# A quantum group for the Einstein equations

Giuseppe Iurato111e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we expose the construction of a possible, simple matrix quantum
group structure (according to Woronowicz), related to elementary formal
aspects of the Einstein field equations of General Relativity, and its
possible symmetries.

Mainly, we present a simple application of the results achieved by M. Dubois-
Violette and G. Launer in [1], where is built up a first matrix quantum group
structure (in the sense of S.L. Woronowics $-$ see [8], 2.1) associated to an
arbitrary non-degenerate bilinear form.

Precisely, we apply, almost verbatim, these considerations to a generalization
of the Einstein field equations (1915), in purely covariant form given
by222According to the Robertson-Noonan sign convention (1968) (see [4]). (see
[6], 1.13.5, and [5], 4.0, 4.1)

$G_{ij}=R_{ij}-\frac{1}{2}Rg_{ij}=-8\pi GT_{ij},\qquad i,j=0,1,2,3,$ $None$

where $G_{ij}$ is the Einstein curvature tensor, $R_{ij}$ is the Ricci
curvature tensor, $g_{ij}$ is the Lorentz metric, $R$ is the Ricci scalar, $G$
is the gravitational constant, and $T_{ij}$ is the so called Hilbert tensor
(see [9], Chapter 7) with $T_{ij}=T_{ij}(g_{lh},\partial
g_{lh};\psi_{k},\partial\psi_{k})$ in the presence of a set of physical fields
$\psi_{k}\ \ k=1,...,p$.
In the geometrized units, it is $G=1$ (see [7]).

We recall that the Einstein field equations (1) may be deduced both by a
variational Palatini’s argument (see [9]) and, inductively, by the newtonian
Poisson’s equation $\Delta\phi=4\pi G\rho$.
Following the latter way, it is assumed that the field equations for the
gravitational field, that we may call generalized Einstein (field) equations,
should have a general form of the type (see [9], Chapter 4, [10], Cap. II, §
2.1, and [4], Chapter 17, § 17.1)

$G_{ij}(g_{lh},g_{lh,r},g_{lh,rt},...)=k\pi T_{ij},\qquad i,j=0,1,2,3,$ $None$

where $G_{ij}$ is a yet to be determined tensor function of the metric tensor
$g_{lh}$ and some of its derivatives, and $k$ is a real constant.

Many physical reasons (see [10], § 2.1, and [4], § 17.1) restricts the class
of the possible functions $G_{ij}$, satisfying (2), to a well-defined tensor,
namely the Einstein curvature tensor mentioned above, obtaining the known
equations (1).

On the other hand, by earlier Weyl’s and Cartan’s results culminated in
Lovelock’s statement (see [16]), if we seek a tensor equation of the form
$G_{ij}=T_{ij}$, where the components $A_{ij}$ involve the metric tensor
$g_{ij}$ and its first and second derivatives (hence, assuring second-order
partial differential equations generalizing the Poisson one), and if $A_{ij}$
have vanishing divergence $A_{ij;j}=0$, then the equation must be of the form
$aG_{ij}+bg_{ij}=-8\pi kT_{ij}$, where $a$ and $b$ are constants; Einstein’s
choice is then $a=1,b=0$ ($b$ is said to be the cosmological constant).

At this point, taking into account the geometrical meaning of the Einstein’s
equations333These arguments shall be the matter of another paper. according to
[17], it is possible to consider the following Einstein bilinear form

$\Omega_{ij}\doteq G_{ij}+8\pi T_{ij},\qquad i,j=0,1,2,3,$ $None$

that, for now, we suppose to be non-degenerate; its zero values are the
generalized Einstein equations (2). In $(2^{\prime})$, we suppose, a priori,
$G_{ij}$ to be an arbitrary bilinear form (of $\mathbb{R}^{4}$), while
$T_{ij}$ is the Hilbert tensor.

