_{1}e_{ii}+\frac{k_{2}-k_{1}}{n}\delta.$

It follows that
$\varphi=k_{1}\cdot\mathrm{id}_{M_{n}(F)}+\frac{k_{2}-k_{1}}{n}\cdot\gamma$.
Thus, $\Delta(M_{n}(F))\subseteq\langle\mathrm{id},\gamma\rangle$. The
converse inclusion is trivial. ∎

######  Theorem 14.

Let $\mathrm{char}(F)=0$ and $n>1$. Then, up to isomorphism, there is only one
non-trivial transposed Poisson algebra structure on $M_{n}(F)$. It is given by

$\displaystyle e_{ii}\cdot e_{jj}=\delta,\ 1\leq i,j\leq n,$ (30)

and it is of Poisson type.

###### Proof.

Let $\cdot$ be a transposed Poisson algebra structure on $M_{n}(F)$. By
Propositions 13 and 4 for all $1\leq i,j\leq n$ there exist $x_{ij}$ and
$y_{ij}\in F$, such that

$\displaystyle e_{ij}\cdot
e_{kl}=x_{ij}e_{kl}+y_{ij}\gamma(e_{kl})=\begin{cases}x_{ij}e_{kl},&k\neq
l,\\\ x_{ij}e_{kk}+y_{ij}\delta,&k=l.\end{cases}$

Let $i\neq j$ and $1\leq k\leq n$. Then $e_{kk}\cdot e_{ij}=x_{kk}e_{ij}$,
while $e_{ij}\cdot e_{kk}=x_{ij}e_{kk}+y_{ij}\delta$, whence

$\displaystyle x_{kk}=0\text{ and }x_{ij}=y_{ij}=0\text{ for }i\neq j\text{
and }1\leq k\leq n.$

It follows from $e_{ii}\cdot e_{jj}=y_{ii}\delta$ and $e_{jj}\cdot
e_{ii}=y_{jj}\delta$ that

$\displaystyle y_{ii}=y_{jj}\text{ for }1\leq i,j\leq n.$

Thus, denoting $y_{ii}$ by $c$, we have the only (possibly) non-zero products

$\displaystyle e_{ii}\cdot e_{jj}=c\delta,\ 1\leq i,j\leq n.$

If $c=0$, then $\cdot$ is trivial. Otherwise, choosing the automorphism $\phi$
of $(M_{n}(F),[\cdot,\cdot])$ given by

$\displaystyle\phi(e_{ij})$ $\displaystyle=e_{ij},\text{ if }i\neq j,$
$\displaystyle\phi(e_{ii})$
$\displaystyle=e_{ii}-\frac{1}{n}\delta+\frac{c}{n}\delta=e_{ii}+\frac{c-1}{n}\delta,$

we obtain the isomorphic structure 30.

It remains to see that 30 is of Poisson type. For all $a,b\in M_{n}(F)$ we
have

$\displaystyle a\cdot b=\sum_{i,j=1}^{n}a(i,i)a(j,j)e_{ii}\cdot
e_{jj}=\sum_{i,j=1}^{n}a(i,i)a(j,j)\delta=\mathrm{tr}(a)\mathrm{tr}(b)\delta=\frac{\mathrm{tr}(a)}{n}\delta\cdot\frac{\mathrm{tr}(b)}{n}\delta.$

Hence, $\cdot$ is the extension by zero of a product on the complement
$Z(M_{n}(F))$ of $[M_{n}(F),M_{n}(F)]$ in $M_{n}(F)$ with values in
$Z(M_{n}(F))$. ∎

######  Remark 15.

The structure 30 is isomorphic to the extension by zero of the product
$\delta\cdot\delta=\delta$.

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