#  Supernova Archaeology with X-Ray Binary Winds - The Case of GRO J1655-40

Noa Keshet  Department of Physics, Technion, Haifa 32000, Israel  Ehud Behar
Department of Physics, Technion, Haifa 32000, Israel  Timothy R. Kallman  NASA
Goddard Space Flight Center, Code 662, Greenbelt, MD 20771, USA

###### Abstract

Supernovae are responsible for the elemental enrichment of the galaxy and some
are postulated to leave behind a black hole. In a stellar binary system the
supernova pollutes its companion, and the black hole can accrete back its own
debris and emit X-rays. In this sequence of events, which is only poorly
understood, winds are ejected, and observed through X-ray absorption lines.
Measuring abundances of elements in the wind can lead to inferences about the
historical explosion and possibly identify the long-gone progenitor of the
compact object. Here, we re-analyze the uniquely rich X-ray spectrum of the
2005 outburst of GRO J1655-40. We reconstruct the absorption measure
distribution (AMD) of the wind, and find that it increases sharply with
ionization from H-like O up to H-like Ca, and then flattens out. The AMD is
then used to measure relative abundances of 18 different elements. The present
abundances are in partial agreement with a previous work with discrepancies
mostly for low-Z elements. The overabundance of odd-Z elements hints at a
high-metallicity, high-mass ($\simeq 25\,M_{\odot}$) progenitor.
Interestingly, the abundances are different from those measured in the
companion atmosphere, indicating that the wind entrains lingering ambient
supernova debris. This can be expected since the current total stellar mass of
the binary ($<10\,M_{\odot}$) is much less than the progenitor mass.

††facilities:  ††software: Xspec (Arnaud, 1996), XSTAR (Bautista & Kallman,
2001)

##  1 Introduction

It has long been understood that the source of most elements in the universe
are stellar cores that explode in a supernova (SN). Only a handful of elements
can be measured in the limited visible band observed immediately after SN
explosions. Measuring abundances in X-ray spectra of SN remnants can reveal
some elements, hint at the progenitor, and test SN models (Siegel et al.,
2020; Vink, 2012). However, a full account of all elements dispersed by the SN
has not been obtained from observations. Here, we indirectly study a
historical SN by observing the low-mass X-ray binary GRO J1655-40 (also known
as Nova Scorpii 1994). GRO J1655-40 is an X-ray binary at a distance of 3.2
kpc with a 6.6 $M_{\odot}$ black hole (BH) and a low mass 2.7 $M_{\odot}$
F-star companion in a 2.6 day period (Casares & Jonker, 2014). The implied
semi-major axis of the binary is $a=1.1\times 10^{12}$ cm.

In 2005, GRO J1655-40 went into outburst and the ensuing outflow produced a
rich X-ray spectrum measured with the Chandra/HETG gratings, which has been
the focus of several works (Miller et al., 2006, 2008; Kallman et al., 2009;
Fukumura et al., 2017; Tomaru et al., 2023). The spectrum features a multitude
of blue shifted lines from many elements, and many ions ranging from H-like O
to H-like Fe. This likely indicates a range of ionization parameter
$\xi=L/(nr^{2})$ values, where $L$ is the ionizing luminosity, $n$ the number
density, and $r$ the distance from the source. $\xi$ represents the balance
between photo-ionization (flux $\propto L/r^{2}$) and recombination ($\propto
n$) rates. Miller et al. (2008) identified most of the lines in the spectrum,
and measured their velocities. Their paper focuses on the absorption lines
from Fe meta-stable levels that provide density diagnostics. The high density
was used (via $\xi$) to rule out thermal wind driving. Other works suggested
that thermal driving might still be viable (Neilsen & Homan, 2012; Tomaru et
al., 2023). Kallman et al. (2009) performed a global spectral fit of a photo-
ionized plasma model. Their best fit model (No. 6) yields an outflow velocity
of $-375$ km s-1, a total column density of $N_{\rm H}=10^{24}$ cm-2, and
mostly solar elemental abundances using a single ionization component of
$\xi=10^{4}$ erg cm s-1. Tomaru et al. (2023) fitted the Chandra/HETG outburst
spectrum with two photoionized components of $\log\xi=4.1,3.4$ (cgs units here
and throughout the paper), and with a total column density $N_{\rm H}=5\times
10^{24}$ cm-2. No global fitted model is statistically satisfactory, due to
the exquisite quality of the Chandra/HETG spectrum.

The present work utilizes a different approach of ion-by-ion column density
fitting (Behar et al., 2003; Holczer et al., 2007; Peretz et al., 2018),
complemented by individual line measurements and curve of growth analysis.
These column densities are used to reconstruct of the ionization distribution
of the wind.

##  2 Observations and Data

GRO J1655-40 was observed with the Chandra/HETG gratings during an outburst on
2005, April 1. The observation, Id 5461, has a total exposure time of 26 ks
and a photon count of $4.2\times 10^{6}$. The data were downloaded from the
TGCAT archive (Huenemoerder et al., 2011) in fully reduced form. The spectrum
is uniquely rich between 1.5 - 20 Å, and includes numerous absorption lines
from eighteen different elements, many of which are rarely seen in
astrophysical spectra, such as Na, Al, P, Cl, and K, see Fig. 1. The complex
spectrum calls for the ion-by-ion and line-by-line analysis employed in this
work. We are able to identify lines from H-like ions of sixteen different
elements, He-like ions of eight elements and the L shell ions Fe21-Fe23,
Ni+25, and Co+24.

Figure 1: Segments from the Chandra/HETG spectrum of the 2005 outflow of GRO
J1655-40, zooming into the H-like Ly$\alpha$ lines of cosmically rare odd-Z
elements. Dashed lines mark laboratory wavelengths.

##  3 Method

Elemental abundances are measured through the relative strengths of the
absorption lines, which are determined by the column density $N$, the amount
of absorbing material expressed as an integral of density $n$ over the line of
sight $N=\int ndr$. The ionic column densities $N_{\rm ion}$ are a product of
the total (mostly H) column $N_{\rm H}$, the elemental abundance $A_{\rm Z}$,
and the ion fractional abundance $f_{\rm ion}$. The outflowing gas includes a
range of ionization ($\xi$), and each ion can emit from different $\xi$
regions. We therefore need to consider the contributions to $N_{\rm ion}$ over
a range of $\xi$ values:

$N_{\rm ion}=A_{\rm Z}\int f_{\rm ion}\frac{dN_{\rm H}}{d\log\xi}d\log\xi$ (1)

where $dN_{\rm H}/d\log\xi$ is the Absorption Measure Distribution (AMD,
Holczer et al., 2007), and $f_{\rm ion}(\xi)$ values need to be taken from
photoionization balance calculations, using the GRO J1655-40 ionizing spectrum
(Kallman et al., 2009). $N_{\rm ion}$ is related to the line optical-depth
profile through the relation $\tau(E)=\sigma(E)N_{\rm ion}$, where $E=h\nu$ is
the line photon energy, and $\sigma(E)$ is the quantum mechanical line-
absorption cross section. The AMD function is recovered by requiring that it
yields the correct $N_{\rm ion}$ in Eq. 1 for all ions.

###  3.1 Ionic column densities

$N_{\rm ion}$ is measured using an ion-by-ion code that can fit simultaneously
all lines of each ion, and all ions at once (Peretz et al., 2018). The free
parameters of the model are $N_{\rm ion}$ of each ion, and the global
parameters of outflow velocity, and thermal and turbulent broadening.

Fitting many lines from the same ion constrains the line broadening through
all of their curves of growth, even when they are unresolved. Additionally,
fitting many ions together accounts for blends of lines of different ions.
Fig. 2 shows our best fit ion-by-ion model in the 6.2-12.4 Å band. This is the
richest part of the spectrum, which includes K-shell lines of Si, Al, Mg, Na,
and Ne as well as L-shell lines of Fe, Ni and Co. The (Balmer) series of Fe+23
2s-np lines is highlighted. Fitting so many observed lines of a single ion
simultaneously tightly constrains $N_{\rm ion}$ (see Table 1). The atomic data
are calculated with the HULLAC code (Bar-Shalom et al., 2001) and corrected
for laboratory wavelengths.

Figure 2: The 6.2-12.4 Å segment of the Chandra/HETG spectrum, fitted with an
ion-by-ion absorption model. The series of Fe+23 2s-np lines is marked, and is
fitted well by the model. Nion values from this model are listed in Table 1.

For Ly$\alpha$ lines of the rare odd-Z elements, which are not included in the
Peretz et al. (2018) database, we measure individual-line equivalent widths.
The equivalent width ($EW$) is the integral over the absorption line profile
$EW=\int(1-e^{-\tau(E)})dE$, which is measured directly for each line. The
relation between $EW$ and $N_{\rm ion}$ is called the curve of growth, and
yields $N_{\rm ion}$. The advantage of individual line fitting is the superior
constraint on the local continuum.

A theoretical curve of growth is calculated based on $\sigma(E)$ of each line
which takes on the Voigt profile. The Voigt profile depends on the natural
width and the measured Doppler velocity width of $v_{\rm
turb}=\sqrt{2}\sigma_{g}=70\,\rm km\,s^{-1}$, where $\sigma_{g}$ is the
Gaussian standard deviation. $v_{\rm turb}$ is fitted with the ion-by-ion
model. As an example, the curve of growth of Ar Ly$\alpha$, a deep and
isolated doublet is given in Fig. 3. The flattening of the curve of growth is
due to the saturation of the center of the line, where $\sigma(E)$ peaks, and
absorption eliminates 100% of the flux. This flattening results in relatively
large uncertainties in $N_{\rm ion}$ (and consequently the elemental
abundances), even when the $EW$ is accurately measured (see Fig. 3). All
currently measured lines are listed in Table 2, along with their identified
ions and $EW$.

Figure 3: Theoretical curve-of-growth plot of the unresolved H-like Ar
Ly$\alpha$ doublet. Each line of the Ly$\alpha$ doublet is calculated
separately with its atomic data and for $v_{\rm turb}$ = 70 km s-1, assuming a
Voigt profile. The sum of both lines is also plotted, and then compared with
the measured EW, represented by the horizontal solid red line; the dashed
lines represent the margin of 90% certainty, which then corresponds to the
$N_{\rm ion}$ confidence region (vertical lines). $N_{\rm ion}$ found by the
ion-by-ion fit (grey region) is consistent with the curve of growth analysis.

Wherever possible, $N_{\rm ion}$ was measured with ion-by-ion fitting (Fig. 2)
and directly (Fig. 3) to validate consistency. The values quoted in Table 1
are those measured with ion-by-ion fitting, when available. Only for Mg+11 and
Si+13, the ion-by-ion fit underestimates the direct measurement, and the
$N_{\rm ion}$ values quoted (Table 1) are those measured directly. These
discrepancies are mainly due to the semi-global fit not accurately capturing
the local continuum. Although fitting a series of lines together can provide
stronger constraints on Nion, the local continuum may deviate from the data,
introducing uncertainties. Fitting line profiles individually allows for a
better fit to the continuum.

All lines in the spectrum share approximately the same kinematics of blue-
shift of $-400\,\rm km\,s^{-1}$ and broadening of $v_{\rm turb}=70\,\rm
km\,s^{-1}$ expect H-like Fe). The third-order HETG spectrum resolves the
Fe+24 resonance line ($r$) from the intercombination ($i$) line ($r$ at 1.85Å
and $i$ at 1.86Å), which is blended with the Fe+23 K$\alpha$ line, see Fig. 4.
When fitting these lines, we allowed separate continua for the two orders, but
confirmed that all lines $EW$s are consistent. This allows for an accurate
measurement of the Fe+24 ion column density. $N_{\rm ion}$ for $\rm Fe^{+23}$
is measured by fitting the series of its 2s-np lines (see Fig. 2). Since Fe+25
has a much higher outflow velocity, at $\sim-1200\,\rm km\,s^{-1}$, and
$v_{\rm turb}=1300\,\rm km\,s^{-1}$, it is likely a separate component of the
outflow (Kallman et al., 2009), and we exclude it from the AMD analysis.
Fitting the Fe+25 doublet with two Voigt profiles in the third order confirms
these higher outflow and turbulent velocities. All final $N_{\rm ion}$ values
are detailed at Tabel 1, and plotted in the top panel of Fig. 5.

