# Wick-rotations of
pseudo-Riemannian Lie groups

Christer Helleland

###### Abstract.

We study Wick-rotations of left-invariant metrics on Lie groups, using results
from real GIT ([1], [2], [3]). An invariant for Wick-rotation of Lie groups is
given, and we describe when a pseudo-Riemannian Lie group can be Wick-rotated
to a Riemannian Lie group. We also prove a general version (for general Lie
algebras) of $\acute{E}$. Cartan’s result, namely the existence and conjugacy
of Cartan involutions.

Faculty of Science and Technology,

University of Stavanger,

N-4036 Stavanger, Norway

<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 A Wick-rotation of a Lie group
  3. 3 An invariant of Wick-rotation of Lie groups
    1. 3.1 The main theorem
    2. 3.2 The Riemannian case
    3. 3.3 Conjugacy of Cartan involutions of a left-invariant metric
    4. 3.4 Wick-rotating to a compact semi-simple Riemannian Lie group
    5. 3.5 Wick-rotating a Lorentzian signature
  4. 4 Some more Wick-rotatable properties
    1. 4.1 Wick-rotating to a Riemannian space
    2. 4.2 Wick-rotating an algebraic soliton
  5. 5 A remark on Wick-rotatable tensors of Lie groups
  6. 6 A Cartan involution of an arbitrary real Lie algebra

##  1\. Introduction

This paper is motivated by the natural example of a Wick-rotation of a pseudo-
Riemannian semi-simple Lie group $(G,-\kappa)\subset(G^{\mathbb{C}},-\kappa)$
equipped with the Killing form. In this case $G$ can be Wick-rotated to a
Riemannian Lie group, thus an interesting question one may ask:

Given a pseudo-Riemannian Lie group $G$, when can it be Wick-rotated to a
Riemannian Lie group $\tilde{G}$?

In view of the results given in ([1], [2], [3]), then certain tensors
restricted to the Lie algebra $\mathfrak{g}$ (so called Wick-rotatable
tensors) must be fixed by the isometry action of some Cartan involution
$\theta\in O(p,q)$ of the metric. However certain tensors for a left-invariant
metric (for instance the Levi-Civita connection, the Riemann tensor and so
on..) are very interlinked with the Lie bracket of the Lie algebra
$\mathfrak{g}$. Thus in the semi-simple case (equipped with the Killing form)
such tensors are naturally fixed by the Cartan involutions of the Lie algebra:
$\theta\in Aut(\mathfrak{g})$. For example the Levi-Civita connection is given
by: $\nabla_{x}y=\frac{1}{2}[x,y]$, thus naturally
$\theta\cdot\nabla_{x}y=\nabla_{x}y$. The author of this paper therefore
pondered about the existence of a Cartan involution: $\theta\in
Aut(\mathfrak{g})$, for a general left-invariant metric (on a general Lie
group $G$) which can be Wick-rotated to a Riemannian Lie group $\tilde{G}$. We
prove an invariant (Theorem 1) for Wick-rotations of Lie groups based on the
observations of the example above, and thus we give a complete answer to the
question above.

######  Remark 1.

In this paper a Riemannian space shall always denote the signature:
$(+,+,\cdots,+)$, and a Lorentzian space shall denote the signature:
$(+,+,\cdots,+,-)$ and so on. The anti-isometry map $g\mapsto-g$ induces an
isomorphism $O(p,q)\cong O(q,p)$. If we change signature via this anti-
isometry map, then our results in this paper will be related precisely via
this map as well.

##  2\. Preliminaries

###  2.1. A Wick-rotation of a Lie group

In this paper a real Lie group $G$ shall be said to be an immersive real form
of a complex Lie group $G^{\mathbb{C}}$, if there is a real immersion
$G\rightarrow G^{\mathbb{C}}$ (of Lie groups) where $G^{\mathbb{C}}$ is viewed
as a real Lie group, such that $\mathfrak{g}$ is embedded as a real form of
$\mathfrak{g}^{\mathbb{C}}$ (the Lie algebra of $G^{\mathbb{C}}$). If the
immersion is also injective then we shall call $G$ a virtual real form. A
virtual real form $G$ which is also an embedding (i.e the image of $G$ is
closed in $G^{\mathbb{C}}$), we shall say that the real form is an embedded
real form. An embedded real form which also satisfies: $G^{\mathbb{C}}=G\cdot
G^{\mathbb{C}}_{0}$ (abstract group product) shall be said to be a real form.

Note that a connected embedded real form is also a real form. All these
specialised ”complexifications” divide the Lie groups into different classes.
For instance if $G$ is a connected semi-simple Lie group, then it is a fact
that $G$ is a virtual real form if and only if $G$ is linear.

One shall note that given any connected real Lie group $G$, then we can
complexify the Lie algebra via an inclusion $i$:
$\mathfrak{g}\hookrightarrow\mathfrak{g}^{\mathbb{C}}$. We can find a complex
connected Lie group: $G^{\mathbb{C}}$ with Lie algebra
$\mathfrak{g}^{\mathbb{C}}$. Thus using the exponential maps of these groups,
we can find a smooth map (of real Lie groups): $G\rightarrow G^{\mathbb{C}}$
with differential $i$, thus $G$ is an immersive real form of $G^{\mathbb{C}}$.

We shall abuse notation and write $G\subset G^{\mathbb{C}}$ for an immersive
real form.

######  Example 1.

Consider the complex orthogonal group: $O(4,\mathbb{C})$, then the map:
$g\mapsto I_{3,1}\overline{g}I_{3,1},$ is a conjugation map (i.e the
differential is a conjugation map), where $(I_{3,1})_{ii}=+1$ for $1\leq i\leq
2,\ (I_{3,1})_{33}=-1$ and zero otherwise. The fix points of this map is just
$O(1,3)$, which is an example of a real form of $O(4,\mathbb{C})$. Consider
the universal covering group $G:=\widetilde{SL_{2}(\mathbb{R})}$ of
$SL_{2}(\mathbb{R})$, then it is a fact that $G$ is not a virtual real form of
any complex Lie group. However $G$ is an immersive real form of
$SL_{2}(\mathbb{C})$.

We note that for a compact Lie group $G$, we can always complexify it to a
complex Lie group: $G^{\mathbb{C}}$, such that $G\subset G^{\mathbb{C}}$ is a
real form, by using the universal complexification group. Thus starting from a
compact Lie group with a left-invariant metric we naturally have a candidate
for a holomorphic Riemannian Lie group.

Following the definitions of Wick-rotations given in [2] then for Lie groups
we define:

######  Definition 1.

Let $G\subset G^{\mathbb{C}}\supset\tilde{G}$ be two immersive real forms
which are Wick-related in $(G^{\mathbb{C}},g^{\mathbb{C}})$ for
$g^{\mathbb{C}}$ a left-invariant holomorphic metric. Then we shall say that
the pseudo-Riemannian Lie group $(G,g)$ is Wick-rotated to
$(\tilde{G},\tilde{g})$.

We call $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ a real slice of Lie
groups, and shall write $(p,q)$ for the signature of $g$. If there is another
real slice $(\tilde{G},\tilde{g})\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ of
Lie groups, then we shall refer to the signature of $\tilde{g}$ as
$(\tilde{p},\tilde{q})$. Note that two Lie groups which are Wick-related are
also Wick-rotated at the identity point $p:=1$.

The definition implies that two Wick-rotatable metrics on real Lie groups are
left-invariant themselves, and also note that a Wick-rotation of Lie groups
induces in the obvious way a Wick-rotation of the identity components.
Moreover the property of bi-invariance for connected groups is an invariant:

######  Proposition 1.

Suppose $(G,g)$ is Wick-rotatable to $(\tilde{G},\tilde{g})$ and they are both
connected. Then $g(-,-)$ is bi-invariant if and only if $\tilde{g}(-,-)$ is
bi-invariant.

###### Proof.

The proofs given in ([5], Lemma 7.1 and 7.2) also hold for pseudo-Riemannian
left-invariant metrics, with $\mathfrak{o}(n)$ replaced with
$\mathfrak{o}(p,q)$. Moreover if the metric $g(-,-)$ is bi-invariant, then
because
$ad(\mathfrak{g})\subset\mathfrak{o}(p,q)\subset\mathfrak{o}(n,\mathbb{C})$,
and $ad(\mathfrak{g})^{\mathbb{C}}=ad(\mathfrak{g}^{\mathbb{C}})$ it follows
that the holomorphic metric must also be bi-invariant, thus also
$\tilde{g}(-,-)$. The converse is identical. ∎

Note that the property of being connected or simply connected are not
necessarily preserved under a Wick-rotation. However under a Wick-rotation of
real forms, then being connected is conserved.

######  Example 2.

Let $SL_{2}(\mathbb{R})\subset SL_{2}(\mathbb{C})\supset SU(2)$ be the natural
inclusions. Then they are real forms, and Wick-rotated w.r.t to the
holomorphic Killing form $\kappa$ on $\mathfrak{sl}_{2}(\mathbb{C})$. Note
that $(SL_{2}(\mathbb{R}),\kappa)$ is Lorentzian and $(SU(2),\kappa)$ has
signature: $(-,-,-)$.