In [1] (see also [11], Example 4.62), it is considered a finite family
$\\{T({\alpha})\\}_{\alpha\in\Xi}$ of $(r_{\alpha},s_{\alpha})$-tensors on
$\mathbb{R}^{n}$ and the group $G$ of the automorphisms of $\mathbb{R}^{n}$
that preserve $T({\alpha})$ in the following sense

$u_{k_{1}}^{i_{1}}...u_{k_{r_{\alpha}}}^{i_{r_{\alpha}}}T(\alpha)_{j_{1}...j_{s_{\alpha}}}^{k_{1}...k_{r_{\alpha}}}=u_{j_{1}}^{k_{1}}...u_{j_{s_{\alpha}}}^{k_{s_{\alpha}}}T(\alpha)_{k_{1}...k_{s_{\alpha}}}^{i_{1}...i_{r_{\alpha}}}\qquad\forall\alpha\in\Xi,$
$None$

supposing invertible the generic matrix $u=\|u_{j}^{i}\|\in G$.

In matrix quantum group theory (see [2]), one can considers the elements
$u_{j}^{i}$ as linear coordinate functions on $G$, which assigns to each $g\in
G$ its matrix elements (respect to a given base), namely
$u_{j}^{i}(g)=g_{j}^{i}$, and that one can also interprets as generating the
unital associative algebra $Fun(G)$ of functions on $G$, under the relations
(3).
The latter is a commutative Hopf algebra, with usual comultiplication given by
$(\Delta(f))(g_{1},g_{2})=f(g_{1}g_{2})$, so that the cocommutativity, or not,
of this algebra, is related to the commutativity, or not, of the group $G$;
furthermore, the coproduct is induced by $\Delta u_{j}^{i}=u_{k}^{i}\otimes
u^{k}_{j}$, since $u_{j}^{i}(g)=g_{j}^{i}$.

Hence, following [1], we could say that (3) defines a first (matrix) quantum
group structure preserving each $T(\alpha)$; moreover, we restricts our study
to the case in which $T(\alpha)$ is a given non-degenerate bilinear form
$\Omega_{ij}$ on $\mathbb{R}^{4}$, with dual $\Omega^{ij}$ (given by the
inverse matrix), that is we suppose $r_{\alpha}=0,s_{\alpha}=2$, $card\ \Xi=1$
and444However, the following considerations holds true also for any $n\geq 2$.
$n=4$.

If $\Omega$ is a bilinear form on $\mathbb{R}^{4}$ with components (respect to
a given base) $\Omega_{ij}$, and $\tilde{\Omega}$ is a bilinear form on its
dual with components (respect to the dual base) $\tilde{\Omega}^{ij}$, then,
as known, $\tilde{\Omega}\otimes\Omega$ is identified with the endomorphisms
of $\mathbb{R}^{4}\otimes\mathbb{R}^{4}$ with components
$\Omega^{i_{1}i_{2}}\Omega_{j_{1}j_{2}}$; likewise, if $u$ and $v$ are
endomorphisms of $\mathbb{R}^{4}$ with components $u_{j}^{i}$ and $v_{j}^{i}$,
then $u\otimes v$ is identified with the endomorphism of
$\mathbb{R}^{4}\otimes\mathbb{R}^{4}$ with components
$u_{j_{1}}^{i_{1}}v_{j_{2}}^{i_{2}}$.

Let $\Omega$ be the non-degenerate bilinear form with components (in the
canonical base) $\Omega_{ij}$ given by (3); the matrix of its components
$\Omega_{ij}$, will be denoted again by $\Omega$. Associated to $\Omega$ is
its dual $\Omega^{-1}$ of $\mathbb{R}^{4}\otimes\mathbb{R}^{4}$, that is the
bilinear form on the dual of $\mathbb{R}^{4}$ ($\cong\mathbb{R}^{4}$), with
components $\Omega^{ij}$ defined by $\Omega^{ik}\Omega_{kj}=\delta_{j}^{i}$;
the matrix of the components $\Omega^{ij}$ will be again denoted by
$\Omega^{-1}$, the inverse of the matrix $\Omega$ (that there exists because
$\Omega$ is non-degenerate).