Figure 4: HEG first- (black) and third- (blue data, model in red) order
spectra. The rest wavelengths (left to right, respectively) of Fe+25
Ly$\alpha$ doublet, Fe+24 He$\alpha$ $r$ and $i$, and Fe+23 K$\alpha$ are
marked. The latter two are blended even in third order. Note there are
differences in the continuum between orders, which is likely an issue of the
absolute calibration at these extreme low wavelengths.

###  3.2 Reconstructing the AMD

Obtaining relative elemental abundances from the various $N_{\rm ion}$ values
requires considering that $N_{\rm H}$ in Eq. (1) can depend on the ionization
parameter $\xi$. The functional form of the AMD is unknown apriori. Here, the
AMD is constructed from local segments of power-laws by requiring that they
give the same element abundance $A_{\rm Z}$ for all pairs of H-like and He-
like ions (see Eq. 1). The individual segments below $\log\xi=3.3$ all
indicate a steeply rising trend. Hence for simplicity, we assumed a broken
power-law AMD (Fig. 5) between $\log\xi=0-5$, where the slopes and break
points are the free parameters. By iterations, each time obtaining $A_{Z}$
from Eq. 1 for all ions, we seek the best ($A_{Z}$) match between ion pairs.
We use $N_{\rm ion}$ measured values from Sec. 3.1, and $f_{\rm ion}(\xi)$
from Kallman et al. (2009) in Eq. 1. This process eventually results in the
AMD presented in the bottom panel of Fig. 5 and the best abundance estimates.

In Fig. 5 (top panel), we plot measured $N_{\rm ion}$ values for all ions in
the spectrum, where each ion is plotted at its maximal formation position
(log$\xi_{\rm max}$, see Table 1). The relative positions of ion pairs on the
(top) plot of Fig. 5 indicate the local slope of the AMD hinting at a steep
distribution. The bottom panel of Fig. 5 shows the AMD that best produces the
$N_{\rm ions}$ values in the top panel. It sharply increases with $\xi$ up to
$\log\xi=3.3$ and then levels off or even slightly decreases. There are no ion
pairs below $\log\xi=1.6$ or above $\log\xi=4.0$, hence the AMD is less
tightly constrained in these regions. The fact that we see no sign of
decreasing AMD towards high ionization is consistent with previous models
(Shidatsu et al., 2016; Neilsen et al., 2016; Fukumura et al., 2017) that
require fully ionized gas in the inner parts of the wind, which does not
absorb in lines, but makes it Compton-thick. In the case that the AMD extends
up to $\log\xi=5$ we get a total column density of $N_{\rm H}=6.2\times
10^{23}$ cm-2 that does not include the fast H-like Fe component. This is
consistent with the models of Kallman et al. (2009). Tomaru et al. (2023)
report $N_{\rm H}$ that is an order of magnitude higher but it refers to the
fast H-like Fe component. They also invoke partial covering of the X-ray
source, which exponentially raises $N_{\rm H}$.

Figure 5: (top panel) Distribution of ionic column densities $N_{\rm ion}$,
each plotted at its $\xi$ of maximal formation $\xi_{\rm max}$, see Table 1.
This outlines the overall distribution of $N_{\rm H}$ with $\xi$ in the GRO
J1655-40 outflow. Dashed lines represent the local slopes from H-like and He-
like ion pairs, which are then used to produce the continuous distribution,
i.e. the AMD, (bottom panel) With no actual H in the spectrum, H-like Ca is
arbitrarily chosen as the reference point. The AMD drastically rises up to
$\log\xi=3.3$, and slightly decreases towards high $\xi$. The integral over
the AMD yields total column density of $6.2\times 10^{23}$ cm-2.

The AMD of the wind points to two different regions. A high-ionization flat
component at $\log\xi>3.3$ and a low-ionization steep one (Fig. 5). High
ionization is typical of X-ray binary outflows, while low ionization is rare
(Diaz Trigo & Boirin, 2016). It has been shown that the slope of the AMD
$\sim\xi^{a}$ maps exactly the line-of-sight radial wind-density profile
$n(r)$ (Behar, 2009, Eq. 7 therein). The high-ionization flat ($a\approx 0$)
distribution of GRO J1655-40 is akin to those observed in active galactic
outflows (Keshet & Behar, 2022). Such an AMD is produced by a shallow
$n(r)\propto r^{-1}$ density profile, which can be due to a magnetic disk wind
(Fukumura et al., 2017). Further out, the wind interacts with circumstellar
gas, and becomes more spherical, approaching $n(r)\propto r^{-2}$ (Fukumura et
al., 2010). The steep AMD ($\log\xi<3$) could be this more extended region,
where $n$ decreases more strongly with $r$.

##  4 Elemental Abundances

Once the AMD profile is solved in terms of $N_{\rm H}$, the relative
abundances are determined by the factor $A_{Z}$ applied to Nion that is
required to match the AMD (see Eq. 1). For each element, the abundance
measurement is taken as the weighted average from all available ions. This
vertical scaling between the various elements represents their relative
abundances. The final abundances of eighteen elements in the outflow of GRO
J1655-40 are listed in Table 3. As far as we know, this sets a record for any
astrophysical measurement, mostly because the visible and UV bands do not
contain spectral lines of as many elements as the soft X-ray band. Abundances
in Table 3 are given relative to their solar values (Asplund et al., 2009).
The X-ray measured abundances of the wind are compared in the table to the
photospheric values of the companion (González Hernández et al., 2008), and to
the values previously measured with a global fit (Kallman et al., 2009). Since
no H lines can be observed in X-rays, and since the Ca/H abundance ratio is
solar in the photosphere, all abundances in Table 3 are listed assuming Ca is
solar.

Table 1: Ions identified in the HETG spectrum of GRO J1655-40 and used for abundance estimates. Values in the last two columns are taken from photo-ionization models (Kallman et al., 2009). All uncertainties represent 90% confidence.  Ion | $N_{\rm ion}$ | log$\xi_{\rm max}$ | $f_{\rm ion}(\xi_{\rm max})$
---|---|---|---
|  $10^{17}$[cm-2] | [erg $\,\rm s^{-1}$cm] |
$\rm O^{+7}$ | $1.1\pm 1.0$ | 1.1 | 0.54
$\rm Ne^{+9}$ | $4.2\pm 4.0$ | 1.5 | 0.55
$\rm Na^{+10}$ | $0.22\pm 0.09$ | 1.6 | 0.66
$\rm Mg^{+11}$ | $2.3\pm 0.2$ | 1.9 | 0.53
$\rm Mg^{+10}$ | $0.016\pm 0.007$ | 1.6 | 0.27
$\rm Al^{+12}$ | $0.40\pm 0.04$ | 2.0 | 0.64
$\rm Si^{+13}$ | $8\pm 2$ | 2.3 | 0.55
$\rm Si^{+12}$ | $0.3\pm 0.1$ | 1.9 | 0.30
$\rm P^{+14}$ | $0.11\pm 0.10$ | 2.3 | 0.66
$\rm S^{+15}$ | $11\pm 3$ | 2.6 | 0.54
$\rm S^{+14}$ | $0.22\pm 0.05$ | 2.2 | 0.34
$\rm Cl^{+16}$ | $0.3\pm 0.1$ | 2.6 | 0.69
$\rm Ar{+17}$ | $2.6\pm 0.3$ | 3.0 | 0.52
$\rm Ar^{+16}$ | $0.24\pm 0.04$ | 2.5 | 0.41
$\rm K^{+18}$ | $0.11\pm 0.10$ | 2.9 | 0.71
$\rm Ca^{+19}$ | $3.6\pm 0.9$ | 3.3 | 0.55
$\rm Ca{+18}$ | $0.5\pm 0.1$ | 2.8 | 0.41
$\rm Ti{+21}$ | $0.39\pm 0.36$ | 3.5 | 0.67
$\rm Cr{+23}$ | $3.0\pm 1.0$ | 4.0 | 0.67
$\rm Cr^{+22}$ | $2.4\pm 1.7$ | 3.4 | 0.35
$\rm Mn^{+24}$ | $1.1\pm 0.5$ | 4.1 | 0.71
$\rm Mn^{+23}$ | $1.6\pm 0.4$ | 3.6 | 0.37
$\rm Fe^{+25}$ | $9.0\pm 1.0$ | 4.3 | 0.52
$\rm Fe^{+24}$ | $20\pm 15$ | 3.7 | 0.69
$\rm Fe^{+23}$ | $12\pm 2$ | 3.3 | 0.19
$\rm Co^{+24}$ | $0.11\pm 0.06$ | 3.6 | 0.14
$\rm Ni^{+25}$ | $3.0\pm 0.6$ | 3.7 | 0.18

The abundances obtained by a different analysis approach of the same X-ray
spectrum were published by Kallman et al. (2009). Most of our abundances agree
with their results within the uncertainties. The significant discrepancies up
to a factor of 2.8 are for low-Z (atomic number) elements below P (Z$<15$).
This is a result of the systematic difference of the two approaches. Kallman
et al. (2009) used a single ionization component with $\log\xi=4$, where the
fractional abundances of low-Z ions are. Therefore, high elemental abundances
were required to fit the spectrum. The current analysis with an AMD extending
to the low $\xi$ region fits the spectrum with lower abundances. The ion-by-
ion method allows more flexibility in fitting the spectrum, particularly weak
lines that the global fit is not sensitive to. This results for the most part
in smaller uncertainties and well constrained odd-Z abundances, which are
important in comparing with SN models (see Sec. 5). Kallman et al. (2009)
mentioned a possible thermal instability at $\log\xi\approx 2$. Considering
the same AMD reconstructed in Sec. 3.2 with a cut-off at $\log\xi=2$ produces
the same measured abundances, with negligible differences.

Abundance measurements from the companion atmosphere based on optical
observations (González Hernández et al., 2008) are also listed in Table 3. The
abundances of O, Mg, Si, and S, which are supersolar in the photosphere, are
subsolar in the wind. An earlier photospheric measurement reports Mg, Si, and
S solar abundances (Foellmi et al., 2007).

Table 2: Lines measured in the HETG spectrum of GRO J1655-40 and used for
abundance estimates. All uncertainties are 90% confidence. Laboratory energies
are taken from the NIST database (Kramida et al., 2023).