Suppose we have a real slice of Lie groups:
$(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$. Let $O(p,q)$ denote the real
Lie group consisting of linear maps: $\mathfrak{g}\xrightarrow{A}\mathfrak{g}$
which preserves:

$g(Ax,Ay)=g(x,y)$

for all $x,y\in\mathfrak{g}$, then $O(p,q)$ is a real form of
$O(n,\mathbb{C})$ (isometries of $g^{\mathbb{C}}$), by noting the anti-
holomorphic involution (real structure):

$A\mapsto\sigma\circ A\circ\sigma,$

where $\sigma$ is the conjugation map of $\mathfrak{g}$ in
$\mathfrak{g}^{\mathbb{C}}$.

We recall from [2], that if $(E,g^{\mathbb{C}})$ is a holomorphic inner
product space, and $V,\tilde{V}$ and $W$ are real slices (of the same
dimension) such that $W$ is a compact real slice (i.e Euclidean signature),
then if they are pairwise compatible, the triple:
$\Big{(}V,\tilde{V},W\Big{)}$, is said to be a compatible triple. Note that
Example 2 is an example of a compatible triple:
$\Big{(}V:=\mathfrak{sl}_{2}(\mathbb{R}),\tilde{V}:=\mathfrak{su}(2),W:=\mathfrak{su}(2)\Big{)}$.

We also recall from [2]:

######  Definition 2.

Let $W\subset(E,g^{\mathbb{C}})$ be a real slice. We say an involution
$W\xrightarrow{\theta}W$, is a Cartan involution of $W$, if
$g_{\theta}(\cdot,\cdot):=g\big{|}_{W}(\cdot,\theta(\cdot))$, is an inner
product on $W$.

We shall call the eigenspace decomposition of a Cartan involution: $\theta$,
for the Cartan decomposition. Thus for a compact real slice: $W\subset E$,
then by definition we see that $\theta=1$. By the uniqueness of a signature
associated to a pseudo-inner product $g$ then all Cartan involutions are
conjugate in $O(p,q)$. In fact given two Cartan involutions: $\theta_{j}$
($j=1,2$) then $g\mapsto\theta_{j}g\theta_{j}$ is a global Cartan involution
of $O(p,q)$. Thus if $g\theta_{1}g^{-1}=\theta_{2}$ for some $g\in O(p,q)$,
then writing $g=k_{2}e^{x}$, where $k_{2}$ commutes with $\theta_{2}$ and
$x\in\mathfrak{o}(p,q)$, we obtain $\theta_{1}=e^{x}\theta_{2}e^{-x}$, and
therefore $\theta_{1},\theta_{2}$ are conjugate by an element $g\in
O(p,q)_{0}$.

Suppose now we have a Wick-rotation of two real Lie groups:
$(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$. Let
$\theta\in O(p,q)$ be a Cartan involution of the metric $g$, and let $W$
denote the corresponding unique compact real slice associated with $\theta$.
Then by [3] it is possible to find a real form
$V\subset\mathfrak{g}^{\mathbb{C}}$ (as vector spaces) and an isomorphism:
$V\xrightarrow{\phi}\tilde{\mathfrak{g}}$ such that $\phi^{\mathbb{C}}\in
O(n,\mathbb{C})$, and $(\mathfrak{g},V,W)$ is a compatible triple. So consider
the triple:
$\Big{(}\mathfrak{o}(p,q),\mathfrak{o}(\tilde{p},\tilde{q}),\mathfrak{o}(n)\Big{)}$,
of Lie algebras of the isometry groups associated with the compatible triple
$(\mathfrak{g},V,W)$.

The following result is important to note:

######  Lemma 1 ([2], Lemma 3.6).

The triple of real forms:
$\Big{(}\mathfrak{o}(p,q),\mathfrak{o}(\tilde{p},\tilde{q}),\mathfrak{o}(n)\Big{)}$,
embedded into $\mathfrak{o}(n,\mathbb{C})$ under a standard Wick-rotation is a
compatible triple of Lie algebras.

Let now
$\mathcal{V}:=\Big{(}\bigotimes_{i=1}^{k}\mathfrak{g}\Big{)}\bigotimes\Big{(}\bigotimes_{i=1}^{m}\mathfrak{g}^{*}\Big{)}$
and
$\tilde{\mathcal{V}}:=\Big{(}\bigotimes_{i=1}^{k}\tilde{\mathfrak{g}}\Big{)}\bigotimes\Big{(}\bigotimes_{i=1}^{m}\tilde{\mathfrak{g}}^{*}\Big{)}$,
be some tensor product space of the real Lie algebras, and let
$O(n,\mathbb{C})$ act by isometries on the complexification:
$\mathcal{V}^{\mathbb{C}}=\tilde{\mathcal{V}}^{\mathbb{C}}$ (induced from the
isometry action of the metric). Thus we also recall the definition of Wick-
rotatable tensors translated to the case of Lie groups at $p:=1$:

######  Definition 3.

Let $(G,g)$ and $(\tilde{G},\tilde{g})$ be two Wick-rotatable Lie groups. Then
two tensors $v\in\mathcal{V}$ and $\tilde{v}\in\tilde{\mathcal{V}}$ are said
to be Wick-rotatable at $p:=1$, if they lie in the same
$O(n,\mathbb{C})$-orbit, i.e

$O(n,\mathbb{C})v=O(n,\mathbb{C})\tilde{v}.$

One also note if $v=\tilde{v}\in\mathcal{V}\cap\tilde{\mathcal{V}}$, then from
above we may take the map: $\phi^{\mathbb{C}}$, such that $v$ and
$\phi^{-1}\cdot v$ are Wick-rotatable, and we have the compatibility
conditions satisfied. More generally if $v$ and $\tilde{v}$ are Wick-rotatable
i.e $O(n,\mathbb{C})v=O(n,\mathbb{C})v$, then also
$O(n,\mathbb{C})v=O(n,\mathbb{C})\phi^{-1}(\tilde{v})$, and thus results from
[3] are applicable.

##  3\. An invariant of Wick-rotation of Lie groups

###  3.1. The main theorem

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$
be a Wick-rotation of Lie groups. Consider $V$ to be the real vector space of
bilinear forms on $\mathfrak{g}$, then $V$ is a real form of $V^{\mathbb{C}}$
(the set of complex bilinear forms on $\mathfrak{g}^{\mathbb{C}}$). In
particular the real Lie bracket $[-,-]\in V$ w.r.t $\mathfrak{g}$, and is a
restriction of the complex Lie bracket of $\mathfrak{g}^{\mathbb{C}}$. Note
that the isometry action of $O(p,q)$ on $\mathfrak{g}$ extends to an action
$\rho$ on $V$, given simply by:

$g\cdot v:=g\Big{(}v(g^{-1}(x),g^{-1}(y))\Big{)}$

for $x,y\in\mathfrak{g}$ and $g\in O(p,q)$. The metric induces a symmetric
non-degenerate bilinear form $b$ on $V$ with Cartan involution $\rho(\theta)$
where $\theta$ is a Cartan involution of $g(-,-)$. This Cartan involution also
balances the action and thus the inner Cartan involutions w.r.t $b$ are those
conjugate to $\rho(\theta)$ by the action of $O(p,q)$ (see [3]). We recall
that a minimal vector $v\in V$ is a vector satisfying $||g\cdot v||\geq||v||$
for all $g\in O(p,q)$, the set of minimal vectors of the action is denoted by
$\mathcal{M}(O(p,q),V)$. Analogously for the Lie algebra:
$\tilde{\mathfrak{g}}$, we have the corresponding action of
$O(\tilde{p},\tilde{q})$ on $\tilde{V}$, and hence we have two real
representations which have the same complexification. So we may lend ourselves
to compatible representations defined in [3], and thus we can prove our main
result:

######  Theorem 1.

Suppose $(G,g)$ is a pseudo-Riemannian Lie group which can be Wick-rotated to
another Lie group $(\tilde{G},\tilde{g})$. Then there exist a Cartan
involution of the metric $g(-,-)$ (of Lie algebras) if and only if there exist
a Cartan involution of the metric $\tilde{g}(-,-)$ (of Lie algebras).