Let $\mathcal{A}_{\mathbb{R}}(\Omega)$ be the unital associative
$\mathbb{R}$-algebra generated by the scalars $t_{j}^{i}\in\mathbb{R}\ \
i,j=0,1,2,3$, with the relations

$\Omega_{ij}t_{k}^{i}t_{l}^{j}=\Omega_{kl},\qquad\Omega^{ij}t^{k}_{i}t_{j}^{l}=\Omega^{kl},\qquad
k,l=0,1,2,3,$

where $\Omega_{kl},\Omega^{kl}\in\mathbb{R}$ are identified, respectively,
with
$\Omega_{kl}1_{\mathcal{A}},\Omega^{kl}1_{\mathcal{A}}\in\mathcal{A}_{\mathbb{R}}(\Omega)$,
if $1_{\mathcal{A}}$ is the unit of $\mathcal{A}_{\mathbb{R}}(\Omega)$.
Hence, it is possible to prove (see [1]) that

  1. 1.

there exists a unique homomorphism of algebras, say
$\Delta:\mathcal{A}_{\mathbb{R}}(\Omega)\rightarrow\mathcal{A}_{\mathbb{R}}(\Omega)\otimes\mathcal{A}_{\mathbb{R}}(\Omega)$,
such that $\Delta t_{j}^{i}=t_{k}^{i}\otimes t_{j}^{k}\ \ i,j=0,1,2,3$;

  2. 2.

there exists a unique homomorphism of algebras, say
$\varepsilon:\mathcal{A}_{\mathbb{R}}(\Omega)\rightarrow\mathbb{R}$, such that
$\varepsilon(t_{j}^{i})=\delta_{j}^{i}\ \ i,j=0,1,2,3$;

  3. 3.

there exists a unique linear antimultiplicative mapping, say
$S:\mathcal{A}_{\mathbb{R}}(\Omega)\rightarrow\mathcal{A}_{\mathbb{R}}(\Omega)$,
such that555Setting $t=\|t_{j}^{i}\|$, it is $S(t)=(\Omega^{-1})^{t}t\Omega$.
$S(t_{j}^{i})=\Omega^{ik}t_{k}^{l}\Omega_{lj}\ \ i,j=0,1,2,3,$ and
$S(1_{\mathcal{A}})=1_{\mathcal{A}}$.

Furthermore, $\Delta$ is a coproduct, $\varepsilon$ is a counit, and $S$ is an
antipode666In general, there is no antipode for a generic tensor $T(\alpha)$.
since $S(t_{k}^{i})t_{j}^{k}=t^{i}_{k}S(t_{j}^{k})$, so that, denoted by $m$
the product of the algebra $\mathcal{A}_{\mathbb{R}}(\Omega)$, we have that
$(\mathcal{A}_{\mathbb{R}}(\Omega),m,1_{\mathcal{A}},\Delta,\varepsilon,S)$ is
a Hopf algebra, called the Hopf algebra of the Einstein bilinear form
$\Omega$.

This Hopf algebra defines (in the terminology of [1]; see also [12], Appendix
2) the quantum group of the non-degenerate bilinear form $\Omega$, that we may
call the Einstein quantum group; hence, as usual, we may think
$\mathcal{A}_{\mathbb{R}}(\Omega)$ as a kind of algebra of ’representative
functions’ on this quantum group.

Besides, this quantum group extends the classical group of the linear
transformations of $\mathbb{R}^{4}$ which preserves $\Omega$, and, therefore,
such a quantum object may represents further generalized symmetries of the
Einstein bilinear form $\Omega$.

The matrix $t=\|t_{j}^{i}\|\in M^{(4,4)}(\mathcal{A}_{\mathbb{R}}(\Omega))$ is
a multiplicative matrix (see [3]) whose entries generates
$\mathcal{A}_{\mathbb{R}}(\Omega)$, obtaining an example of matrix quantum
group.

Note. All the above considerations about $\mathcal{A}_{\mathbb{R}}(\Omega)$,
holds for an arbitrary non-degenerate bilinear form $\Omega$ of
$\mathbb{R}^{n}$, with $n\geq 2$.