Line | Measured Energy | Laboratory Energy | $EW$
---|---|---|---
| [keV] | [keV] |  $10^{-3}$[keV]
$\rm O^{+7}$ Ly$\alpha$ | 0.6547 | 0.6537 | $0.8\pm 0.3$
$\rm O^{+7}$ Ly$\beta$ | 0.7756 | 0.7747 | $0.7\pm 0.1$
$\rm Ne^{+9}$ Ly$\alpha$ | 1.0238 | 1.0218 | $2\pm 1$
$\rm Ne^{+9}$ Ly$\beta$ | 1.2124 | 1.2111 | $1.38\pm 0.05$
$\rm Na^{+10}$ Ly$\alpha$ | 1.2382 | 1.2368 | $0.7\pm 0.2$
$\rm Na^{+10}$ Ly$\beta$ | 1.4680 | 1.4658 | $0.21\pm 0.04$
$\rm Mg^{+11}$ Ly$\alpha$ | 1.4743 | 1.4723 | $2.45\pm 0.06$
$\rm Mg^{+11}$ Ly$\beta$ | 1.7480 | 1.7450 | $2.0\pm 0.8$
$\rm Mg^{+10}$ He$\alpha$ | 1.3533 | 1.3538 | $0.13\pm 0.12$
$\rm Al^{+12}$ Ly$\alpha$ | 1.7318∗ | 1.7285 | $2.69\pm 0.04$
$\rm Si^{+13}$ Ly$\alpha$ | 2.0087 | 2.0054 | $5.2\pm 0.2$
$\rm Si^{+13}$ Ly$\beta$ | 2.3809 | 2.3768 | $2.6\pm 0.2$
$\rm Si^{+12}$ He$\alpha$ | 1.8673 | 1.8671 | $1.3\pm 0.5$
$\rm P^{+14}$ Ly$\alpha$ | 2.3061 | 2.3031 | $0.46\pm 0.45$
$\rm P^{+14}$ Ly$\beta$ | 2.7340 | 2.7296 | $0.11\pm 0.10$
$\rm S^{+15}$ Ly$\alpha$ | 2.6249 | 2.6215 | $8.0\pm 0.7$
$\rm S^{+15}$ Ly$\beta$ | 3.1122 | 3.1069 | $2.8\pm 0.3$
$\rm S^{+14}$ He$\alpha$ | 2.4649 | 2.4635 | $1.3\pm 0.2$
$\rm Cl^{+16}$ Ly$\alpha$ | 2.9659 | 2.9608 | $1.2\pm 0.3$
$\rm Cl^{+16}$ Ly$\beta$ | 3.5050 | 3.5090 | $0.8\pm 0.5$
$\rm Ar{+17}$ Ly$\alpha$ | 3.3262 | 3.3209 | $5.7\pm 0.3$
$\rm Ar{+17}$ Ly$\beta$ | 3.9404 | 3.9359 | $2.6\pm 0.5$
$\rm Ar^{+16}$ He$\alpha$ | 3.1447 | 3.1435 | $1.5\pm 0.2$
$\rm K^{+18}$ Ly$\alpha$ | 3.700 | 3.7020 | $0.5\pm 0.4$
$\rm Ca^{+19}$ Ly$\alpha$ | 4.1123 | 4.1042 | $8\pm 1$
$\rm Ca{+19}$ Ly$\beta$ | 4.8715 | 4.8642 | $4.1\pm 0.8$
$\rm Ca{+18}$ He$\alpha$ | 3.9086 | 3.9076 | $3.1\pm 0.4$
$\rm Ti{+21}$ Ly$\alpha$ | 4.9739 | 4.9718 | $1.6\pm 1.4$
$\rm Cr{+23}$ Ly$\alpha$ | 5.9381 | 5.9243 | $8\pm 2$
$\rm Cr^{+22}$ He$\alpha$ | 5.6902 | 5.6907 | $10\pm 4$
$\rm Mn^{+24}$ Ly$\alpha$ | 6.4440 | 6.4326 | $4\pm 2$
$\rm Mn^{+24}$ Ly$\beta$ | 7.6120 | 7.6243 | $8\pm 5$
$\rm Mn^{+23}$ He$\alpha$ | 6.1770 | 6.1901 | $8\pm 1$
$\rm Fe^{+25}$ Ly$\alpha$ | 7.0047∗∗ | 6.9693 | $30\pm 10$
$\rm Fe^{+24}$ He$\alpha$ | 6.6990 | 6.7112 | $50\pm 20$
$\rm Co^{+24}$ K$\alpha$ | 1.2673 | 1.2660 | $0.3\pm 0.1$
$\rm Ni^{+25}$ K$\alpha$1 | 1.3702 | 1.3685 | $1.9\pm 0.1$
$\rm Ni^{+25}$ K$\alpha$2 | 1.3636 | 1.3620 | $1.0\pm 0.3$

**footnotetext: blended lines

****footnotetext: higher velocity component

Table 3: Elemental abundances in the X-ray wind of GRO J1655-40 compared to those measured in the visible waveband for the companion star (González Hernández et al., 2008) and to those measured in a global fit (Kallman et al., 2009). X-ray measurements only provide relative abundances, and are thus listed relative to Ca, while companion measurements are relative to H, but normalized here relative to Ca for the comparison. All abundances are given with respect to their solar values.  Element | Measured $A_{Z}$/Ca | Companion $A_{Z}$/Ca | Global fit $A_{Z}$/Ca
---|---|---|---
O | $0.5\pm 0.1$ | $9\pm 4$ | $1.4\pm 0.9$
Ne | $1.9\pm 0.3$ | - | $5.4\pm 0.9$
Na | $1.0\pm 0.1$ | $2.5\pm 1.8$ | $0.9\pm 0.2$
Mg | $0.27\pm 0.04$ | $2.9\pm 1.4$ | $0.55\pm 0.05$
Al | $0.48\pm 0.08$ | $1.2\pm 0.6$ | $0.8\pm 0.2$
Si | $0.23\pm 0.04$ | $4\pm 2$ | $0.6\pm 0.1$
P | $0.7\pm 0.6$ | - | $0.0\pm 0.2$
S | $0.6\pm 0.1$ | $6\pm 2$ | $0.8\pm 0.2$
Cl | $0.8\pm 0.3$ | - | $0.10\pm 0.09$
Ar | $0.7\pm 0.2$ | - | $1.0\pm 0.2$
K | $0.57\pm 0.53$ | - | $0.50\pm 0.45$
Ca | $1.0\pm 0.2$ | $1.0\pm 0.5$ | $1.0\pm 0.9$
Ti | $1.6\pm 1.5$ | $2.0\pm 1.0$ | $1.1\pm 0.5$
Cr | $2.2\pm 0.9$ | - | $2.6\pm 2.5$
Mn | $2.6\pm 0.8$ | - | $14\pm 12$
Fe | $0.6\pm 0.3$ | $0.8\pm 0.3$ | $0.96\pm 0.05$
Co | $3\pm 2$ | - | $7\pm 6$
Ni | $4\pm 1$ | $1.1\pm 0.6$ | $0.61\pm 0.05$

##  5 Discussion

It is puzzling that the GRO J1655-40 wind elemental abundances are
significantly different from those of the companion star. This could be a
result of fractionation in the accretion process or contamination by ambient
material. We compare the measured abundances with SN model yields of massive
stars ($\geq 25M_{\odot}$), the progenitor candidates of the BH. Fig. 6
presents a comparison of the measured abundances with core-collapse SN model
yields from Nomoto et al. (2013), for 25 $M_{\odot}$ stars of different
metallicites. Models with higher mass progenitors are less compatible with our
results, therefore herewith we focus on $25\,M_{\odot}$ models. The measured
elemental abundances of GRO J1655-40, and in particular the showing of the
odd-Z elements hint to the wind being contaminated by debris from a SN
explosion of a $25\,M_{\odot}$ star, with solar or higher metallicity. Odd-Z
elements are produced from even-Z proton-rich isotopes by electron capture, or
positron decay (Nomoto et al., 2013). The present peaks of odd-Z abundances
Na, Al, P, Cl, Mn, and Co (Fig. 6) confirm the role of neutronization
processes, which is an important component in understanding the abundances of
elements in the universe. Different SN models vary in their abundance yields.
Fig. 7 shows three models for the same progenitor mass and metallicity. Since
the GRO J1655-40 abundance pattern does not exactly match the SN models, we
suspect the wind is a mixture of the SN debris and the companion composition.

High mass SNe are predicted to leave behind a BH of only $\sim 3\,M_{\odot}$,
with $\sim 2\,M_{\odot}$ falling back (Israelian et al., 1999; Podsiadlowski
et al., 2002; González Hernández et al., 2008), while ejecting most of their
mass into the circumstellar medium (Nomoto et al., 2013). Indeed, the total
mass of the GRO J1655-40 binary is $<10\,M_{\odot}$, leaving at least
$15\,M_{\odot}$ in the ambient circumstellar region. Corroborating evidence
for this excess mass comes from the optical/infrared thermal emission that
coincided with the GRO J1655-40 outburst (Neilsen et al., 2016; Shidatsu et
al., 2016). This excess was ascribed by Neilsen et al. (2016) to a cool,
Compton thick photosphere, with a size of $\sim 5\times 10^{11}$ cm, (c.f.,
Fukumura et al., 2017) of the order of the Roche lobe radius ($\sim 0.5a$).
The mass loss in the Neilsen et al. (2016) model is highly (20-40 times)
super-Eddington. Our current abundance measurements suggest that perhaps the
extended photosphere does not all need to originate in the disk, but can
comprise in-situ ambient SN debris, which is heated by the X-ray source, thus
alleviating the need for such high Eddington mass loss rates.

One of the unique features of the GRO J1655-40 outflow is its Fe+21 absorption
lines that enable density diagnostics (Miller et al., 2006). The density of a
few $n\approx 10^{13}$ cm-3 (Miller et al., 2008; Mitrani & Behar, 2023, but
see Tomaru et al. (2023)) for an ion that forms at $\log\xi\approx 3$
indicates a distance from the ionizing source of a few $10^{10}$ cm. This
ionization region is in the steep low-$\xi$ part of the AMD, just below the
break (Fig. 5). Lower ionization (low-Z) gas would be more extended in the
wind (as long as the density gradient is shallower than $n\sim r^{-2}$). It
would therefore make sense that the SN contamination of the wind be in the
outer regions, where the lower-$\xi$ (lower-$Z$, H-like) ions reside. Indeed,
the abundances of elements with $Z\leq 16$ seem to be under-represented by
comparison the their theoretical SN yields (Figs. 6, 7).

Can we say anything about the age of the SN remnant based on the accretion
rate? The $6.6\,M_{\odot}$ BH in GRO J1655-40 grew either by a stellar merger,
or by gradual mass accretion. The Eddington mass accretion rate is
$\dot{m}_{\rm acc}=L_{\rm Edd}/(\eta c^{2})$, where $L_{\rm Edd}$ is the
Eddington luminosity, $\eta\approx 10\%$ is the accretion efficiency, and $c$
is the speed of light. If the BH was formed at $5\,M_{\odot}$ (Israelian et
al., 1999; González Hernández et al., 2008), for the current $6.6\,M_{\odot}$
BH, $L_{\rm Edd}\sim 10^{39}$ erg s-1 and $\dot{m}_{\rm acc}\sim
10^{-7}\,M_{\odot}\,{\rm yr}^{-1}$. Hence, it would take the BH, accreting at
the Eddington rate, $16$ million years to gain $1.6\,M_{\odot}$, dating the SN
to at least that far back. However, the uncertainty on the initial BH mass,
and the fact that accretion was not at the Eddington rate all the time,
imposes large uncertainties on this age.

Figure 6:  Comparison of the presently measured abundances in the GRO J1655-40
outflow to different SN model yields from Nomoto et al. (2013), for a
$25\,M_{\odot}$ progenitor with different metallicity values, given in
fractional mass units; Z=0.02 is solar metallicity. The high relative
abundances of Na, Al, P, Cl, Mn, and Co indicate a tendency towards models
with a progenitor of solar metallicity or higher. Note that all data points
are scaled to a solar Ca abundance, but can be (uniformly) vertically shifted.
Figure 7:  Three different SN model yields for a $25\,M_{\odot}$ progenitor
with solar metallicity ($Z=0.02$), demonstrating the theoretical uncertainties
of such models, none of which is in perfect agreement with the GRO J1655-40
abundances. The small peaks of odd-Z elements predicted by Nomoto et al.
(2013) qualitatively follow the measurements. Statistically, the yields of
Limongi et al. (2000) give a somewhat better agreement.

##  6 Conclusions

This paper revisits the spectacular Chandra/HETG spectrum of the GRO J1655-40
2005 outburst. Our analysis adds to the previous rich literature on this
source several key results:

  * •

We report individual ionic column densities Nion of 27 different ions (Table
1), pertaining to 18 different elements.

  * •

We construct an absorption measure distribution ($AMD=dN_{\rm H}/d\log\xi$)
that shows a sharp increase from low-$\xi$ that levels off at $\log\xi\approx
3.3$ (Fig. 5).

  * •

The AMD recovers the measured Nion values and yields relative abundances of
all 18 elements, which are fairly different from those of the binary companion
(Table 3).

  * •

The high abundances of odd-Z elements in the outflow are evidence for
neutronization processes, which take place in high-mass ($\geq 25\,M_{\odot}$)
high-metallicity ($\geq Z_{\odot}$) stars.

The measured abundances are thus a mix of a 25 $M_{\odot}$ SN debris and the
accretion disk material feeding the BH from its companion. This kind of
analysis opens the door to more SN archaeology with X-ray binary outbursts.