###### Proof.

The real Lie bracket of $\mathfrak{g}$:
$\mathfrak{g}^{2}\xrightarrow{[-,-]}\mathfrak{g}$, is a restriction of the
complex Lie bracket $[-,-]^{\mathbb{C}}$ of $\mathfrak{g}^{\mathbb{C}}$.
Similarly the Lie bracket $\widetilde{[-,-]}$ of $\tilde{\mathfrak{g}}$ is
also a restriction of the complex Lie bracket on $\mathfrak{g}^{\mathbb{C}}$.
Thus as vectors: $[-,-]\in V$, and $\widetilde{[-,-]}\in\tilde{V}$ with
$[-,-]^{\mathbb{C}}\in V^{\mathbb{C}}=\tilde{V}^{\mathbb{C}}$, then these two
real vectors can be embedded into the same complex orbit:

$O(n,\mathbb{C})\cdot[-,-]\ni\widetilde{[-,-]},$

in such way that we may lend our selves to the results of [3]. Now if
$\theta\cdot[-,-]=[-,-]$, then $[-,-]\in\mathcal{M}(O(p,q),V)$ is a minimal
vector of our action. Hence the real orbit $O(p,q)\cdot[-,-]\subset V$ is
closed, therefore the real orbit
$O(\tilde{p},\tilde{q})\cdot\widetilde{[-,-]}\subset\tilde{V}$ must also be
closed. It follows by ([3], Cor. 5.3), that there exist a Cartan involution
$\tilde{\theta}$ of the metric $\tilde{g}$ such that
$\tilde{\theta}\cdot\widetilde{[-,-]}=\widetilde{[-,-]}$, i.e
$\tilde{\theta}\in Aut(\tilde{\mathfrak{g}})$. The converse is identical. The
theorem is proved. ∎

We find it useful to define for future exploration:

######  Definition 4.

A property of a pseudo-Riemannian Lie group $(G,g)$ is said to be Wick-
rotatable if it is an invariant under a Wick rotation of Lie groups.

######  Corollary 1.

The existence of a Cartan involution (of Lie algebras) is Wick-rotatable.

Other Wick-rotatable properties include: being semi-simple, abelian,
nilpotent, solvable, reductive. Note that being simple, is not Wick-rotatable,
indeed as an example consider the Lie group $O(1,3)$ with the left-invariant
metric being the Killing form. Then $\mathfrak{o}(1,3)$ is simple, but we may
Wick-rotate $O(1,3)$ to $O(2,2)$ which is semi-simple but not simple, as
$\mathfrak{o}(2,2)\cong\mathfrak{sl}_{2}(\mathbb{R})^{2}$.

###  3.2. The Riemannian case

We can now answer the question for when an arbitrary left-invariant metric can
be Wick-rotated to a Riemannian left-invariant metric. One should compare the
result with semi-simple Lie groups equipped with the Killing form:
$g:=-\kappa$.

######  Corollary 2.

Suppose $(G,g)$ is a pseudo-Riemannian Lie group which can be Wick-rotated to
a Riemannian Lie group $(\tilde{G},\tilde{g})$. Then there exist a Cartan
involution $\mathfrak{g}\xrightarrow{\theta}\mathfrak{g}$ (of Lie algebras) of
the metric $g(-,-)$.

###### Proof.

The identity map $\tilde{\mathfrak{g}}\xrightarrow{1}\tilde{\mathfrak{g}}$ is
a Cartan involution of Lie algebras of the metric: $\tilde{g}$. Thus by
Theorem 1 the corollary follows. ∎

In view of Remark 1 with the signature change $g\mapsto-g$, if $(G,g)$ can be
Wick-rotated to a signature $(-,-,\cdots,-)$, then $(G,-g)$ can be Wick-
rotated to a Riemannian space, thus there would exist a Cartan involution (of
Lie algebras) of $-g$. We can also easily supply the converse of the previous
corollary, and thus obtain the equivalence result:

######  Corollary 3.

Suppose $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ is a real slice of Lie
groups. Then $G$ can be Wick-rotated to an immersive Riemannian real form
$\tilde{G}\subset G^{\mathbb{C}}$ if and only if there exist a Cartan
involution $\mathfrak{g}\xrightarrow{\theta}\mathfrak{g}$ of Lie algebras of
the metric $g(-,-)$.

###### Proof.

$(\Rightarrow)$ has been proved. Conversely suppose $\theta$ is a Cartan
involution of Lie algebras of the metric $g$, and write
$\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{p}$, for the Cartan decomposition.
Then is is not difficult to show that
$\tilde{\mathfrak{g}}:=\mathfrak{t}\oplus i\mathfrak{p}$ is a Lie algebra and
is a real form of $\mathfrak{g}^{\mathbb{C}}$. Moreover the complex metric
$g^{\mathbb{C}}(-,-)$ restricts to an inner product on $\tilde{\mathfrak{g}}$
by construction. Thus if we let $\tilde{G}$ be the unique connected Lie
subgroup of $G^{\mathbb{C}}$ (the real Lie group) with Lie algebra
$\tilde{\mathfrak{g}}$, then the corollary follows. ∎

We note in the Corollary that w.r.t to the existing Cartan involution of the
metric, then the Wick-rotated Riemannian Lie group may be chosen to be a
virtual real form. It may be the case that a pseudo-Riemannian Lie group
$(G,g)$ can be Wick-rotated to more than one Riemannian Lie group, in any such
case we have the following:

######  Proposition 2.

Suppose there exist two Riemannian Wick-rotatable Lie groups: $(G,g)$ and
$(\tilde{G},\tilde{g})$. Then $(G_{0},g)$ and $(\tilde{G}_{0},\tilde{g})$ are
locally isometric Lie groups. In particular if moreover $G$ and $\tilde{G}$
are both simply connected then $G$ and $\tilde{G}$ are isometric Lie groups.

###### Proof.

Choose a map $g\in O(n,\mathbb{C})$ mapping
$\mathfrak{g}\mapsto\tilde{\mathfrak{g}}$. Using the map $g$ the real Lie
brackets of the Lie algebras are embedded into the same complex orbit. Indeed
if $[-,-]\in V$ and $\widetilde{[-,-]}\in\tilde{V}$ are the real Lie brackets
respectively, then the bilinear form: $g^{-1}\cdot\widetilde{[-,-]}:=v\in V$,
and thus:

$g\cdot[-,-]=v.$

But by [4], since $O(n)$ (the isometries of $g(-,-)$) is compact, then we may
choose $h\in O(n)$ such that the Lie brackets lie in the same $O(n)$-orbit,
i.e

$gh^{-1}\cdot[-,-]=[-,-].$

Now since $h$ maps $\mathfrak{g}$ to $\mathfrak{g}$ and $gh^{-1}$ fixes the
complex Lie bracket. Then $gh^{-1}\in O(n,\mathbb{C})$ is an automorphism of
complex Lie algebras, and it maps $\mathfrak{g}\mapsto\tilde{\mathfrak{g}}$.
Therefore since the metrics are left-invariant we can conclude that
$(G_{0},g)$ and $(\tilde{G}_{0},\tilde{g})$ are locally isometric Lie groups
as required. ∎

In the case of a complex semi-simple Lie group: $(G^{\mathbb{C}},-\kappa)$,
equipped with the Killing form, then any compact real form:
$\mathfrak{u}\subset\mathfrak{g}$, gives rise to a real form: $U\subset
G^{\mathbb{C}}$ (thus is by definition a Riemannian real slice of Lie groups).
It follows by the theory of semi-simple Lie groups that any two compact real
forms of $G^{\mathbb{C}}$ are isomorphic Lie groups, and thus also isometric
Lie groups.

######  Example 3.

Let $H_{3}(\mathbb{R})\subset H_{3}(\mathbb{C})$ be the 3-dimensional real and
complex Heisenberg groups. The Lie algebra of $H_{3}(\mathbb{R})$ denoted:
$\mathfrak{h}_{3}(\mathbb{R})$, is the set of strictly upper triangular
$3\times 3$ matrices. A basis of the Lie algebra is given by
$\\{e_{1},e_{2},e_{3}\\}$ with,

$[e_{1},e_{2}]=e_{3},\ \ [e_{1},e_{3}]=[e_{2},e_{3}]=0.$

We may identify $\\{e_{j}\\}_{j}$ with the standard basis of $\mathbb{R}^{3}$.
Let $g(-,-)$ be the standard Lorentzian pseudo-inner product on
$\mathbb{R}^{3}$, i.e of signature $(+,+,-)$. Thus $(H_{3}(\mathbb{R}),-g)$ is
a real slice (of Lie groups) of $(H_{3}(\mathbb{C}),-g^{\mathbb{C}})$. Note
that $g(-,-)$ is not bi-invariant, since $g([e_{1},e_{2}],e_{3})=-1\neq
g(e_{1},[e_{2},e_{3}])=0.$ Define the linear map: $\theta\in
End(\mathfrak{h}_{3}(\mathbb{R}))$ by:

$\lambda_{1}e_{1}+\lambda_{2}e_{2}+\lambda_{3}e_{3}\mapsto-\lambda_{1}e_{1}-\lambda_{2}e_{2}+\lambda_{3}e_{3}.$

Then it is easy to show that this is an involution of Lie algebras, and
moreover $\theta$ is a Cartan involution of $-g(-,-)$, thus by Corollary 2 it
follows that $H_{3}(\mathbb{R})$ can be Wick-rotated to a Riemannian Lie group
$\tilde{G}$. Note that $\tilde{G}$ is the real form of $H_{3}(\mathbb{C})$
consisting of matrices of the form: $\begin{bmatrix}1&ix&iy\\\ 0&1&z\\\
0&0&1\\\ \end{bmatrix}$ for $x,y,z\in\mathbb{R}$.

We also give an example of a pseudo-Riemannian Lie group $(G,g)$ which can not
be Wick-rotated to a Riemannian Lie group.