Given a non-degenerate bilinear form $\Omega$ on $\mathbb{R}^{n}$ ($n\geq 2$),
with components $\Omega_{ij}$ (respect to the canonical base), we may define
the quadratic homogeneous algebra777For brief recalls on homogeneous algebras,
see [13] or [12], Appendix 1, and references therein.
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ generated by the elements $x^{j}\ \
j=1,...,n$, with the relations $\Omega_{ij}x^{i}x^{j}=0$.
In [12], § 2. (see also [13]), it is proved as
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ be a Gorenstein and Koszul algebra of
global dimension 2. Conversely, it is possible to prove that any quadratic
algebra generated by $n$ elements $x^{j}$, finitely generated in degree 1 and
finitely presented with relations of degree $\geq 2$, which is Gorenstein and
Koszul of low global dimension 2, is an algebra of the type
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ for a certain non-degenerate bilinear form
$\Omega$.

Moreover, if $\Omega\stackrel{{\scriptstyle\chi}}{{\rightarrow}}\Omega\circ M$
is the action given by $(\Omega\circ M)(x,y)=\Omega(Mx,My)$ for each $M\in
GL_{n}(\mathbb{R})$ and $x,y\in\mathbb{R}^{n}$, then it follows that $\chi$
preserves the non-degeneracy of bilinear forms, and
$\mathcal{Q}_{\mathbb{R}}(\Omega)\cong\mathcal{Q}_{\mathbb{R}}(\Omega^{\prime})$
if and only if $\Omega$ and $\Omega^{\prime}$ belong to the same
$GL_{n}(\mathbb{R})$-orbit of $\chi$, that is, if and only if
$\Omega^{\prime}=\Omega\circ M$ for some $M\in GL_{n}(\mathbb{R})$.
Therefore, since the action of $\chi$ corresponds to a change of generators in
$\mathcal{A}_{\mathbb{R}}(\Omega)$, it follows that
$\mathcal{A}_{\mathbb{R}}(\Omega)$ only depends by the orbit of $\Omega$ under
$\chi$.
So, we may define the moduli space
$\mathcal{M}(\mathcal{Q}_{\mathbb{R}}(\Omega))$ of
$\mathcal{Q}_{\mathbb{R}}(\Omega)$, to be the space of all
$GL_{n}(\mathbb{R})$-orbits of $\chi$.

Furthermore, taking into account what has been said above about
$\mathcal{A}_{\mathbb{R}}(\Omega)$ in $\mathbb{R}^{n}$, by Proposition 20 of
[12], Appendix 2, follows that there is a unique algebra homomorphism
$\Delta_{t}:\mathcal{Q}_{\mathbb{R}}(\Omega)\rightarrow\mathcal{A}_{\mathbb{R}}(\Omega)\otimes\mathcal{Q}_{\mathbb{R}}(\Omega)$
such that $\Delta_{t}(x^{j})=t^{j}_{i}\otimes x^{i}$ for all $j=1,...,n$,
endowing $\mathcal{Q}_{\mathbb{R}}(\Omega)$ of a
$\mathcal{A}_{\mathbb{R}}(\Omega)$-comodule structure.
Hence, the quantum group of $\Omega$ coacts on the quantum space corresponding
to $\mathcal{Q}_{\mathbb{R}}(\Omega)$, that is
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ corresponds to the natural quantum space
for the coaction of $\mathcal{A}_{\mathbb{R}}(\Omega)$.

Come back to the case $n=4$, in [1] the Hopf algebra
$\mathcal{A}_{\mathbb{R}}(\Omega)$ is also endowed with a particular quasi-
triangular structure through a $R$-matrix, say
$\mathcal{R}:\mathbb{R}^{4}\otimes\mathbb{R}^{4}\rightarrow\mathbb{R}^{4}\otimes\mathbb{R}^{4}$,
given by $\mathcal{R}_{a}=\tau+a(\Omega^{-1})^{t}\otimes\Omega$, where
$a\in\mathbb{R}\setminus\\{0\\}$ and $\tau$ is the flip map.