We thank J. Neilsen for useful comments on the manuscript. We thank the
referee for comments on the manuscript that helped to clarify the text. This
work was supported by a Center of Excellence of The Israel Science Foundation
(grant No. 2752/19). N.K. is supported by the Ramon scholarship sponsored by
the Ministry of Science & Technology, Israel. This paper employs a list of
Chandra data sets, obtained by the Chandra X-ray Observatory, contained in
doi:https://doi.org/10.25574/cdc.211.

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for $g\in\GL(2n,\CC)$.

$$\chi(A_\R)=\{(m,\bar m)\mid m\in \Mat(n,\CC\{i\})\},$$

the quaternionic structure model for $\UU(n,n)\cong\Sp_2(A_\R,\bar\sigma)$ can be seen as a subset of $\mathfrak C(\GL(2n,\CC))$ stabilizing $\chi(A_\R)$. $(J_1,J_2)\in \mathfrak C(\GL(2n,\CC))$ stabilizes $\chi(A_\R)^2$ if and only if for all $m\in\Mat(n,\CC\{i\})^2$,

$$(J_1,J_2)(m,\bar m)=(J_1(m),J_2(\bar m))=(m',\bar m'),$$

for some $m'\in\Mat(n,\CC)^2$, i.e. $J_1=J_2$. Therefore,

$$\mathfrak C(\UU(n,n))\cong\{(J,J)\in\mathfrak C(\GL(2n,\CC))\}.$$

We can also see $\mathfrak C(\UU(n,n))$ directly as the space complex structures on $A_\R^2$:

$$\mathfrak C(\UU(n,n))=\left\{J\in\Mat(2n,\CC)\mid \bar J^T\Ome{\Id_n}\in\Herm^+(2n,\CC), J\bar J=-\Id_{2n}\right\}.$$

Consider the real algebra $A_\R:=\Mat(n,\HH\{i,j,k\})$. Then

We use the map $\psi$ from the Section <ref>, to identify $A$ with $A':=\Mat(2n,\CC\{i\})$. The induced by $\Id\otimes\bar\sigma$ involution

on $\Mat(2n,\CC)$ acts in the following way:

$$m\mapsto -\Omega\bar m\Omega$$

where $\Omega=\Ome{\Id_{n}}\in \Mat(2n,\CC)$. We also denote by $\Omega_0:=\diag(\Omega,\Omega)\in\Mat(4n,\CC)$.

If we take a quaternionic structure $J$ on $A^2$ then we define

$$J':=\psi\circ J\circ\psi^{-1}.$$

We can see $J'$ as a complex $4n\times 4n$-matrix acting on $(A')^2$ in the following way: for $x\in (A')^2$,

$$J'(x):=J'\sigma'(x)=-J'\Omega_0\bar x\Omega_0.$$

$J'$ is a quaternionic structure, therefore,

$$-x=(J')^2(x)=J'\Omega_0\overline{J'\Omega_0\bar x\Omega_0}\Omega_0=-J'\Omega_0\bar J'\Omega_0 x.$$

So we obtain, $J'$ is a quaternionic structure on $A$ if and only if

$$J'\Omega_0\bar J'\Omega_0=\Id_{4n}.$$

The induced by $\sigma_1\otimes\Id$ anti-involution

on $\Mat(2n,\CC)$ acts in the following way:

$$m\mapsto -\Omega m^T\Omega.$$

So we define the standard symplectic form $\omega$ on $(A')^2$ with respect to this anti-involution: for $x,y\in (A')^2$,

$$\omega(x,y):=-\Omega_0 x^T\Omega_0\Ome{\Id_{2n}}y$$

and for a quaternionic structure $J'$, we define

$$h_{J'}(x,y):=\omega(J'(x),y)=\bar x^T\Omega_0(J')^T\Omega_0\Omega_0\Ome{\Id_{2n}}y=$$

$$=-\bar x^T\Omega_0(J')^T\Ome{\Id_{2n}} y.$$

The quaternionic structure model for $\GL(2n,\CC)$ is then:

$$\mathfrak C(\OO(4n,\CC)):=\left\{J'\in\Mat(4n,\CC)\left|\begin{array}{l}
J'\Omega_0\bar J'\Omega_0=\Id_{4n},\\-\Omega_0(J')^T\Ome{\Id_{2n}}\in\Herm^+(4n,\CC)
\end{array}
\right.\right\}.$$

The model for the symmetric space of $\mathfrak C(\SO^*(4n))$ can be seen as a subset of $\mathfrak C(\OO(4n,\CC))$ whose elements commute with $\sigma'$ i.e. $\sigma'(J'(x))=J'(\sigma'(x))$. Therefore:

$$\sigma'(J'(x))=-\Omega\overline{J'(x)}\Omega=\Omega\overline{J'\Omega\bar x\Omega}\Omega=-\Omega\bar J'\Omega x,$$

$$J'(\sigma'(x))=-J'\Omega \overline{\sigma'(x)} \Omega=J'\Omega \overline{\Omega \bar x\Omega} \Omega=-J'x$$

and we obtain:

$$\mathfrak C(\SO^*(4n))\cong\{J'\in \mathfrak C(\OO(4n,\CC))\mid J'=\Omega\bar J'\Omega\}.$$

The space $\mathfrak C(\SO^*(4n))$ can be also seen directly as complex structures $J$ on $A_\R^2$ such that the form:

is positive definite. So we obtain:

$$\mathfrak C(\SO^*(4n))=\left\{J'\in\Mat(2n,\HH\{i,j,k\}\left|
\begin{array}{l}
\sigma_1(J)^T\Omega_0\in\Herm^+(2n,\HH\{i,j,k\}),\\
\end{array}\right.\right\}.$$

§ CLASSIFICATION OF HERMITIAN ALGEBRAS

The goal of this section is to classify all Hermitian algebras. To do this, we consider a more general class of algebras that we call pre-Hermitian, and classify them.

Let $(A,\sigma)$ be a ring  with an anti-involution $\sigma$. As usual, we say that $a\in A$ is symmetric if $\sigma(a)=a$, denote the set of all symmetric elements in $A$ by $A^\sigma$. Clearly, if $2\in A^\times$, then  $A^\sigma$ is a (unital) Jordan ring under the operation $a\circ b=2^{-1}(ab+ba)$.

If  $A$ is an algebra over a commutative ring $F$ and $\sigma$ is $F$-linear, then we will refer to $(A,\sigma)$ as an $F$-algebra.

The Jacobson radical $J(A)$ of a unital ring $A$ is the set of all $x\in A$ such that $1+AxA\subseteq A^\times$. In particular, $1 + J(A)$ is a subgroup of $A^\times$.

It is  well-known (see e.g., [17]) that $J(A)$ is a nilpotent ideal for any (left or right) Artinian ring $A$. Moreover, such a ring
is semisimple if and only if $J(A)=\{0\}$. In particular, this holds for finite dimensional algebras over any field.

$J(A)$ is invariant under any anti-involution of any ring $A$.

Clearly, if $\sigma$ is any anti-involution of $A$ then  $1+A\sigma(x)A\subset A^\times$ for all $x\in J(A)$ hence $\sigma(x)\in J(A)$.

We say that a ring $(A,\sigma)$ is pre-Hermitian if $A^\sigma\cap J(A)=\{0\}$.

If $J(A)=\{0\}$, $A$ is sometimes called Jacobson semisimple (in particular, any $C^*$-algebra is Jacobson semisimple as a consequence of Gelfand-Naimark theorem). Also note that any $\RR$-subalgebra of $\Mat (n,\CC)$ invariant under the Hermitian transposition is semisimple and, therefore, Jacobson semisimple (see [20], Exercise 18, p. 168).
By definition, any Jacobson semisimple $(A,\sigma)$ is pre-Hermitian.

We say that a ring $(A,\sigma)$ is Hermitian if $a^2+b^2=0$ for $a,b\in A^\sigma$ implies that  $a=b=0$.

In particular, nonzero symmetric elements of Hermitian rings are not nilpotent.

In contrast to the main part of this paper, we do not assume in this appendix that a Hermitian ring is an algebra over a real closed field.

If $(A,\sigma)$ is a Hermitian ring such that $J(A)=\{0\}$, then, similarly to the Proposition <ref>, we can show that $-a^2\in A^\sigma_{\geq 0}$ (See definition <ref>) for all $a\in A^{-\sigma}$.

Similarly to  $A^\sigma_{\geq 0}$, for any ring $(A,\sigma)$ denote by $A^\sigma_{>0}$ the set of all sums $a_1^2+\cdots +a_n^2$, $n\ge 1$, where all $a_i$ are nonzero elements of $A^\sigma$. By definition, $A^\sigma_{>0}$ is an additive sub-semigroup of $A$, which may or may not contain $0$. Clearly, $A^\sigma_{\ge 0}=A^\sigma_{>0}\cup \{0\}$. Also, it is immediate that if $0\notin A^\sigma_{>0}$, then $(A,\sigma)$ is Hermitian. It would be interesting to classify those rings in which the opposite implication holds.

Any Hermitian ring $A$ with nilpotent $J(A)$ is pre-Hermitian.

Assume, there exists $0\neq x\in J(A)\cap A^\sigma$. Since $J(A)$ is a nilpotent ideal, there exist $x\in J(a)$ such that $a:=x^n\neq 0$ and $a^2=0$ for some $n>0$.

Therefore, $a=0$, which is a  contradiction.

Let $(A,\sigma)$ be a pre-Hermitian ring.

Then  $\sigma(x)=-x$ and $xy=-yx$ for any $x,y\in J(A)$. In particular, $2x^2=0$.

Let $x\in J(A)$. Since $x+\sigma(x)\in J(A)\cap A^\sigma=\{0\}$, $\sigma(x)=-x$. Furthermore, $-xy=\sigma(xy)=\sigma(y)\sigma(x)=yx$ for $x,y\in J(A)$.

If $(A,\sigma)$ is pre-Hermitian then:

(a) $\sigma(x)=-x$ for all $x\in J(A)$ and
for all $x\in J(A)$, $a\in A$.

$yx=-xy$ for all $x,y\in J(A)$ and $xyz=0$ for all $x,y,z\in J(A)$.

(c) $(\sigma(a)-a)xy=xy(\sigma(a)-a)=0$ all $a\in A$, $x,y\in J(A)$

Prove (a). Since $\sigma(x)\in J(A)$ for all $x\in J(A)$ by Proposition <ref>,  $x+\sigma(x)\in J(A)\cap A^\sigma=\{0\}$. This proves the fist assertion. To prove the second assertion, using the fact that $xa\in J(A)$ for all $x\in J(A)$, $a\in A$, we obtain

$$xa=-\sigma(xa)=-\sigma(a)\sigma(x)=\sigma(a)x\ .$$

This proves (a).

To prove (b) note that $xy=-\sigma(xy)=-yx$
for all $x,y\in J(A)$. To prove the second assertion, note that

for all $x,y,z\in J(A)$ hence $xyz=0$.
This proves (b).

To prove (c) note that on the one hand, $xya=x\sigma(a)y=axy$ and on the other hand, $(xy)a=\sigma(a)(xy)$ for all $a\in A$ and $x,y\in J(A)$. This proves (c).

The proposition is proved.

If $(A,\sigma)$ and $(B,\sigma')$ are pre-Hermitian algebras over $F$, $char ~F \ne 2$ and $(A,\sigma)\otimes (B,\sigma')$ is also pre-Hermitian, then either $J(A)=\{0\}$ or $J(B)=\{0\}$.

Indeed, If $x\in J(A)$, $y\in J(B)$, then $x\otimes y\in (A\otimes B)^{\sigma\otimes \sigma'}$ by Proposition <ref>.

Let  $V$ be a finite-dimensional vector space over a field $F$, $char ~F\ne 2$,   $\hat A=\Lambda(V)$ be the exterior algebra of $V$, and $\hat \sigma$ be the unique anti-involution of $\hat A$ such that $\hat \sigma(v)=-v$ for all $v\in V$. Denote by $A$ the quotient of $\hat A$ by the ideal generated by $\Lambda^3 V$, so that $A= F\oplus V\oplus \Lambda^2 V$ as a vector space. Since $\hat \sigma$ preserves the ideal generated by $\Lambda^3 V$, it induces a well-defined  anti-involution  $\sigma$ on $A$. Clearly, $J(A)=V\oplus \Lambda^2 V$ and $\sigma(x)=-x$ for all $x\in J(A)$. Thus, $(A,\sigma)$ is pre-Hermitian with $A^\sigma=F$ and $A^\sigma_{> 0}=F_{> 0}$.