######  Example 4.

Consider the real form: $G:=SL_{2}(\mathbb{R})^{2}\subset
G^{\mathbb{C}}:=SL_{2}(\mathbb{C})^{2}$. Then we can equip $G$ with a left-
invariant metric $g(-,-)$ of signature $(3,3)$, by equipping one copy with
$-\kappa$ and the other copy with $\kappa$. The real forms up to isomorphism
of $\mathfrak{sl}_{2}(\mathbb{C})^{2}$ are:

$\mathfrak{sl}_{2}(\mathbb{R})^{2},\
\mathfrak{sl}_{2}(\mathbb{R})\oplus\mathfrak{su}(2),\ \mathfrak{su}(2)^{2},\
\mathfrak{o}(1,3).$

Let $\tilde{\mathfrak{g}}$ be one of these real forms (except the last one),
then we may Wick-rotate $G$ to the corresponding real forms $\tilde{G}$ of
$SL_{2}(\mathbb{C})^{2}$ of signature either: $(3,3)$ or $(1,5)$. In the case
of Wick-rotating to $(SU(2)^{2},\tilde{g})$ we get a signature of $(3,3)$. Now
note that if $G$ can be Wick-rotated to a signature: $(0,6)$ or Riemannian:
$(6,0)$, then we can find (by Corollary 2) a Cartan involution of Lie algebras
of $-\tilde{g}$ or $+\tilde{g}$ respectively:

$\mathfrak{su}(2)^{2}\xrightarrow{\theta}\mathfrak{su}(2)^{2}.$

Suppose the Cartan involution is of $\tilde{g}$. Then if
$\mathfrak{su}(2)^{2}=\mathfrak{t}\oplus\mathfrak{p},$ is the Cartan
decomposition w.r.t $\theta$, we have $\mathfrak{g}_{1}:=\mathfrak{t}\oplus
i\mathfrak{p}\cong\mathfrak{o}(1,3)$. Indeed $\theta^{\mathbb{C}}$ is a Cartan
involution of $\mathfrak{g}_{1}$ (i.e of $-\kappa$), thus $-\kappa$ has
signature $(3,3)$, hence it must be the case that
$\mathfrak{g}_{1}\cong\mathfrak{o}(1,3)$. We recall that
$\mathfrak{t}\cong\mathfrak{su}(2)$, and that the Killing form of
$\mathfrak{t}$ is just:

$(X,Y)\mapsto 4Tr(XY).$

Now we note that $\mathfrak{t}^{\mathbb{C}}\cong\mathfrak{sl}_{2}(\mathbb{C})$
is simple and naturally
$\mathfrak{t}^{\mathbb{C}}\subset\mathfrak{sl}_{2}(\mathbb{C})^{2}$ simply
because $\mathfrak{t}\subset\mathfrak{su}(2)^{2}$ is contained in a real form
of $\mathfrak{sl}_{2}(\mathbb{C})^{2}$. Thus when restricting $\tilde{g}$ on
$\mathfrak{t}$ we must get
$\tilde{g}(X,Y)=-\lambda\kappa_{\mathfrak{t}}(X,Y)=-4\lambda Tr(XY)$ for some
$\lambda>0$. Now if $X:=(x,y)\in\mathfrak{t}$, then:

$\displaystyle\tilde{g}(X,X)$
$\displaystyle:=-\kappa(x,x)+\kappa(y,y)=-4Tr(x^{2})+4Tr(y^{2})$
$\displaystyle=-4\lambda Tr(X^{2})=-4\lambda(Tr(x^{2})+Tr(y^{2})).$

Thus we conclude that $y=0$, and therefore:

$\mathfrak{t}=\\{(x,0)|x\in\mathfrak{su}(2)\\}\subset\mathfrak{su}(2)^{2},\ \
\ \mathfrak{p}=\\{(0,y)|y\in\mathfrak{su}(2)\\}\subset\mathfrak{su}(2)^{2}.$

This is impossible since then $\theta=1\oplus-1$ which is not a Lie
homomorphism. The argument for the signature case: $(0,6)$, is identical with
the change: $\tilde{g}\mapsto-\tilde{g}$. We conclude that $(G,g)$ can not be
Wick-rotated to a Riemannian Lie group nor of signature $(0,6)$.

One shall note that Proposition 2 does not hold for a general non-Riemannian
signature. Indeed consider the previous example then $SL_{2}(\mathbb{R})^{2}$
has signature $(3,3)$ and can be Wick-rotated to $SU(2)^{2}$ also of signature
$(3,3)$, but they are not locally isometric (since their Lie algebras are non-
isomorphic).

###  3.3. Conjugacy of Cartan involutions of a left-invariant metric

Given a pseudo-Riemannian Lie group $(G,g)$, with two Cartan involutions
$\theta_{j}$ ($j=1,2$) of Lie algebras of the metric $g$, one may wonder if
they are conjugate in $Aut(\mathfrak{g})$. This is in fact true as we will
show here, and we note the resemblance with semi-simple Lie groups and Cartan
involutions of the Killing form ($-\kappa$).

######  Theorem 2.

Suppose $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ is a pseudo-Riemannian
Lie group. Assume there exist two Cartan involutions: $\theta_{1},\theta_{2}$
of the metric, which are homomorphisms of Lie algebras. Then $\theta_{1}$ is
conjugate to $\theta_{2}$ in $Aut(\mathfrak{g})_{0}\cap O(p,q)_{0}$.

###### Proof.

Write
$\mathfrak{g}=\mathfrak{t}_{1}\oplus\mathfrak{p}_{1}=\mathfrak{t}_{2}\oplus\mathfrak{p}_{2}$
for the Cartan decompositions w.r.t $\theta_{1}$ and $\theta_{2}$
respectively. Denote also: $\mathfrak{t}_{j}\oplus
i\mathfrak{p}_{j}:=\mathfrak{u}_{j}$ ($j=1,2$) for the real forms of
$\mathfrak{g}^{\mathbb{C}}$. There exist Wick-rotations of $G$ to connected
virtual real forms: $U_{j}\subset G^{\mathbb{C}}$ with Lie algebras
$\mathfrak{u}_{j}$ which are Riemannian (by Corollary 3). If $\sigma$ denotes
the conjugation map w.r.t $\mathfrak{g}$, and $\tau_{j}$ denotes the
conjugation map of $\mathfrak{u}_{j}$, then we have
$\theta^{\mathbb{C}}_{j}=\sigma\tau_{j}$. Now since $\theta_{1}$ is conjugate
to $\theta_{2}$ in $O(p,q)_{0}$, as linear maps, say by $\phi$ then
$\phi^{\mathbb{C}}$ sends $\mathfrak{u}_{1}\mapsto\mathfrak{u}_{2}$. Thus
following the notation in the proof of Proposition 2 we may take $g\in
O(p,q)_{0}$. The Lie brackets of the real Lie algebras are restrictions of the
same complex Lie bracket: $[-,-]^{\mathbb{C}}$, on
$\mathfrak{g}^{\mathbb{C}}$. Denote $[-,-]$ for the Lie bracket on
$\mathfrak{g}$. Thus if we let $\tilde{v}:=g^{-1}\cdot[-,-]^{\mathbb{C}}$,
then $\tilde{v}$ is a bilinear form on $\mathfrak{u}_{1}$, and:

$\tilde{v}\in O(n)_{0}\cdot[-,-]\cap O(p,q)_{0}\cdot[-,-]=K_{0}\cdot[-,-],$

by the compatibility conditions of the pair:
$(\mathfrak{g},\mathfrak{u}_{1})$. Here

$K:=\\{g\in O(p,q)|g\theta_{1}=\theta_{1}g\\}\subset O(p,q),$

is the maximal compact subgroup associated with the fixed Cartan involution of
$O(p,q)$: $g\mapsto\theta_{1}g\theta_{1}$. Thus the element $h$ from the proof
of Proposition 2, leaves invariant $\mathfrak{g}$ also. It follows that
$[\sigma,gh^{-1}]=0$, i.e

$\theta_{2}=gh^{-1}\circ\theta_{1}\circ hg^{-1},$

for some $gh^{-1}\in Aut(\mathfrak{g})_{0}\cap O(p,q)_{0}$. The corollary is
proved. ∎

Let us give an example where there is a unique Cartan involution of a metric
(of Lie algebras):

######  Example 5.

If we consider again the real Heisenberg group: $(H_{3}(\mathbb{R}),-g)$ and
follow Example 3, then calculating the derivation algebra of
$\mathfrak{h}_{3}(\mathbb{R})$ w.r.t to the basis $\\{e_{j}\\}$, then the
matrices have the form: $\begin{bmatrix}a&c&0\\\ e&b&0\\\ f&l&a+b\\\
\end{bmatrix}$, for $a,b,c,e,l,f\in\mathbb{R}$. Thus
$Dim\Big{(}\mathfrak{der}(\mathfrak{h}_{3}(\mathbb{R}))\Big{)}=6$. Now if such
a derivation $D$ belongs to $\mathfrak{o}(1,2)$, then an easy calculation
shows that $D=\begin{bmatrix}0&0&0\\\ 0&b&0\\\ 0&0&b\\\ \end{bmatrix}$, i.e
the Lie algebra of $Aut(\mathfrak{h}_{3}(\mathbb{R}))\cap O(1,2)$ has
dimension 1. Now in view of the previous theorem if $\theta_{1}$ is another
Cartan involution of $-g(-,-)$ of Lie algebras, there would exist an element
$A\in Aut(\mathfrak{h}_{3}(\mathbb{R}))_{0}\cap O(1,2)_{0}$ such that $A\theta
A^{-1}=\theta_{1}$, i.e $\theta_{1}=\theta$ since $A$ is diagonal w.r.t our
basis. We conclude that there exist a unique Cartan involution of $-g(-,-)$,
namely $\theta$.