Indeed, for $a\neq 0$, we have the following homogeneous defining relations of
$\mathcal{A}_{\mathbb{R}}(\Omega)$:

$\mathcal{R}^{i_{1}i_{2}}_{k_{1}k_{2}}t^{k_{1}}_{j_{1}}t^{k_{2}}_{j_{2}}=t^{i_{1}}_{k_{1}}t^{i_{2}}_{k_{2}}\mathcal{R}^{k_{1}k_{2}}_{j_{1}j_{2}},\qquad
i_{l},j_{l}=0,1,2,3,\quad l=1,2,$

so that such $\mathcal{R}$ is a $R$-matrix because it satisfy the following,
well-known Yang-Baxter equation
$\mathcal{R}_{12}\mathcal{R}_{13}\mathcal{R}_{23}=\mathcal{R}_{23}\mathcal{R}_{13}\mathcal{R}_{12}$
when and only when $a\in\mathbb{R}\setminus\\{0\\}$ verify the braid relation
$a(1+a\Omega^{ij}\Omega_{ij}+a^{2})=0$, equivalent (since $a\neq 0$) to
$a+a^{-1}+\Omega^{ij}\Omega_{ij}=a+a^{-1}+tr(\Omega^{-1}\Omega^{t})=0$.
Thus, we have a $R$-matrix for $\mathcal{A}_{\mathbb{R}}(\Omega)$, given by
$\mathcal{R}=\tau+a(\Omega^{-1})^{t}\otimes\Omega$ with
$a\in\mathbb{R}\setminus\\{0\\}$ such that
$a+a^{-1}+\Omega^{ij}\Omega_{ij}=0$.

In [14], it is proved that the representation category of
$\mathcal{A}_{\mathbb{R}}(\Omega)$ (in $\mathbb{R}^{n},\ \ n\geq 2$) is
monoidally equivalent to the representation category of the quantum group
$\mathcal{O}_{a}(SL_{2}(\mathbb{R}))$ of functions over $SL_{2}(\mathbb{R})$,
if $a\in\mathbb{R}\setminus\\{0\\}$ verify the above braid relation, so that
$Comod(\mathcal{A}_{\mathbb{R}}(\Omega))\cong^{\otimes}Comod(O_{q}(SL_{2}(\mathbb{R}))$.
Moreover, in the § 5. of [14] it is also presented the following isomorphic
classification of the Hopf algebra $\mathcal{A}_{\mathbb{R}}(\Omega)$: if
$\Omega$ and $\Omega^{\prime}$ are non-degenerate bilinear forms respectively
in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ with $n,m\geq 2$, then
$\mathcal{A}_{\mathbb{R}}(\Omega)$ and
$\mathcal{A}_{\mathbb{R}}(\Omega^{\prime})$ are isomorphic if and only if
$m=n$ and there exists $M\in GL_{n}(\mathbb{R})$ such that
$\Omega^{\prime}=M^{t}\Omega M$.
Then, in the § 6 of [14], the Author determines the possible Hopf
$\ast$-algebra structures and CQG (compact quantum group) algebra structures
on $\mathcal{A}_{\mathbb{C}}(\Omega)$ (that is, in the complex case).
Following the results of [14], T. Aubriot, in [15], studies the possible
Galois and bi-Galois objects over $\mathcal{A}_{\mathbb{R}}(\Omega)$.

At last, the paper [1] finishes with some remarks; in particular, the Authors
notices that, in dimension $n\geq 3$, there is no $\Omega$ such that
$\mathcal{A}_{\mathbb{R}}(\Omega)$ be commutative, that is to say, a such Hopf
algebra is necessarily non-commutative.

On the other hand, we remember that in $\mathbb{R}^{4}$ may be establish a
standard canonical complex structure as follows.
Respect to the canonical base of $\mathbb{R}^{4}$, if $J_{0}\in End\
(\mathbb{R}^{4})$ is defined putting $J_{0}(e_{j})=e_{n+j}$ for $1\leq j\leq
2$ and $J_{0}(e_{j})=-e_{j-n}$ for $3\leq j\leq 4$, then it follows that such
a $J_{0}$ is a complex structure888Since $J_{0}^{2}=-id_{\mathbb{R}^{4}}$. on
$\mathbb{R}^{4}$, and if $\mathbb{R}^{4}_{\mathbb{C}}(J_{0})$ is the resulting
linear complex space structure induced by $J_{0}$ on $\mathbb{R}^{4}$, then we
have the canonical isomorphism
$\mathbb{R}^{4}_{\mathbb{C}}(J_{0})\cong\mathbb{C}^{2}$.
From here, it is possible to construct the following faithful representation
$\rho:M^{(2,2)}(\mathbb{C})\rightarrow M^{(4,4)}(\mathbb{R})$ defined by