Let $A$ be a pre-Hermitian ring. Then

$$A^\circ \cdot J(A)=J(A)\cdot A^\circ =0$$

where $A^\circ=A\cdot [A,A]=[A,A]\cdot A$ is the ideal of $A$  generated by all commutators $[a,b]=ab-ba$, $a,b\in A$.

It follows from Proposition <ref> that

for all $a,b\in A$, $x\in J(A)$ hence $[a,b]x=0$. Also, $x[a,b]=[\sigma(b),\sigma(a)]x=0$.

The proposition is proved.

$A^\circ=A$ if $A$ is simple noncommutative and
$(\Mat_n (A))^{\circ}=\Mat_n (A)$ if $A^{\circ}=A$.

The following definition is motivated by Proposition <ref>.

We say that a (unitless) ring $J$ is nilpotent pre-Hermitian if:

$\bullet$ $2x=0$ implies that $x=0$ (this is relevant only in characteristic $2$).

$\bullet$ $yx=-xy$  and  $xyz=0$ for all $x,y,z\in J$.

Clearly, for any nilpotent pre-Hermitian ring $J$ with $0\notin 2(J\setminus \{0\})$, the assignments $j\mapsto -j$ define an anti-involution on $J$ with no fixed points in $J\setminus \{0\}$.

Let $J$ be a ring and $K$ be a commutative ring that acts on $J$ from the left. We denote the action by:

\act \colon & K\times J & \to & J\\
& (k,j) & \mapsto & k\act j.
\end{matrix}$$

Similarly, if $K$ acts in $J$ from the right, we denote the action by:

\ract \colon & J\times K & \to & J\\
& (j,k) & \mapsto & j\ract k.
\end{matrix}$$

We say that a ring $J$  is a left $K$-algebra if $J$ is a left $K$-module with respect to $\act$ and

$$k\act (jj')=(k\act j) j'$$

for all $j,j'\in J$, $k\in K$ (so we will sometimes denote it simply by $k\act jj'$).

We say that a ring $J$  is a right $K$-algebra if $J$ is a right $K$-module with respect to $\ract$ and

$$(jj')\ract k=j(j'\ract k)$$

for all $j,j'\in J$, $k\in K$ (so we will sometimes denote it simply by $jj'\ract k$).

Let $K$ be a commutative unital ring with an involution $\overline{\cdot}$, $J$ be a unitless $K$-algebra and $\gamma:K\to J$ be a homomorphism of abelian groups. We say that $J$ is a $(K,\overline{\cdot},\gamma)$-algebra if:

\begin{equation}
\label{eq:Kgamma algebra}
%~(k-\overline k)\act (jj')=\gamma(k)jj',
\begin{split}
j(k\act j')=(\overline k\act j-j\gamma(k))j',(k-\overline k)\act jj'=j\gamma(k)j'
\end{split}
\end{equation}

for all $j,j'\in J$, $k,k'\in K$.

When $J$ satisfies $J^3=0$, e.g., when $J$ is  pre-Hermitian, the conditions
(<ref>) simplify to

\begin{equation}
\label{eq:Kgamma algebra simplified}
\begin{split}
%~(k-\overline k)\act (jj')=\gamma(k)jj',
j(k\act j')=\overline k\act jj'=k\act jj'
\end{split}
\end{equation}

for all $j,j'\in J$, $k\in K$.

Let $K$ be a commutative unital ring and $J$ be a $(K,\overline{\cdot},\gamma)$-algebra. Then:

(a) $J\oplus K$ has a structure of an associative unital ring with the multiplication given by

\begin{equation}
\label{eq:Kgamma product}
(j+k)(j'+k')=jj'+k\act j'+\overline {k'}\act j-j\gamma(k')+kk'
\end{equation}

for all $j,j'\in J$, $k,k'\in K$ (we denote this ring by $J\rtimes_\gamma K$ and refer to  as semidirect sum of $J$ and $K$ over $\gamma$).

(b) $J$ is a two-sided ideal in $J\rtimes_\gamma K$, moreover, the projection to the second factor is a surjective homomorphism $J\rtimes_\gamma K \twoheadrightarrow K$ of rings whose kernel is $J$.

(c) Suppose that $2\in K^\times$ and $\gamma(kk')=\overline k\act \gamma(k')+k'\act \gamma(k)-\gamma(k)\gamma(\overline {k'})+\frac{\gamma(k)\gamma(k')}{2}$ for all $k,k'\in K$. Then the assignments $k\mapsto \iota(k):=\overline k+\frac{\gamma( k)}{2}$  define an injective ring homomorphism $\iota:K\hookrightarrow J\rtimes_\gamma K$. Suppose additionally that $j\gamma(k)=2j\gamma(\overline k)+\gamma(\overline k)j$  for all $j\in J$, $k\in K$. Then  $J\rtimes_\gamma K=J\rtimes_{\bf 0} \iota(K)$.

Define a map $\ract:J\times K\to J$ by

$$j\ract k:=\overline k\act j-j\gamma(k)\ .$$

$\ract$ is an action of $K$ on $J$ commuting with $\act$ and
$(jj')\ract k=j(j'\ract k)$ for all  $j,j'\in J$, $k\in K$, (i.e., $J$ is a right $K$-algebra).

First, show that $\ract$ commute with $\act$. Indeed,

$$(k\act j)\ract k'=\overline {k'}\act(k\act j) - (k\act j)\gamma(k')=k \act(\overline {k'}\act j) -k\act j\gamma(k') =k\act (j\ract k')$$

for all $j\in J$, $k,k'\in K$ because $J$ is a left $K$-algebra.

$(j\ract k)\ract k'=\overline {k'}\act (j\ract k)-(j\ract k)\gamma(k')=\overline {kk'} \act j-\overline {k'}\act j\gamma(k)-(j\ract k)\gamma(k')$

$$\overline {kk'} \act j-(j\ract k')\gamma(k)-j\gamma(k)\gamma(k')-(j\ract k)\gamma(k')
%$$=\overline {k'}\act (\overline k\act j+\gamma(k)j)+\gamma(k')(\overline k\act j+\gamma(k)j)$$
%$$=\overline {kk'}\act j+\overline {k'}\act\gamma(k)j+\overline k\act\gamma(k')j=\overline {kk'}\act j+k'\act\gamma(k)j+k\act\gamma(k')j
=\overline {kk'}\act j-j\gamma(k'k)=j\ract (kk')$$

for all $k,k'\in K$, $j\in J$  by the commutation of $\ract$ with $\act$ and  the first condition of (<ref>).

Since $J\gamma(1)=\{0\}$ by the first condition (<ref>), i.e., $\ract 1=Id_J$, this proves that $J$ is a  $K$-module under $\ract$.

$$j(j'\ract k)=j(\overline k\act j'-j'\gamma(k))=k\act j j'-j\gamma(\overline k)j'-jj'\gamma(k)=\overline k\act jj'-jj'\gamma(k)=(jj')\ract k$$

for all $k,k'\in K$, $j\in J$ by the second and third conditions of (<ref>).
This proves that $J$ is a right $K$-algebra under $\ract$.

The lemma is proved.

Note that the identity

\begin{equation}
\label{eq:middle action}
(j\ract k)j'=j(k\act j')
\end{equation}

for all $k\in K$, $j,j'\in J$ is equivalent to the second condition (<ref>).

The following is immediate and well-known.

Let $R$ and $S$ be associative ring, $R$ is unitless, $S$ is unital  and $R$ is an $S$-bimodule such that

\begin{equation}
\label{eq:associative biaction}
s\act (rr')=(s\act r)r',~(rr')\ract s=r(r'\ract s),~(r\ract s)r'=r(s\act r')
\end{equation}

for all $r,r'\in R$, $s,s'\in S$.
Then  $A:=R\oplus S$ is a unital associative ring with the product given by

$$(r+s)(r'+s')=rr'+s\act r'+r\ract s'+ss'$$

for all $r,r'\in R$, $S,s'\in S$.

Thus, (<ref>) and Lemma <ref> guarantee that all assumptions of Lemma <ref> hold for $R=J$, $S=K$,
therefore, $J\oplus K$ is a unital associative ring.  This proves (a).

Part (b) is obvious.

Prove (c). Indeed,
$\iota(k)\iota(k')=(\overline k+\frac{\gamma(k)}{2})(\overline{k'}+\frac{\gamma(k')}{2})=\overline{kk'}+\frac{\overline k\act \gamma(k')+\gamma(k)\ract \overline{k'}}{2}+\frac{\gamma(k)\gamma(k')}{4}$

=\overline{kk'}+\frac{\overline k\act \gamma(k')+k'\act \gamma(k)-\gamma(k)\gamma(\overline {k'})}{2}+\frac{\gamma(k)\gamma(k')}{4}

for all $k,k'\in K$ by the first relation (<ref>). Therefore, $\iota$ is a homomorphism of rings. Its injectivity follows because $\iota$ splits the canonical homomorphism from (b). Finally,

$$j\iota(k)=j(\overline k+\frac{\gamma(k)}{2})=kj-j\gamma(\overline k)+\frac{j\gamma(k)}{2}=k j+\frac{\gamma(\overline k)}{2}j=\iota(\overline k)j$$

for all $j\in J$, $k\in K$. This proves that $J$ is a $(\iota(K),\widetilde {\cdot},{\bf 0})$-algebra with the trivial $\gamma={\bf 0}$ and:

$\bullet$ The involution $\widetilde {\cdot}$ defined by $\widetilde{\iota(k)}=\iota(\overline k)$ for all $k\in K$.

$\bullet$  The left action of $\iota(k)$ on $J$  by the left multiplication in $J\rtimes_\gamma K$.

In particular, $J\rtimes_\gamma K=J\rtimes_{\bf 0} \iota(K)$. Part (c) is proved.

The proposition is proved.

Given a  commutative ring $(K,\overline{\cdot})$ with anti-involution, we say that  a left  $K$-algebra  $J$ is a $(K,\overline{\cdot})$-algebra if

$$\overline k\act jj'=k\act jj'=j(k\act j')$$

for all $j,j'\in J$,  $k\in K$. Clearly, $(K,\overline{\cdot})$-algebras are same as $(K,\overline{\cdot},{\bf 0})$-algebras. Also, in view of (<ref>), any nilpotent pre-Hermitian $(K,\overline{\cdot},\gamma)$-algebra  is automatically a $(K,\overline{\cdot})$-algebra.

Let $(K,\overline{\cdot})$ be any commutative ring with anti-involution and let $J$ be any nilpotent pre-Hermitian ring. Suppose that  $J$ is a $(K,\overline{\cdot})$-algebra and let $\gamma:K\to J$ be any homomorphism of abelian groups such that:
$\gamma(kk')=\overline k\act \gamma(k')+k'\act \gamma(k)$ for all $k,k'\in k$.

(a) $J$ is a $(K,\overline{\cdot},\gamma)$-algebra.

(b) Suppose additionally that  $\gamma(\overline k)=\gamma(k)$ for all $k\in K$.  Then the assignments $j+k\mapsto \gamma(k)-j+\overline k$ define an  anti-involution $\sigma$ on $J\rtimes_\gamma K$.  Moreover, if $K$ is semisimple (i.e., is a direct sum of fields), then $(J\rtimes_\gamma K,\sigma)$ is pre-Hermitian.

Prove (a). Indeed, the conditions (<ref>) hold automatically for this choice of $\gamma$ and because $J$ is $(K,\overline{\cdot})$-algebra. This proves (a).

Prove (b). First,  let us verify that $\sigma$ is an anti-involution.

$$\sigma(\sigma(j+k))=\sigma(\gamma(k)-j+\overline k)=j-\gamma(k)+\sigma(\overline k)=j-\gamma(k)+k+\gamma(\overline k)=j+k$$

for all  $j\in J$, $k\in K$. That is, $\sigma^2=1$.