###  3.4. Wick-rotating to a compact semi-simple Riemannian Lie group

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ be a real slice, and
$\mathfrak{g}^{\mathbb{C}}$ be simple. If $g$ is bi-invariant, then $g$ is
proportional to the Killing form. Assume w.l.o.g that $g=-\lambda\kappa(-,-)$
($\lambda>0$). Then $G$ can be Wick-rotated to a compact Riemannian Lie group,
and any Cartan involution of the metric is just a Cartan involution of the Lie
algebra $\mathfrak{g}$. If $g(-,-)$ is a general left-invariant metric on a
semi-simple Lie group with such a Wick-rotation, then we similarly have:

######  Proposition 3.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ be a real slice, and $G$ be
semi-simple. Then $(G,g)$ can be Wick-rotated to a Riemannian compact Lie
group if and only if there exist a Cartan involution $\theta$ of the metric
which is also a Cartan involution of $\mathfrak{g}$ (i.e of $-\kappa$).

###### Proof.

$(\Rightarrow)$. If $(G,g)$ is Wick-rotated to a Riemannian Lie group, then by
Corollary 2, we can choose a Cartan involution (of Lie algebras) $\theta$ of
the metric. Denote $\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{p}$ for the
Cartan decomposition. Then following the proof of Corollary 3, then we can
find a Riemannian Lie group $\tilde{G}$ with Lie algebra:
$\tilde{\mathfrak{g}}:=\mathfrak{t}\oplus i\mathfrak{p}$, which is Wick-
rotated to $G$ in $(G^{\mathbb{C}},g^{\mathbb{C}})$. By Proposition 2,
$\tilde{\mathfrak{g}}$ is compact, since we can Wick-rotate $G$ to a
Riemannian compact Lie group (by assumption). But since $\theta$ is a Cartan
involution of $\mathfrak{g}$ if and only if $\tilde{\mathfrak{g}}$ is compact,
then the direction is proved. $(\Leftarrow)$. Suppose $\theta$ is a Cartan
involution (of Lie algebras) of $g(-,-)$ which is also a Cartan involution of
$-\kappa$. Thus if $\mathfrak{u}:=\mathfrak{t}\oplus i\mathfrak{p}$ is the
compact real form of $\mathfrak{g}^{\mathbb{C}}$ associated with $\theta$,
then there exist a compact real form $U\subset G^{\mathbb{C}}$ with Lie
algebra $\mathfrak{u}$. The proposition follows. ∎

Thus since we may lift a local Cartan involution:
$\mathfrak{g}\xrightarrow{\theta}\mathfrak{g}$, to a global Cartan involution:
$G\xrightarrow{\Theta}G$, then in view of the previous proposition, there is a
$\Theta$ which is an isometry of $(G,g)$, i.e $\Theta\in Isom(G)$. Observe
also that if there exist a real slice of Lie groups of $G^{\mathbb{C}}$ which
is compact Riemannian, then the possible signatures $(p,q)$ w.r.t
$g^{\mathbb{C}}$ is a subset of the possible signatures of $-\kappa$ (of
$\mathfrak{g}^{\mathbb{C}}$).

Note in general that if $(G,g)$ has a Cartan involution $\theta_{g}$ (of Lie
algebras) of the metric $g$, then (by the theory of semi-simple Lie algebras)
there exist a Cartan involution of $\mathfrak{g}$ (i.e of $-\kappa$) such that
$[\theta_{g},\theta]=0$.

It is tempting to think that if $(G,g)$ can be Wick-rotated to a compact
Riemannian Lie group, then it would be locally isometric to
$(G,-\lambda\kappa)$ ($\lambda>0$). However this is false, indeed consider
$G:=SL_{2}(\mathbb{R})^{2}$ equipped with the metric
$g:=-\kappa\oplus-\kappa$. We can Wick-rotate to the compact Riemannian Lie
group: $SU(2)^{2}$. Consider the real form $\cong\mathfrak{o}(1,3)$ identified
with the set:

$\\{(x,x)|x\in\mathfrak{sl}_{2}(\mathbb{C})_{\mathbb{R}}\\}\subset\mathfrak{sl}_{2}(\mathbb{C})^{2}.$

If $(G,g)$ is locally isometric to $\lambda\kappa$ for some
$\lambda\in\mathbb{R}$, then we can Wick-rotate to a $\tilde{G}\subset
G^{\mathbb{C}}$ with Lie algebra $\mathfrak{o}(1,3)$. However
$\mathfrak{o}(1,3)$ on $g^{\mathbb{C}}$ is not a real slice. We thus conclude
that $(G,g)$ is not locally isometric to $(G,\lambda\kappa)$ for any
$\lambda\in\mathbb{R}$.

###  3.5. Wick-rotating a Lorentzian signature

If we assume our metric on our Lie group $G$ is Lorentzian or of signature
$(+,-,\cdots,-)$, then being able to Wick-rotate to a Riemannian space puts
some constraints on the structure of the Lie algebra (in view of Corollary 2).
Now since a Wick-rotation is a local condition, it would be interesting to
know what type of Lie algebra allows for a Wick-rotation to a Riemannian Lie
group.

We recall by the fundamental Levi-Malcev theorem that our Lie algebra
$\mathfrak{g}$ can be written as a semi-direct sum
$\mathfrak{g}=\mathfrak{s}\ltimes\mathfrak{h}$, where $\mathfrak{h}$ is the
radical of $\mathfrak{g}$ and $\mathfrak{s}\subset\mathfrak{g}$ is either
trivial or a semi-simple subalgebra of $\mathfrak{g}$ called the Levi-factor.

It is clear that
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{s}^{\mathbb{C}}\ltimes\mathfrak{h}^{\mathbb{C}}$,
and if $\tilde{\mathfrak{g}}$ is another real form of
$\mathfrak{g}^{\mathbb{C}}$, then writing a Levi-decomposition:
$\tilde{\mathfrak{g}}=\tilde{\mathfrak{s}}\ltimes\tilde{\mathfrak{h}}$, then
$\tilde{\mathfrak{h}}$ is a real form of $\mathfrak{h}^{\mathbb{C}}$. To see
that $\tilde{\mathfrak{s}}$ is a real form of $\mathfrak{s}^{\mathbb{C}}$, we
note that there exist a $k\geq 1$ such that

$\mathfrak{s}^{\mathbb{C}}=[{\mathfrak{g}^{\mathbb{C}}}^{(k)},{\mathfrak{g}^{\mathbb{C}}}^{(k)}]\supset[\tilde{\mathfrak{g}}^{(k)},\tilde{\mathfrak{g}}^{(k)}]=\tilde{\mathfrak{s}}.$

In view of the existence of an involution of Lorentzian decomposition we can
say the following:

######  Proposition 4.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ be a real slice of Lie
groups. Then the following statements hold:

  1. (1)

Suppose $g(-,-)$ has Lorentzian signature. If $(G,g)$ can be Wick-rotated to a
Riemannian Lie group $(\tilde{G},\tilde{g})$ then $\mathfrak{s}=0$ or
$\mathfrak{h}\neq 0$. Moreover if $\tilde{\mathfrak{s}}$ is a Levi-factor of
$\tilde{\mathfrak{g}}$, then $\tilde{\mathfrak{s}}\cong\mathfrak{s}$.

  2. (2)

Suppose $g(-,-)$ has signature $(+,-,\cdots,-)$. If $(G,g)$ can be Wick-
rotated to a Riemannian Lie group, then either $\mathfrak{s}=0$ or
$\mathfrak{s}\cong\mathfrak{sl}_{2}(\mathbb{R})$.