$\rho(A+iB)=\left(\begin{array}[]{cc}A&-B\\\ B&A\end{array}\right),$

that it is a $\mathbb{R}$-algebra monomorphism such that
$\rho(iH)=J_{0}\rho(H)$ for any $H\in M^{(2,2)}(\mathbb{C})$, extending the
usual immersion999For any $n\geq 2$, we remember that there exists a well-
known immersion $GL_{n}(\mathbb{C})\hookrightarrow GL_{2n}(\mathbb{R})$.
$GL_{2}(\mathbb{C})\hookrightarrow GL_{4}(\mathbb{R})$.

Hence, if $\Omega_{ij}\in M^{(4,4)}(\mathbb{R})$ of $(2^{\prime})$, is such
that $\Omega_{ij}\in\rho(M^{(2,2)}(\mathbb{C}))$, let
$\tilde{\Omega}_{ij}=\rho^{-1}(\Omega_{ij})\in M^{(2,2)}(\mathbb{C})$; whence,
we may identifies $\Omega_{ij}$ with $\tilde{\Omega}_{ij}$, that it is a non-
degenerate (if such is $\Omega_{ij}$) bilinear form of $\mathbb{C}^{2}$.
Therefore, if $\mathcal{A}_{\mathbb{C}}(\tilde{\Omega})$ is the Hopf algebra
associated to $\tilde{\Omega}$, then it is immediate to prove that
$\mathcal{A}_{\mathbb{R}}(\Omega)\cong\mathcal{A}_{\mathbb{C}}(\tilde{\Omega})$.
In [1], § 6., there is a complete classification of the moduli space of
$\tilde{\Omega}$, according to the rank of $\tilde{\Omega}$. Precisely

  * •

if $rk\ \tilde{\Omega}=0$, then there is only one orbit of which one
representative element is $\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)$, this case corresponding to $SL_{2}(\mathbb{C})$ with
$R$-matrix the identity $R_{0}$ of $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$;

  * •

if $rk\ \tilde{\Omega}=1$, then there is only one orbit of which one
representative element is $\left(\begin{array}[]{cc}0&-1\\\
1&\lambda\end{array}\right)$ with $\lambda\neq 0$ (these are all equivalent
among them), this case corresponding to the so called Manin’s jordanian (that
it is a special quantum deformation of $SL_{2}(\mathbb{C})$; see [3]), with
equivalent $R$-matrices $R_{\lambda}$ such that $\lim_{\lambda\rightarrow
0}R_{\lambda}=R_{0}$;

  * •

if $rk\ \tilde{\Omega}=2$, then there are many orbits, each represented by
$\tilde{\Omega}_{q}=\left(\begin{array}[]{cc}0&-1\\\ q&0\end{array}\right)$
for every $q\in\mathbb{C}\setminus\\{0,1\\}$, with
$\mathcal{A}_{\mathbb{C}}(\tilde{\Omega}_{q})\cong SL_{2,q}(\mathbb{C})$,
$R$-matrix corresponding to that of $M_{2,q}(\mathbb{C})$ (quantum deformation
of $M^{(2,2)}(\mathbb{C})$; see [3]), and
$\mathcal{Q}_{\mathbb{C}}(\tilde{\Omega}_{q})$ corresponding to the Manin
plane (that it is the natural quantum space for the coaction of
$SL_{2,q}(\mathbb{C})$).

The considerations of this paper, may have physical interpretations in view of
the possible physical meaning of $\Omega$ (and of $\tilde{\Omega}$, when
$\tilde{\Omega}$ exists).

$\bf References.$

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[15] T. Aubriot, ”On the classification of Galois objects over the quantum
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[16] D. Lovelock, ”The four-dimensionality of space and the Einstein’s
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[17] T. Frankel, Gravitational Curvature, W.H. Freeman and Comp., San
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