Furthermore, by definition (<ref>),
$j  k=\overline kj-j\gamma(k)=\overline kj+\gamma(k)j=k\sigma(k)j$
hence $kj=j\sigma(k)$ for all $j\in J$, $k\in K$.

$$\sigma(kk')=\overline {kk'}+\gamma(kk')=\overline {kk'}+\overline k\gamma(k')+k'\gamma(k)=\overline {kk'}+\overline k(\sigma(k')-\overline k')+k'\gamma(k)=$$

$$=\overline k\sigma(k')+\gamma(k)\sigma(k')=\sigma(k)\sigma(k')$$

for all $k,k'\in K$. Clearly,

for all $j,j'\in J$. Also,

$$\sigma(kj)=\sigma(k\act j)=-k\act j=-kj=-j\sigma(k)=\sigma(j)\sigma(k)$$

for all $j\in J$, $k\in K$.

This proves the first assertion.  To prove the second assertion, note that semisimplicity of $K$ and Proposition <ref>(b) imply that Jacobson radical of $J\rtimes_\gamma K$ is $J$. This finishes proof of (b).

The proposition is proved.

The following is an immediate corollary of Proposition <ref>.

Let $(A,\sigma)$ be any pre-Hermitian ring and $K$ be any commutative unital subring of $A$ such that $K\cap J(A)=\{0\}$ and $\sigma(K)\subset K+J(A)$. Then

(a) $J(A)$ is both a $(K,\overline{\cdot})$-algebra and a $(K,\overline{\cdot},\gamma)$-algebra, where:

$\bullet$ $J(A)$ is a $K$-algebra via  left multiplication.

$\bullet$ $\overline{\cdot}:K\to K$  and  $\gamma:K\to J(A)$ are determined by $\sigma(k)=\gamma(k)+\overline k$ for all $k\in K$.

(b) The subring of $A$ generated by $K$ and $J(A)$ is naturally isomorphic to $J(A)\rtimes_\gamma K$.

Recall that $F$ is a perfect field if every irreducible polynomial over $F$ has distinct roots. In particular, all fields of characteristic zero and all finite fields are perfect.

Let $R$ be a finite dimensional algebra over a perfect field $F$. Then

(a) There is a splitting $\iota$ of the short exact sequence $J(R)\to R\to S:=R/J(R)$, e.g., $R=\iota(S)\oplus J(R)$.

(b) The images of all splittings $\iota:S\hookrightarrow R$ are conjugate in $R$ by $1+J(R)$.

In fact, the multiplication in $R$ in Theorem <ref> is as in Lemma <ref> since $J(R)$ is naturally a bimodule over $S\iota(S)$.

The following is immediate.

In the assumptions of Theorem <ref>  suppose that $\sigma(\iota(S))=\iota(S)$ for an anti-involution $\sigma$ of $R$ and a splitting $\iota:S\hookrightarrow R$. Then  $R^\sigma=\iota(S)^\sigma\oplus J(R)^\sigma$.

The following is well-known (cf. <cit.>, Theorem 25C.17).

In the assumptions of Theorem <ref>, there exists a faithful $n$-dimensional representation $\rho$ of $R$ into the algebra of ${\bf n}$-block upper triangular matrices (for some partition ${\bf n}$ of $n$) such that  $\rho(J(R))$ is in the block-strictly upper triangular part of $Mat ({\bf n}, F)$ and the image under $\rho$ of at least one splitting $\iota:S\hookrightarrow R$ is in the block-diagonal part of $Mat ({\bf n}, F)$.

We will use  Theorem <ref> to finish the classification of finite-dimensional pre-Hermitian algebras over perfect fields as follows.

Let $(A,\sigma)$ be a finite-dimensional pre-Hermitian algebra over a perfect field $F$, denote by $K$ the maximal abelian ideal  of the semisimple quotient $S=A/J(A)$ and by $B$ its complement so that $S=B\oplus K$.
Then there is a unique copy of $B$ in $A$ splitting the  canonical homomorphism $\pi:A\twoheadrightarrow S$, that is, $A= B\oplus  \tilde K$, $\sigma(B)=B$, $\sigma(\tilde K)=\tilde K$, where $\tilde K=\pi^{-1}(K)$. More precisely, $\tilde K=J(A)$ if $K= \{0\}$ and $\tilde K\cong J(A)\rtimes_\gamma K$ otherwise (in the notation of Corollary <ref>).

Indeed, $B^\circ=B$ because each simple component $C$ of $B$ satisfies $C^\circ=C$ since $[C,C]\ne 0$. Furthermore, in the notation of Theorem <ref>(a), fix a splitting $\iota:S\hookrightarrow A$ of homomorphism $\pi:A\twoheadrightarrow S$.
Then $\iota(B)=\iota(B^\circ)=\iota(B)^\circ\subset A^\circ$ hence $\iota(B)J(A)=J(A)\iota(B)=\{0\}$ by Proposition <ref>. To prove the first assertion note that, by Theorem <ref>(b), the images of $B$ under any splitting $S\hookrightarrow A$ are conjugate to $\iota(B)$ by $1+ J(A)$. Since
for all $b\in B$, we see that $\iota(B)$ is a unique copy of $B$ in $A$. In particular, $\sigma(\iota(B))=\iota(B)$.

Furthermore, by definition, $\sigma(\tilde K)=\tilde K$ and $\tilde K=\iota(K)+J(A)$ as a vector space over $F$. Since $\iota(B) \iota(K)= \iota(K)\iota(B)=\{0\}$, we see that
$\iota(B)\tilde K=\tilde K \iota(B)=\{0\}$.
Therefore, $A=B\oplus \tilde K$, as an algebra.

If $K=\{0\}$, then, clearly, $\tilde K=J(A)$. Suppose that $K\ne \{0\}$.  Since $\sigma(\iota(K))\subset \tilde K=\iota(K)+J(A)$, we see that $\iota(K)$ satisfies the hypotheses of Corollary <ref>. Therefore,
$\tilde K=J(A)\rtimes_\gamma \iota(K)$.

The theorem is proved.

The following is immediate.

In the assumptions of Theorem <ref>,  one has

(a) If $K=\{0\}$ then $A$ is Hermitian if and only if $B$ is Hermitian. Otherwise, $A$ is Hermitian if and only if both $B$ and $\tilde K$ are Hermitian.

(b) If $A$ is Hermitian, then

$A^\sigma_{\geq 0}=\begin{cases}
B^\sigma_{\geq 0} & \text{if $\tilde
B^\sigma_{\geq 0}\oplus \tilde K^\sigma_{\geq 0} &\text{otherwise}
\end{cases}$ and  $A^\sigma_{+}=\begin{cases}
B^\sigma_{+} & \text{if $\tilde
B^\sigma_{+}\oplus K^\sigma_+ &\text{otherwise}
\end{cases}$.

If $(A,\sigma)$ and $(B,\sigma')$ are algebras with anti-involutions over a field $F$, then their tensor product (over $F$) is

$$(A,\sigma)\otimes (B,\sigma'):=(A\otimes B,\sigma\otimes \sigma')\ .$$

If $(A,\sigma)$ and $(B,\sigma')$ are pre-Hermitian algebras over $F$ and and  $(A,\sigma)\otimes (B,\sigma')$ is also pre-Hermitian, then either $J(A)=\{0\}$ or $J(B)=\{0\}$.

Indeed, If $x\in J(A)$, $y\in J(B)$, then $x\otimes y\in (A\otimes B)^{\sigma\otimes \sigma'}$ by Proposition <ref>.

Therefore, the same question Hermitian algebras reduces to the following one.

Is the tensor product over $\RR$ of an Hermitian algebra with $\CC$ and $\HH$ also Hermitian?

Note that the tensor product of semisimple algebras over a field of characteristic zero is semisimple
(see <cit.>, Corollary 2.37).

We discuss now anti-involutions on simple rings.

Clearly, if  $(A,\sigma)$ is a Hermitian algebra over a field $F$, then so is $F$, i.e., $F$ is a formally real field. In particular, if $A^\sigma=F$, then $(A,\sigma)$ is Hermitian if and only if $F$ is a formally real.

The following is an immediate consequence of the  Skolem-Noether theorem.

For any ring $(A,\tau)$ with an anti-involution $\tau$ denote by $A^{[\tau]}$ the set of all $v\in A^\times$ such that $\tau(v)=v z$, for some $z\in Z(A)^\times$ such that $\tau(z)=z^{-1}$ (In particular, if each element of the center $Z(A)$ is fixed under $\tau$, then $\tau(v)\in \{v,-v\}$, i.e., ). Clearly, the assignments $(w,v)\mapsto w\act v:=w v\tau(w)$ define an action of the multiplicative group $A^\times$ on $A^{[\tau]}$.

Let $(A,\tau)$ be a simple Artinian ring. Then

(a) For any anti-involution $\sigma:A\to A$
there is an element $v\in A^{[\tau]}$ (unique up to multiplication by elements of the field $Z(A)$)  such that
for all $a\in A$.

(b) For any $v\in A^{[\tau]}$ the assignments $a\mapsto v\tau(a)v^{-1}$ define an involution $\sigma_v$ on $A$.

(c) For any $w\in A^\times$ the assignments $a\mapsto waw^{-1}$ define an isomorphism  of rings with anti-involution $(A,\sigma_v)\widetilde \to (A,\sigma_{w\act v})$.

We say that an algebra $(A,\sigma)$ over $F$ is thin if $A^\sigma=F$.

The first example of a thin  algebra is given by Example <ref>. Another one is a (generalized) quaternion algebra (see Remark <ref> below).

Clearly every thin algebra $(A,\sigma)$ over $F$ is pre-Hermitian and the direct sum  of two $F$-algebras is never thin because $(A\oplus A')^{\sigma\oplus \sigma'}=A^\sigma\oplus {A'}^{\sigma'}$. Moreover, the following is an immediate consequence of Theorem <ref>.

If a finite-dimensional algebra  $(A,\sigma)$ over a perfect field $F$ with $char(F)\ne 2$ is thin then it is either simple noncommutative, or direct sum of two copes of a simple algebra interchanged by $\sigma$, or pre-Hermitian whose semisimple quotient $A/J(A)$ is a field extension of $F$.

Let $F$ be a perfect field with $char(F)\ne 2$ and $(A,\sigma)$ be a thin semisimple finite-dimensional  algebra over  $F$. Then:

(a) $A=F\oplus A^{-\sigma}$ where $A^{-\sigma}=\{a\in A:\sigma(a)=-a\}$ is the space of skew-symmetric elements.

(b) $A^{-\sigma}$ admits a unique nonsingular symmetric bilinear form $\beta$  such that $aa'+a'a=-\beta(a,a')$ for all $a\in A^{-\sigma}$.

(c) If $a$ is a nilpotent element in $A$ then $a\in A^{-\sigma}$ and $a^2=0$.

(d) $A^{-\sigma}$ is a Lie algebra with respect to the commutator bracket $[a,a']=aa'-a'a=2aa'+\beta(a,a')$

(e) $2a''a'a-2aa'a''=\beta([a,a'],a'')=\beta(a,[a',a''])$
for $a,a',a''\in A^{-\sigma}$.

Part (a) is obvious because $A^\sigma=F$. To prove (b) note that $\hat \beta(a,a'):=-aa'-a'a$ is fixed by $\sigma$ and thus belongs to $F$ for all $a,a'\in A$. This is obviously a symmetric bilinear form on $A$, denote by $\beta$ its restriction to $A^{-\sigma}$. If $z$ is in the radical of $\beta$, then $za+az=0$ for all $a\in A^{-\sigma}$, which implies that $z=0$ by Lemma  <ref>.  This proves (b).

Prove (c). Let $a\in A$ be a nilpotent and suppose that $a\notin A^{-\sigma}$. Then there exists $c\in F^\times$ such that the nilpotent element $b:=ca$ satisfies $\sigma(b)=1-b$. Therefore, $(1-b)^n=0$ for some $n\ge 2$. This contradicts to the non-invertibility of $b$. Thus, $\sigma(a)=-a$ and $a^2$ is a nilpotent in $A^\sigma=F$, that is, $a^2=0$.

Part (d) is obvious because $\sigma([a,a'])=-[a,a']$ for all $a,a'\in A^\sigma$.