###### Proof.

For case (1), assume $\mathfrak{g}=\mathfrak{s}\ltimes\mathfrak{h}$ for
$\mathfrak{s}\neq 0$, and choose a Cartan involution $\theta$ of $g(-,-)$
which is also a homomorphism of Lie algebras. Write
$\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{p}$ for the Cartan decomposition.
Then $\theta$ leaves $\mathfrak{s}$ invariant:
$\theta(\mathfrak{s})\subset\mathfrak{s}$. Indeed note that since
$\mathfrak{h}$ is solvable, then there exist $k\geq 1$ such that the
$k^{th}$-derived algebra satisfies:
$[\mathfrak{g}^{(k)},\mathfrak{g}^{(k)}]=\mathfrak{s},$ thus it follows that
$\theta$ must leave $\mathfrak{s}$ invariant, and hence we can write,

$\mathfrak{s}=(\mathfrak{s}\cap\mathfrak{t})\oplus(\mathfrak{s}\cap\mathfrak{p}).$

We claim that $\mathfrak{s}\cap\mathfrak{p}=0$, indeed suppose not, i.e
$\mathfrak{p}\subset\mathfrak{s}$ thus
$[\mathfrak{s},\mathfrak{p}]\subset\mathfrak{p}$ so $\mathfrak{p}$ is an
abelian non-trivial ideal of $\mathfrak{s}$, contradicting the semi-simplicity
of $\mathfrak{s}$. Thus $\theta$ fixes $\mathfrak{s}$ point wise. Moreover
$\mathfrak{p}\lhd\mathfrak{g}$ is an abelian ideal, and so therefore
$\mathfrak{p}\subset\mathfrak{h}$, i.e $\mathfrak{h}\neq 0$. Finally since
$g(-,-)$ restricted to $\mathfrak{s}$ and $\tilde{g}(-,-)$ restricted to
$\tilde{\mathfrak{s}}$ is positive definite, then $\mathfrak{s}$ and
$\tilde{\mathfrak{s}}$ give rise to a Wick-rotation of two Riemannian Lie
groups, thus by Proposition 2 it follows that
$\mathfrak{s}\cong\tilde{\mathfrak{s}}$, and case (1) is proved. For case (2)
suppose $\mathfrak{g}$ is non-solvable (i.e $\mathfrak{s}\neq 0$), then again
w.r.t $\theta$ we see that

$\mathfrak{s}=(\mathfrak{s}\cap\mathfrak{t})\oplus(\mathfrak{s}\cap\mathfrak{p}),$

where $\mathfrak{s}\cap\mathfrak{t}\neq 0$, since if not then
$\mathfrak{s}\subset\mathfrak{p}$, i.e
$\mathfrak{s}=[\mathfrak{s},\mathfrak{s}]\subset\mathfrak{t}$, which is a
contradiction. Now since $\theta^{\mathbb{C}}$ is a Cartan involution of a
real form $\tilde{\mathfrak{g}}\subset\mathfrak{s}^{\mathbb{C}}$, then
$-\kappa$ on $\tilde{\mathfrak{g}}$ must also have the signature
$(+,-,\cdots,-)$, this follows since $\mathfrak{t}^{\mathbb{C}}$ is
1-dimensional. Now finally if
$\tilde{\mathfrak{g}}=\tilde{\mathfrak{t}}\oplus\tilde{\mathfrak{p}}$ is a
Cartan decomposition, then $\tilde{\mathfrak{t}}$ is abelian and
1-dimensional, thus it follows that
$\tilde{\mathfrak{g}}\cong\mathfrak{sl}_{2}(\mathbb{R})$. We conclude that
$\mathfrak{s}^{\mathbb{C}}\cong\mathfrak{sl}_{2}(\mathbb{C})$, and hence also
$\mathfrak{s}\cong\mathfrak{sl}_{2}(\mathbb{R})$. The proposition is proved. ∎

Thus restricting to the class of semi-simple Lie algebras it is impossible to
Wick-rotate a Lorentzian metric to a Riemannian metric. However even for a
nilpotent Lie algebra the converse of (1) is not necessarily true, indeed
consider the nilpotent Lie algebra $\mathfrak{h}_{3}(\mathbb{R})$ of $3\times
3$ strictly upper triangular matrices. Then if $\theta$ is an involution with
$Dim(\mathfrak{p})=1$, we must be able to find a basis
$\\{x_{1},x_{2},x_{3}\\}$ such that

$[x_{1},x_{2}]=C^{1}_{12}x_{1}+C^{2}_{12}x_{2},\ \
[x_{1},x_{3}]=C^{3}_{13}x_{3},\ \ [x_{2},x_{3}]=C^{3}_{23}x_{3}.$

But since $[\mathfrak{h}_{3}(\mathbb{R}),\mathfrak{h}_{3}(\mathbb{R})]$ has
dimension 1, then it follows that $C^{1}_{12}=0=C^{2}_{12}$. Moreover since
$\mathfrak{h}_{3}(\mathbb{R})$ is nilpotent of class 2, then we conclude also
that $C^{3}_{12}=0=C^{3}_{23}$, i.e $\mathfrak{h}_{3}(\mathbb{R})$ would have
to be abelian, thus it is not possible to Wick-rotate a Lorentzian metric on
$H_{3}(\mathbb{R})$ to a Riemannian metric.

In view of case (2), there are examples of metrics (of signature
$(+,-,\cdots,-)$) that are Wick-rotatable to a Riemannian metric within:
$\mathfrak{g}=\mathfrak{sl}_{2}(\mathbb{R})$ or
$\mathfrak{g}=\mathfrak{h}_{3}(\mathbb{R})$ and even
$\mathfrak{g}=\mathfrak{sl}_{2}(\mathbb{R})\oplus\mathfrak{h}_{3}(\mathbb{R})$.

##  4\. Some more Wick-rotatable properties

###  4.1. Wick-rotating to a Riemannian space

The Levi-Civita connection $\nabla_{x}y$ of a real slice of a holomorphic
Riemannian manifold at $p$: $(M,g)\subset(M^{\mathbb{C}},g^{\mathbb{C}})$,
restricts from the complex Levi-Civita connection: $\nabla^{\mathbb{C}}_{x}y$,
thus the same is true for the real Riemann curvature tensor $R$ (see also
[1],[2],[3]). In particular if $ric_{g}$ denotes the real Ricci curvature:
$T_{p}M^{2}\rightarrow\mathbb{R}$, given by:

$ric_{g}(x,y):=Tr\Big{(}z\mapsto R(z,y)(x)\Big{)},$

then using a real basis of $T_{p}M$ for $T_{p}M^{\mathbb{C}}$ we see that
restricting the complex Ricci curvature: $ric_{g^{\mathbb{C}}}$ on
$M^{\mathbb{C}}$ to $T_{p}M$ we get $ric_{g}$. Thus in terms of Wick-rotations
of pseudo-Riemannian manifolds at a common point $p$:
$(M,g)\subset(M^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{M},\tilde{g}),$ we
see that the real Ricci curvatures $ric_{g}$ and $ric_{\tilde{g}}$ are
examples of Wick-rotatable tensors, i.e we can embed them into the same
complex orbit:

$ric_{\tilde{g}}\in O(n,\mathbb{C})\cdot ric_{g},$

in such a way that results from ([2], [3]) can be applied. Thus also the same
applies to the Ricci operators $Ric_{g}$ and $Ric_{\tilde{g}}$, recall that
the Ricci operator is defined by $g(Ric_{g}(x),y)=ric_{g}(x,y)$, where
$ric_{g}$ is the Ricci curvature. In particular by using the isometry action
of $O(n,\mathbb{C})$ on these tensors respectively we have:

######  Theorem 3.

Let $(M,g)\subset(M^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{M},\tilde{g})$
be a Wick-rotation of at a common point $p\in M\cap\tilde{M}$. Assume
$(\tilde{M},\tilde{g})$ is a Riemannian space. Then the following statements
hold:

  1. (1)

There exist a Cartan involution $\theta$ of $g$ such that
$\nabla_{\theta(x)}\theta(y)=\nabla_{x}y$ for all $x,y\in T_{p}M$.

  2. (2)

There exist a Cartan involution $\theta$ of $g$ such that
$ric_{g}\Big{(}\theta(x),\theta(y)\Big{)}=ric_{g}(x,y)$ for all $x,y\in
T_{p}M$.

  3. (3)

There exist a Cartan involution $\theta$ of $g$ such that
$[\theta,Ric_{g}]=0$.

  4. (4)

There exist a Cartan involution $\theta$ of $g$ such that
$R\Big{(}\theta(x),\theta(y)\Big{)}\Big{(}\theta(z)\Big{)}=R(x,y)(z)$ for all
$x,y,z\in T_{p}M$.

###### Proof.

Follows from ([3], Thm 5.4). ∎

We shall strengthen Theorem 3 for Lie groups in the next section.

###  4.2. Wick-rotating an algebraic soliton

A pseudo-Riemannian Lie group $(G,g)$, such that the Ricci operator
$Ric_{g}\in\mathfrak{gl}(\mathfrak{g})$ has the form: $Ric_{g}=\lambda\cdot
1_{\mathfrak{g}}+D$, where $\lambda\in\mathbb{R}$ and
$D\in\mathfrak{der}(\mathfrak{g})$ (a derivation) is called an algebraic
soliton (defined in [8]). If $D$ can be taken to be $D=0$, then $(G,g)$ is
said to be Einstein, and moreover if the Lie algebra is also nilpotent (resp.
solvable), then an algebraic soliton $(G,g)$, is said to be a Ricci nilsoliton
(resp. solsoliton). For a discussion of Riemannian Ricci nilsolitons we refer
to for example [6].

We shall prove a result regarding the existence of a Wick-rotation of an
algebraic soliton to a Riemannian Lie group.

######  Lemma 2.

The property of being an algebraic soliton is Wick-rotatable.

###### Proof.

Let $(G,g)$ be Wick-rotatable to $(\tilde{G},\tilde{g})$. Suppose $(G,g)$ is
an algebraic soliton. The Ricci operator:
$\mathfrak{g}\xrightarrow{Ric_{g}}\mathfrak{g}$ on $G$ is also a restriction
of the Ricci operator on $G^{\mathbb{C}}$. So if $Ric_{g}=\lambda\cdot
1_{\mathfrak{g}}+D$ for some $\lambda\in\mathbb{R}$ and
$D\in\mathfrak{der}(\mathfrak{g})$, then also

$Ric_{g^{\mathbb{C}}}=(\lambda\cdot
1_{\mathfrak{g}})^{\mathbb{C}}+D^{\mathbb{C}}=\lambda
1_{\mathfrak{g}^{\mathbb{C}}}+D^{\mathbb{C}}.$

Note that $D^{\mathbb{C}}$ is a derivation of $\mathfrak{g}^{\mathbb{C}}$,
thus when restricting to $Ric_{\tilde{g}}$ we see that $D^{\mathbb{C}}$ must
leave $\tilde{\mathfrak{g}}$ invariant and is thus a derivation $\tilde{D}$ of
$\tilde{\mathfrak{g}}$ as required. The lemma follows. ∎

Note from the lemma that $D^{\mathbb{C}}=\tilde{D}^{\mathbb{C}}$ are also
Wick-rotatable tensors.

######  Corollary 4.

The property of being a Ricci nilsoliton (resp. solsoliton) is Wick-rotatable.

######  Corollary 5.

The property of being Einstein is Wick-rotatable.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})$ be a real slice of Lie
groups. We can extend the isometry action of $O(p,q)$ on $\mathfrak{g}$ to
$\mathfrak{gl}(\mathfrak{g})$ by conjugation, i.e $g\cdot f:=gfg^{-1}$. The
Cartan involution $\theta$ of $g(-,-)$ (and the metric) is extended to a
Cartan involution $f\mapsto\theta f\theta$. Note that when restricted to
acting on $\mathfrak{o}(p,q)\subset\mathfrak{gl}(\mathfrak{g})$, then the
induced left-invariant metric on $O(p,q)$ (for $p+q\geq 3$) is just
proportional to the Killing form (see for example [1]).

######  Theorem 4.

Suppose $(G,g)$ is an algebraic soliton, with $Ric_{g}=\lambda\cdot
1_{\mathfrak{g}}+D$, which can be Wick-rotated to a Riemannian algebraic
soliton: $(\tilde{G},\tilde{g})$ with $Ric_{\tilde{g}}=\lambda\cdot
1_{\tilde{g}}+\tilde{D}$. Then there exist a Cartan involution $\theta$ of the
metric $g(-,-)$ such that $[\theta,D]=0$.

###### Proof.

Since $D^{\mathbb{C}}=\tilde{D}^{\mathbb{C}}$ (see the proof of Lemma 2), and
by [3] then $D$ and $\tilde{D}$ can be embedded such that they lie in the same
complex orbit, and moreover an embedding may be chosen such that the real
orbits intersect:

$O(n)\cdot D\cap O(p,q)\cdot\tilde{D}\neq\emptyset.$

Thus it follows by ([3], Theorem 5.4) that there exist a Cartan involution
$\theta$ of the metric such that $[\theta,D]=0$. The theorem is proved. ∎

We shall show in the next section that one may in fact choose $\theta$ in
Theorem 4 to also be a homomorphism of Lie algebras.

######  Example 6.

We follow Example 3. Thus consider the real 3-dimensional Heisenberg group
$(H_{3}(\mathbb{R}),-g)$, then this is a Ricci nilsoliton of signature
$(+,-,-)$. Indeed one can calculate with respect to the basis
$\\{e_{1},e_{2},e_{3}\\}$, that the Ricci operator can be written uniquely as:

$Ric_{-g}=-\frac{3}{2}\cdot I_{3}+D,$

where $D(e_{1})=e_{1},\ D(e_{2})=e_{2},\ D(e_{3})=2e_{3}.$ Note that the
Cartan involution $\theta$ commutes with $D$, i.e $[\theta,D]=0$. Thus when
restricting to the Wick-rotated Riemannian Ricci nilsoliton
$(\tilde{G},\tilde{g})$ with Lie algebra: $\tilde{\mathfrak{g}}:=\langle
ie_{1},ie_{2},e_{3}\rangle\subset\mathfrak{h}_{3}(\mathbb{C})$, we get the
corresponding Ricci operator expressed as:

$Ric_{\tilde{g}}=-\frac{3}{2}\cdot I_{3}+\tilde{D},$

with $\tilde{D}(ie_{1})=ie_{1},\ \tilde{D}(ie_{2})=ie_{2},\
\tilde{D}(e_{3})=2e_{3}$.

##  5\. A remark on Wick-rotatable tensors of Lie groups

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$
be a Wick-rotation of Lie groups. Assume $(\tilde{G},\tilde{g})$ is
Riemannian. By Corollary 2 there exist a Cartan involution $\theta$ of
$g(-,-)$ which is also a homomorphism of Lie algebras. We prove that if
$\tilde{v}=v\in\mathcal{V}\cap\tilde{\mathcal{V}}$ are two tensors on the Lie
algebras, then they are Wick-rotatable with respect to an embedding
$\phi^{-1}\in H^{\mathbb{C}}$ into the same $H^{\mathbb{C}}$-orbit for

$H^{\mathbb{C}}:=Aut(\mathfrak{g}^{\mathbb{C}})\cap O(n,\mathbb{C})\subset
O(n,\mathbb{C}),$

such that $(\mathfrak{g},\phi^{-1}(\tilde{\mathfrak{g}}))$ is a compatible
pair (i.e also a compatible triple). Denote $H:=Aut(\mathfrak{g})\cap O(p,q)$.
Note that $H\subset H^{\mathbb{C}}$ is a real form. Indeed the conjugation map
of $O(n,\mathbb{C})$ fixing $O(p,q)$ given by $A\mapsto\sigma A\sigma$ where
$\sigma$ is the conjugation map w.r.t $\mathfrak{g}$, leaves $H^{\mathbb{C}}$
invariant, and thus fixes $H$. Note also that a global Cartan involution
$\Theta$: $A\mapsto\theta A\theta$ of $O(p,q)$ where $\theta$ is a Cartan
involution (of Lie algebras) of $g(-,-)$, also leave $H$ invariant. Thus
$\Theta$ is a global Cartan involution of $H$.

######  Lemma 3.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$
be a Wick-rotation of Lie groups. Assume $(\tilde{G},\tilde{g})$ is
Riemannian. Then any two tensors
$v=\tilde{v}\in\mathcal{V}\cap\tilde{\mathcal{V}}$ can be embedded into the
same $H^{\mathbb{C}}$-orbit for some $\phi^{-1}\in H^{\mathbb{C}}$ such that
$\Big{(}\mathfrak{g},\phi^{-1}(\tilde{\mathfrak{g}})\Big{)}$ is a compatible
pair.

###### Proof.

By Theorem 2, we can also choose a Cartan involution $\theta$ (of Lie
algebras) of the metric: $g(-,-)$. Moreover by Proposition 2 there is an
isomorphism of Lie algebras: $\phi\in O(n,\mathbb{C})\cap
Aut(\mathfrak{g}^{\mathbb{C}})$ sending
$\mathfrak{u}\mapsto\tilde{\mathfrak{g}}$, where
$\mathfrak{u}=\mathfrak{t}\oplus i\mathfrak{p}$ w.r.t the Cartan decomposition
of $\theta$. Thus $\phi^{-1}(\tilde{\mathfrak{g}})=\mathfrak{u}$ and
$\mathfrak{g}$ are compatible. Let now
$\tilde{v}=v\in\mathcal{V}\cap\tilde{\mathcal{V}}$ be two Wick-rotatable
tensors, using the isometry action of $\phi^{-1}$ on $v$, then $\phi^{-1}\cdot
v$ and $v$ are Wick-rotatable tensors which lie in the same
$H^{\mathbb{C}}$-orbit as required. ∎

######  Theorem 5.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$
be a Wick-rotation of Lie groups. Assume $(\tilde{G},\tilde{g})$ is
Riemannian. Let $\tilde{v}=v\in\mathcal{V}\cap\tilde{\mathcal{V}}$ be two
tensors. There exist a Cartan involution $\theta$ (of Lie algebras) of the
metric $g(-,-)$ such that $\theta\cdot v=v$.

###### Proof.

Consider the real forms: $H\subset H^{\mathbb{C}}\supset\tilde{H}$. Then one
simply note that $H^{\mathbb{C}}$ is naturally algebraic, and moreover is a
linearly complex reductive Lie group, simply because $\tilde{H}$ is a compact
real form. Now since $O(n,\mathbb{C})$ and $Aut(\mathfrak{g}^{\mathbb{C}})$
are naturally algebraic groups defined over $\mathbb{R}$, then so is
$H^{\mathbb{C}}$. Moreover $H$ and $\tilde{H}$ are the real points of
$H^{\mathbb{C}}$ (respectively). Thus the groups naturally satisfy ([3], Thm
2.15, Thm 2.17, Thm 2.23), by the setup of ([4]). The theorem follows by Lemma
3 and ([3], Thm 5.4). ∎

Thus we can restate a stronger version of Theorem 3 for Lie groups:

######  Theorem 6.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$
be a Wick-rotation of Lie groups. Assume $(\tilde{G},\tilde{g})$ is a
Riemannian Lie group. Then the following statements hold:

  1. (1)

There exist a Cartan involution $\theta$ (of Lie algebras) of $g$ such that
$\nabla_{\theta(x)}\theta(y)=\nabla_{x}y$ for all $x,y\in\mathfrak{g}$.