Prove (e). Indeed,

for all $a,a',a''\in A^\sigma$. This proves the first assertion. To prove the second one, compute

$$=-2(aa'+a'a)a'a-2aa'(a'a+aa')+4a{a'}^2a+4a'a^2a'= $$

$$=2\beta(a,a')(aa'+a'a)-2a^2\beta(a',a')-2{a'}^2\beta(a,a)=\beta(a,a)\beta(a',a') -\beta(a,a')^2$$

for all $a,a'\in A^{-\sigma}$. This proves (d).

The proposition is proved.

Let $F$ be a perfect field with $char(F)\ne 2$ and $(A,\sigma)$ be a thin semisimple finite-dimensional  algebra over  $F$. Then $A$ is either  a division algebra over $F$ with  $\dim_F A\in \{1,2,4\}$ or

$A=F\oplus F$  (so that $\sigma$ is the permutation of summands).

Indeed, any simple component of $A$ is $\Mat_n(D)$, where $D$ is a division algebra over $F$. Note that if $n\ge 3$, then $A$ contains an element $e=e_{12}+e_{23}$ which contradicts to  Proposition <ref>(c). Thus, $n\le 2$ and

$A\in\{D,D\oplus D,\Mat_2(D),\Mat_2(D)\oplus \Mat_2(D)\}$ by Lemma <ref>.
Note, however, that the last two algebras are not thin. If $A=D\oplus D$, then $(x,\sigma(x))\in A^\sigma$ for all $x\in D$, i.e. we always have a copy of $D$ in $A^\sigma=F$, that is, $D=F$.

The theorem is proved.

For any ring $F$ and $\alpha,\beta\in F$ denote by $\HH_{\alpha,\beta}$ the $F$-algebra with a presentation:

$$\HH_{\alpha, \beta}=\langle i,j\ |\ i^2=\alpha,~j^2=\beta,~ji=-ij\rangle\ .$$

Clearly, it admits a unique anti-involution $\overline{\cdot}$ such that $\overline i=-i$, $\overline j=-j$. By  construction,
$a\overline a=\overline a a\in F$
for all $a\in \HH_{\alpha, \beta}$. In particular, $\HH_{\alpha, \beta}$ is a division algebra if and only if $F$ is a field and  $a \overline  a \in F^\times$ for all nonzero $a\in \HH_{\alpha, \beta}$.

Any 4-dimensional division algebra $D$ over a field $F$ with $\ch F\ne 2$ is isomorphic to $\HH_{\alpha,\beta}$ for some non-squares $\alpha,\beta\in F^\times$.
Clearly, it is thin with the above anti-involution $\overline {\cdot}$. Note that if  both $-\alpha$ and $-\beta$ are complete squares in $F$, then $\HH_{\alpha,\beta} \cong \HH_{-1,-1}$ is the ordinary algebra of quaternions.

Suppose that $\ch F\ne 2$ and the algebraic closure $\overline F$ of $F$ is  a quadratic extension of $F$. Then any division algebra over $F$ is isomorphic to  either $\overline F$ or to some $\HH_{\alpha,\beta}$.

Generalizing the above observations, we construct some twisted group algebras of abelian groups with anti-involutions. Recall that, given a group $G$ and its linear action $\triangleright$ on a commutative ring $K$, a map $\chi:G\times G\to K$ is called a $2$-cocycle on $G$ (in $K$) if

$$(g\triangleright \chi(g',g''))\chi(g,g'g'')=\chi(g,g')\chi(gg',g'')$$

for all $g,g',g''\in G$.

Furthermore, denote by $K_\chi G$ the $\chi$-twisted group algebra of $G$, i.e., a $K$-algebra with a free $K$-basis $\{[g], g\in G\}$ and the multiplication table:

$$[g][g']=\chi(g,g')\cdot [gg'], [g]k=g(k)\cdot g$$

for all $g,g'\in G$, $k\in K$. It is well-known that any central simple (e.g., a division) algebra $A$ over a field $F=K^G$  is isomorphic to some $K_\chi G$ so that $K$ is a Galois field extension of $F$.

The following is immediate.

Let $\overline{\cdot}$ be an involution on $K$. Then the following statements are equivalent for a $2$-cocycle $\chi:G\times G\to K$:

(a) The assignments $k\cdot [g]\mapsto [\sigma(g)]\cdot \overline k$ for $g\in G$, $k\in K$ define an anti-involution on $K_\chi G$.

(b)  $\sigma(g)\triangleright \overline k=\overline{g^{-1}\act  k}$ and $\overline {\chi(g,g')}=\overline{gg'}\triangleright \chi(\sigma(g'),\sigma(g))$ for all $g,g'\in G$, $k\in K$.

In particular, if the $G$-action   $\triangleright$ is trivial, the cocycle conditions simplifies:

for all $g,g',g''\in G$ (e.g., $\chi$ is a bicharacter of $G$).

The following is immediate.

Let $G$ be an abelian group, trivially acting on $K$, $\overline{\cdot}$ be an anti-involution on $K$ and $\varepsilon:G\to K^\times$, $g\mapsto \varepsilon_g$ be a map. Then the following are equivalent:

(a) The assignments $k\cdot [g]\mapsto \overline k\varepsilon_g\cdot [g]$ for $g\in G$, $k\in K$ define an anti-involution $\sigma_\varepsilon$ on $K_\chi G$.

(b)  $\overline{\varepsilon_g}=\varepsilon_g^{-1}$ and $\chi(g',g)=\overline{\chi(g,g')}\varepsilon_{gg'}\varepsilon_g^{-1}\varepsilon_{g'}^{-1}$ for all $g,g'\in G$.

In particular, $(K_\chi G)^{\sigma_\varepsilon}=\bigoplus\limits_{g\in G}K_g\cdot [g]$,
where $K_g=\{k\in K:\overline k=k\cdot \varepsilon_g\}$ for all $g\in G$.

Now we classify  anti-involutions on $4$-dimensional division algebras.

In the notation as above, suppose that   $\HH_{\alpha,\beta}$ is a noncommutative division algebra (over a field $F$). Then any anti-involution on $\HH_{\alpha,\beta}$ is either $\overline{\cdot}$ or is given by
$$\sigma(a)= v\overline xv^{-1}$$

for some nonzero  imaginary $v$, i.e., such that $\overline v=-v$ (in what follows, we denote the latter anti-involution  by $\sigma_v$).

By Skolem-Noether theorem, any anti-involution $\sigma$ on $\HH_{\alpha,\beta}$ can be written as
$\sigma (x)=v\overline xv^{-1}$ where $v\in \HH_{\alpha,\beta}^{\times}$. Then $\sigma^2(a)=uau^{-1}$ where
$u=v\overline v^{-1}$. Since $\sigma^2=1$, we have $a=uau^{-1}$ for all $a\in $, i.e. $u$ is central in $\HH_{\alpha,\beta}$. That is $u=c\in F^\times$.  This implies that $\overline v=c^{-1} v$ hence either $c=1$, $v\in F^\times$ or $c=-1$, $\overline v=-v$.

In the assumptions of Proposition <ref>, for any nonzero imaginary $v\in \HH_{\alpha,\beta}$ there exist  imaginary  $w,w'$ such that  $[w,w']=v$  and $\{1,w,w',v\}$ is a basis of $\HH_{\alpha,\beta}$.
In particular, $vw=-wv$, $vw'=-w'v$, $ww'+w'w\in F$, $(\HH_{\alpha,\beta})^{\sigma_v}=F+Fw+Fw'$ and $(\HH_{\alpha,\beta})^{-\sigma_v}=F\cdot v$.

Denote by $V$ the set of all imaginary elements of $\HH_{\alpha,\beta}$.
For an imaginary $v$ define an $F$-linear map  $f_v:V\to \HH_{\alpha,\beta}$   by $f_v(a):= va+av$.
Clearly, the range of $f_v$ is $F$, in particular, $Ker~f_v$ is a $2$-dimensional  subspace of the (3-dimensional) space $V$.

$$\sigma_v(a)=v\overline av^{-1}=-vav^{-1}=avv^{-1}=a$$

for any  $a\in Ker~f_v$.
That is, $F+Ker~f_v\subset (\HH_{\alpha,\beta})^{\sigma_v}$. Since $v\in (\HH_{\alpha,\beta})^{-\sigma_v}$, we see that $F+Ker~f_v= (\HH_{\alpha,\beta})^{\sigma_v}$ and $(\HH_{\alpha,\beta})^{-\sigma_v}=Fv$.

Therefore, $[w,w']\in F^\times v$ for any   basis $\{w,w'\}$ of $Ker~f_v$.

The proposition is proved.

Note that if $\HH=\HH_{-1,-1}$ is the ordinary quaternion algebra over $\RR$, then $w^2=-w\overline w<0$ for any imaginary $w\in \HH$, hence $(\sqrt{-w^2})^2+w^2=0$ and we obtain the following immediate corollary of Proposition <ref>.

The algebra $(\HH, \sigma)$ is  Hermitian if and only if $\sigma=\overline{\cdot}$.

We conclude the discussion with the standard definition of Hermitian algebras from Function Analysis (see <cit.>). How the algebraic and the analytic definition are related?

(see <cit.>)
Let $A$ be an unital complex associative algebra.
For any $a\in A$ define $\text {spec}(a):=\{\lambda \in \CC: (\lambda\cdot 1-a)\notin  A^{\times}\}$.
An algebra $A$ with an anti-involution $*$ is
called Hermitian if and only if $\sigma (a)\subseteq \RR$ whenever $a=a^*$.

In terms the above definition of Hermitian algebras:

1) If $(A,*)$ is Hermitian then its tensor product with the algebra of complex $n\times n$-matrices is Hermitian.

2) Any $C*$-algebra is Hermitian (see <cit.>).

We conclude the section with recalling the Cayley-Dickson construction of (generalized) octonions ${\mathbb O}_{\alpha,\beta}:=\HH_{\alpha,\beta}\oplus \HH_{\alpha,\beta}$, as a free module over a ring $F$ with the following multiplication table

$$(a,b)(a',b')=(aa'-\overline {b'}b,b'a+b\,\overline{a'})\ .$$

This is a non-associative $F$-algebra  with the anti-involution given by

$$\overline {(a,b)}=(\overline  a,-b)$$

so that
$\overline {(a,b)}(a,b)=(a,b)\overline {(a,b)}=(a\overline a+\overline  bb,0)$ for all $a,b\in {\mathbb O}_{\alpha,\beta}$.
In particular,
if $\HH_{\alpha,\beta}$ is a division algebra, then ${\mathbb O}_{\alpha,\beta}$ is a non-associative division algebra as well.

§ ISOMORPHISMS AND EMBEDDINGS OF MATRIX ALGEBRAS

Some tensor products and direct products of matrix algebras are related in a way we want to discuss in this section. The described in here isomorphisms and embeddings are used in the main part of the paper as a tool to construct symmetric spaces and study their properties.

As usual, for every algebra $A$ and (anti-)involution $\sigma$, we
denote by $A^\sigma$ the set of fixed points of $\sigma$ in $A$.

We also remind our standard notation of anti-involutions on matrix algebras.

* If $A$ is $\Mat(n,\R)$ or $\Mat(n,\CC)$, then the anti-involution given by the transposition of the matrix is denoted by $\sigma$.

* If $A$ is $\Mat(n,\CC)$, then by $\bar\sigma$ is denoted the anti-involution given by $\sigma$ composed with the complex conjugation.

* If $A$ is $\Mat(n,\HH\{i,j,k\})$, then the anti-involution $\sigma_0\colon A\to A$ is given by the rule $\sigma_0(r_1+r_2j)=\sigma(r_1)+\bar\sigma(r_2)j$ where $r_1,r_2\in\Mat(n,\CC\{i\})$. Another anti-involution $\sigma_1\colon A\to A$ is given by the rule $\sigma_1(r_1+r_2j)=\bar\sigma(r_1)-\sigma(r_2)j$ where $r_1,r_2\in\Mat(n,\CC\{i\})$.

We also identify the algebra of $1\times 1$ matrices with the corresponding (skew-)field and use the same notation for the (anti-)involution in a (skew-)field as in the matrix algebra over this (skew-)field.