  2. (2)

There exist a Cartan involution $\theta$ (of Lie algebras) of $g$ such that
$ric_{g}\Big{(}\theta(x),\theta(y)\Big{)}=ric_{g}(x,y)$ for all
$x,y\in\mathfrak{g}$.

  3. (3)

There exist a Cartan involution $\theta$ (of Lie algebras) of $g$ such that
$[\theta,Ric_{g}]=0$.

  4. (4)

There exist a Cartan involution $\theta$ (of Lie algebras) of $g$ such that
$R\Big{(}\theta(x),\theta(y)\Big{)}\Big{(}\theta(z)\Big{)}=R(x,y)(z)$ for all
$x,y,z\in\mathfrak{g}$.

Any of the properties 1-4 of the previous theorem, are (using also Theorem 1)
Wick-rotatable. We recall that a pseudo-Riemannian space is said to be Riemann
purely electric (RPE) at a point $p$, if the property $(4)$ of the previous
theorem is satisfied for some Cartan involution (linear) of the metric at $p$.
Thus we state as a result (compare with [1]):

######  Corollary 6.

Let $(G,g)$ be a pseudo-Riemannian Lie group. Then the property of being
Riemann purely electric (RPE) at $1$ w.r.t to a Cartan involution $\theta$ (of
Lie algebras) is Wick-rotatable.

###### Proof.

Follows by Theorem 1 and Theorem 6. ∎

We may also replace the Cartan involution in Theorem 4 with a Cartan
involution (of Lie algebras). Thus we have a stronger version:

######  Theorem 7.

Suppose $(G,g)$ is a Ricci soliton, with $Ric_{g}=\lambda\cdot
1_{\mathfrak{g}}+D$, which can be Wick-rotated to a Riemannian Ricci soliton:
$(\tilde{G},\tilde{g})$ with $Ric_{\tilde{g}}=\lambda\cdot
1_{\tilde{g}}+\tilde{D}$. Then there exist a Cartan involution $\theta$ (of
Lie algebras) of the metric $g(-,-)$ such that $[\theta,D]=0$.

We also note the following:

######  Corollary 7.

Let $(G,g)\subset(G^{\mathbb{C}},g^{\mathbb{C}})\supset(\tilde{G},\tilde{g})$
be a Wick-rotation of Lie groups. Assume $(\tilde{G},\tilde{g})$ is
Riemannian. Then

$\Big{(}\forall
x\in\mathfrak{g}\cap\tilde{\mathfrak{g}}\Big{)}\Big{(}\exists\theta\in
Aut(\mathfrak{g})\Big{)}\Big{(}\theta(x)=x\Big{)},$

where $\theta$ is a Cartan involution of the metric $g(-,-)$.

###### Proof.

Consider the isometry action of $O(n,\mathbb{C})$ (restricted to
$H^{\mathbb{C}}$ defined above) on the complex Lie algebra:
$\mathfrak{g}^{\mathbb{C}}$. Let
$x=\tilde{x}\in\mathfrak{g}\cap\tilde{\mathfrak{g}}$. Then $x=\tilde{x}$ are
two Wick-rotatable tensors such that we may apply Lemma 3, and thus by Theorem
5 the corollary follows. ∎

######  Remark 2.

One shall observe that the proofs of Theorem 1 and Corollary 2 also hold for a
Wick-rotation of Lie groups (at $p:=1$) where the metrics are not necessarily
left-invariant.

##  6\. A Cartan involution of an arbitrary real Lie algebra

Consider a pair $(\mathfrak{g},g)$, where $\mathfrak{g}$ is an arbitrary real
Lie algebra equipped with a pseudo-inner product: $g(-,-)$. In similarity to
semi-simple Lie algebras, define
$\mathfrak{u}\subset\mathfrak{g}^{\mathbb{C}}$ to be a compact real form
(w.r.t to $g^{\mathbb{C}}$) precisely when $g^{\mathbb{C}}(-,-)$ is real on
$\mathfrak{u}$ and is positive definite. We observe that if there exist a
compact real form of $\mathfrak{g}^{\mathbb{C}}$, then by our results there
exists a Cartan involution (of Lie algebras) of $g$, thus we have proved a
general version of $\acute{E}$. Cartan’s result: ([7], Thm 7.1). Note also
that the proof given there for the semi-simple case w.r.t to the Killing form
is not valid for a general pair: $(\mathfrak{g},g)$ as above, indeed following
the notation of the proof, it is not obvious that $N:=\sigma\tau\in
O(n,\mathbb{C})\cap Aut(\mathfrak{g}^{\mathbb{C}})$. We also note that Theorem
2 is a generalised version of ([7], Thm 7.2).

We define for a general real Lie algebra:

######  Definition 5.

Let $\mathfrak{g}\subset(\mathfrak{g}^{\mathbb{C}},g^{\mathbb{C}})$ be a real
form and a real slice. A Cartan involution $\theta$ of $\mathfrak{g}$ is a
Cartan involution of $g^{\mathbb{C}}_{|_{\mathfrak{g}}}(-,-)$ which is also a
homomorphism of Lie algebras.

Note that if $\mathfrak{g}$ is semi-simple and $g:=\kappa(-,-)$, then there
are no Cartan involutions of $\mathfrak{g}$, however there are w.r.t $-g$.
Thus as an example consider $\mathfrak{g}:=\mathfrak{sl}_{2}(\mathbb{R})^{2}$
with $g:=-\kappa\oplus\kappa$, then there are no Cartan involutions of
$\mathfrak{g}$ neither w.r.t $g$ nor $-g$ (see Example 4). However if
$g:=-\kappa\oplus-\kappa$, then $\theta\oplus\theta$ is a Cartan involution of
$\mathfrak{g}$.

######  Theorem 8.

Let $(\mathfrak{g}^{\mathbb{C}},g^{\mathbb{C}})$ be a holomorphic inner
product space, where $\mathfrak{g}^{\mathbb{C}}$ is a complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{g}^{\mathbb{C}}$ be a real form which is a real
slice. Assume there exist a compact real form
$\mathfrak{u}\subset\mathfrak{g}^{\mathbb{C}}$. Let $\sigma$ be the
conjugation map of $\mathfrak{g}$. Then the following statements hold:

  1. (1)

There exist an automorphism $\phi\in Aut(\mathfrak{g}^{\mathbb{C}})\cap
O(n,\mathbb{C})$ such that
$\sigma(\phi(\mathfrak{u}))\subset\phi(\mathfrak{u})$.

  2. (2)

Any two Cartan involutions of $\mathfrak{g}$ are conjugate in
$Aut(\mathfrak{g})_{0}\cap O(p,q)_{0}$.

Discussion. As we have shown in this paper, Wick-rotating a pseudo-Riemannian
Lie group $G$ to a Riemannian Lie group relies on the existence of a Cartan
involution of $\mathfrak{g}$. Moreover in view of the existence of such a
Cartan involution $\theta$, then there exist naturally a Wick-rotation to a
Riemannian Lie group $\tilde{G}$ such that $G$ and $\tilde{G}$ are compatible,
i.e on Lie algebra level the conjugation maps of the Lie algebras commute. In
order to understand a Wick-rotation of a signature $(p,q)$ to a non-Riemannian
signature, then it would be interesting to know whether the following
conjecture is true, which is not obvious (to the author) even for a semi-
simple Lie algebra equipped with the Killing form: $-\kappa$. The conjecture
is true for abelian Lie algebras, and many examples for where the conjecture
is true are naturally found within the class of semi-simple Lie algebras,
typically among the classical ones, for example real forms of
$\Big{(}\mathfrak{o}(p+q,\mathbb{C}),-\kappa\Big{)}$.

######  Conjecture 1.

Let $(\mathfrak{g}^{\mathbb{C}},g^{\mathbb{C}})$ be a holomorphic inner
product space, where $\mathfrak{g}^{\mathbb{C}}$ is a complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{g}^{\mathbb{C}}\supset\tilde{\mathfrak{g}}$ be
real forms which are also real slices. Assume there exist a compact real form
$\mathfrak{u}\subset\mathfrak{g}^{\mathbb{C}}$. Then there exist automorphisms
$\phi_{1},\phi_{2},\phi_{3}\in Aut(\mathfrak{g}^{\mathbb{C}})\cap
O(n,\mathbb{C})$ such that:
$\Big{(}\phi_{1}(\mathfrak{g}),\phi_{2}(\tilde{\mathfrak{g}}),\phi_{3}(\mathfrak{u})\Big{)}$,
is a compatible triple of Lie algebras.

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