§.§ Three isomorphisms of matrix algebras

In this section, we describe three well-known matrix algebras isomorphisms.

§.§.§  and

The following map is an isomorphism of $\CC\{i\}$-algebras:

\chi\colon & \Mat(n,\CC\{I\})\otimes_\R\CC\{i\} & \to & \Mat(n,\CC\{i\})\times\Mat(n,\CC\{i\})\\ & a+bI &
\mapsto & (a+bi , a-bi)
\end{matrix}$$

where $a,b\in\Mat(n,\CC\{i\})$. In particular,

$$\chi(\Id I\otimes 1)=(i,-i),\;\chi(\Id\otimes i)=(i,i).$$

The induced by $\sigma\otimes\Id$ anti-involution

on $\Mat(n,\CC\{i\})\times\Mat(n,\CC\{i\})$ acts in the following way:

The induced by $\bar\sigma\otimes\Id$ anti-involution

on $\Mat(n,\CC\{i\})\times\Mat(n,\CC\{i\})$ acts in
the following way:

The induced by $\Id\otimes\bar\sigma$ involution

on $\Mat(n,\CC\{i\})\times\Mat(n,\CC\{i\})$ acts in the following way:

$$(m_1,m_2)\mapsto(\bar m_2,\bar m_1).$$

$$\chi((\Mat(n,\CC\{I\})\otimes_\R\CC\{i\})^{\bar\sigma\otimes\Id})=\{(m,m^T)\mid m\in \Mat(n,\CC\{i\})\},$$

$$\chi(\Mat(n,\CC\{I\}))=\chi((\Mat(n,\CC\{I\})\otimes_\R\CC\{i\})^{\Id\otimes\bar\sigma})=\{(m,\bar m)\mid m\in \Mat(n,\CC\{i\})\}.$$

§.§.§  and

The following map is an isomorphism of $\CC\{I\}$-$\CC\{i\}$-algebras:

\psi\colon & \Mat(n,\HH\{i,j,k\})\otimes_\R\CC\{I\} & \to & \Mat(2n,\CC\{i\})\\
& (q_1+q_2j) + (p_1+p_2j)I & \mapsto &
\begin{pmatrix}
q_1+p_1i & q_2+p_2i\\ -\bar q_2-\bar p_2i & \bar q_1+\bar p_1i
\end{pmatrix}.
\end{matrix}$$

where $q_1,q_2,p_1,p_2\in\Mat(n,\CC\{i\})$. In particular,

$$\chi(\Id i\otimes 1)=\begin{pmatrix}
\Id i & 0\\ 0 & -\Id i
\end{pmatrix},\;\chi(\Id\otimes j)=\begin{pmatrix}
0 & \Id\\ -\Id & 0
\end{pmatrix},$$

$$\chi(\Id k\otimes 1)=\begin{pmatrix}
0& \Id i\\ \Id i & 0
\end{pmatrix},\;\chi(\Id\otimes I)=\Id i.$$

The induced by $\sigma_0\otimes\Id$ anti-involution

on $\Mat(2n,\CC)$ acts in the following way:

$$m\mapsto \begin{pmatrix}
\Id & 0\\ 0 & -\Id
\end{pmatrix}m^T\begin{pmatrix}
\Id & 0\\ 0 & -\Id
\end{pmatrix}.$$

The induced by $\sigma_1\otimes\Id$ anti-involution

on $\Mat(2n,\CC)$ acts in the following way:

$$m\mapsto -\Ome{\Id}m^T\Ome{\Id}=\Ome{i}m^T\Ome{i}.$$

The induced by $\Id\otimes\bar\sigma$ involution

on $\Mat(2n,\CC)$ acts in the following way:

$$m\mapsto -\Ome{\Id}\bar m\Ome{\Id}=\Ome{i}\bar m\Ome{i}.$$

The induced by $\sigma_0\otimes\bar\sigma$ anti-involution

on $\Mat(2n,\CC)$ acts in the following way:

$$m\mapsto \begin{pmatrix}
0 & \Id\\ \Id & 0
\end{pmatrix}\bar m^T\begin{pmatrix}
0 & \Id\\ \Id & 0
\end{pmatrix}.$$

$$=\left\{m\in\Mat(2n,\CC\{i\})\midwd m=-\Ome{\Id}m^T\Ome{\Id}\right\}=$$

$$=\left\{m\in\Mat(2n,\CC\{i\})\midwd m=\begin{pmatrix}
0 & \Id\\ \Id & 0
\end{pmatrix}\bar m^T\begin{pmatrix}
0 & \Id\\ \Id & 0\end{pmatrix}\right\},$$

$$=\left\{m\in\Mat(2n,\CC\{i\})\midwd m=-\Ome{\Id}\bar m\Ome{\Id}\right\}=$$

q_1 & q_2\\ -\bar q_2 & \bar q_1
\end{pmatrix}\midwd q_1,q_2\in\Mat(n,\CC\{i\})\right\}.$$

§.§.§  and

The following map:

$$\phi\colon \Mat(n,\HH\{I,J,K\})\otimes_\R\HH\{i,j,k\}\to \Mat(4n,\R)$$

defined on generators of $A_\HH$ as follows:

$$\phi(a\otimes i)=\begin{pmatrix}
0 & a & 0 & 0 \\ -a & 0 & 0 & 0 \\ 0 & 0 & 0 & -a \\ 0 & 0 & a & 0
\end{pmatrix},\;
\phi(a\otimes j)=\begin{pmatrix} 0 & 0 & a & 0 \\ 0 & 0 & 0 & a \\ -a & 0 & 0 & 0 \\ 0 & -a & 0 & 0
\end{pmatrix},$$

$$\phi(a\otimes k)=\begin{pmatrix}
0 & 0 & 0 & a \\ 0 & 0 & -a & 0 \\ 0 & a & 0 & 0 \\ -a & 0 & 0 & 0
\end{pmatrix},\;
\phi(aI\otimes 1)=\begin{pmatrix} 0 & -a & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & 0 & 0 & -a \\ 0 & 0 & a &
\end{pmatrix},$$

$$\phi(aJ\otimes 1)=\begin{pmatrix}
0 & 0 & -a & 0 \\ 0 & 0 & 0 & a \\ a & 0 & 0 & 0 \\ 0 & -a & 0 & 0
\end{pmatrix},\;
\phi(aK\otimes 1)=\begin{pmatrix} 0 & 0 & 0 & -a \\ 0 & 0 & -a & 0 \\ 0 & a & 0 & 0 \\ a & 0 & 0 &
\end{pmatrix}$$

where $a\in\Mat(n,\R)$ is an $\R$-algebra isomorphism.

The anti-involution $\sigma_1\otimes\sigma_0$ corresponds under $\phi$ to the following

on $\Mat(4n,\R)$: $m\mapsto -\Xi m^T \Xi$ where

0 & 0 & 0 & \Id_n \\ 0 & 0 & -\Id_n & 0 \\ 0 & \Id_n & 0 & 0 \\ -\Id_n & 0 & 0 & 0
\end{pmatrix}.$$

The anti-involution $\sigma_1\otimes\sigma_1$ corresponds under $\phi$ to the transposition on

$$\phi((\Mat(n,\HH\{I,J,K\})\otimes_\R\HH\{i,j,k\})^{\sigma_1\otimes\sigma_0})=\left\{m\in\Mat(4n,\R)\mid m=-\Xi m^T \Xi\right\}=$$

The real locus $\Mat(n,\HH\{I,J,K\})$ of $\Mat(n,\HH\{I,J,K\})\otimes_\R\HH\{i,j,k\}$ is mapped by $\phi$

a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a
\end{pmatrix}\midwd a,b,c,d\in\Mat(n,\R)\right\}.$$

§.§ Embeddings between matrix algebras

In this section, we consider the following two embeddings:

$$\Mat(n,\CC\{I\})\otimes\CC\{j\}\hookrightarrow \Mat(n,\CC\{I\})\otimes\HH\{i,j,k\},$$

$$\Mat(n,\HH\{I,J,K\})\otimes\CC\{j\}\hookrightarrow \Mat(n,\HH\{I,J,K\})\otimes\HH\{i,j,k\}.$$

We use these embedding to see the symmetric space for a real group inside the symmetric space for a complexified group.

§.§.§ Embedding

In the previous sections, we have seen the isomorphisms:

\chi\colon & \Mat(n,\CC\{I\})\otimes_\R\CC\{j\} & \to & \Mat(n,\CC\{j\})\times\Mat(n,\CC\{j\})\\ & a+bI &
\mapsto & (a+bj , a-bj)
\end{matrix}$$

where $a,b\in\Mat(n,\CC\{j\})$ and

\psi\colon & \Mat(n,\CC\{I\})\otimes_\R\HH\{i,j,k\} & \to & \Mat(2n,\CC\{i\})\\
& (q_1+q_2j) + (p_1+p_2j)I & \mapsto &
\begin{pmatrix}
q_1+p_1i & q_2+p_2i\\ -\bar q_2-\bar p_2i & \bar q_1+\bar p_1i
\end{pmatrix}.
\end{matrix}$$

where $q_1,q_2,p_1,p_2\in\Mat(n,\CC\{i\})$.

We want to describe the map

$$\psi\circ\iota\circ\chi^{-1}\colon \Mat(n,\CC\{j\})\times\Mat(n,\CC\{j\})\to \Mat(2n,\CC\{i\})$$

$$\iota\colon\Mat(n,\CC\{I\})\otimes\CC\{j\}\hookrightarrow \Mat(n,\CC\{I\})\otimes\HH\{i,j,k\}$$

is the natural embedding. Let $(a,b):=(a_1+a_2j,b_1+b_2j)\in\Mat(n,\CC\{j\})\times\Mat(n,\CC\{j\})$ for
$a_1,a_2,b_1,b_2\in\Mat(n,\R)$, then

a_1+b_1+(a_2-b_2)i & a_2+b_2-(a_1-b_1)i\\ -(a_2+b_2)+(a_1-b_1)i & a_1+b_1+(a_2-b_2)i
\end{pmatrix}.$$

We also describe the image of the map $\psi\circ\iota\circ\chi^{-1}$:

q & p\\ -p & q
\end{pmatrix}\midwd p,q\in \Mat(n,\CC\{i\})\right\}=$$

$$=\left\{m\in\Mat(2n,\CC\{i\})\midwd m=-\Ome{\Id}m\Ome{\Id}\right\}.$$

§.§.§ Embedding

In the previous sections, we have seen the isomorphisms:

\psi\colon & \Mat(n,\HH\{I,J,K\})\otimes_\R\CC\{j\} & \to & \Mat(2n,\CC\{I\})\\
& (q_1+q_2J) + (p_1+p_2J)j & \mapsto &
\begin{pmatrix}
q_1+p_1I & q_2+p_2I\\ -\bar q_2-\bar p_2I & \bar q_1+\bar p_1I
\end{pmatrix}.
\end{matrix}$$

where $q_1,q_2,p_1,p_2\in\Mat(n,\CC\{I\})$ and

$$\phi\colon \Mat(n,\HH\{I,J,K\})\otimes_\R\HH\{i,j,k\}\to \Mat(4n,\R)$$

defined as in the Section <ref>.

We want to describe the image of the map

$$\phi\circ\iota\circ\psi^{-1}\colon \Mat(2n,\CC\{I\})\to\Mat(4n,\R)$$

$$\iota\colon\Mat(n,\HH\{I,J,K\})\otimes\CC\{j\}\hookrightarrow \Mat(n,\HH\{I,J,K\})\otimes\HH\{i,j,k\},$$

is the natural embedding.

Note that an element $x\in
\Mat(n,\HH\{I,J,K\})\otimes\HH\{i,j,k\}$ is contained in the subalgebra $\Mat(n,\HH\{I,J,K\})\otimes\CC\{j\}$ if and only if $x$
commutes with $1\otimes j$. So we obtain:

$$\Imm(\psi\circ\iota\circ\chi^{-1})=\left\{m\in\Mat(4n,\R)\mid m=
-\phi(\Id_n\otimes j)m\phi(\Id_n\otimes j) \right\}.$$

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