Wick-rotations of pseudo-Riemannian Lie groups

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Journal of Geometry and Physics

journal homepage:www.elsevier.com/locate/geomphys

Wick-rotations of pseudo-Riemannian Lie groups

Christer Helleland

Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway

a r t i c l e i n f o

Article history:

Received 21 July 2019

Received in revised form 23 April 2020 Accepted 31 August 2020

Available online 5 September 2020

Keywords:

Real GIT Wick-rotations Cartan involutions Lie groups

a b s t r a c t

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (Helleland and Hervik, 2018; Helleland and Hervik, 2019). An invariant for Wick- rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group (a Lie group with a left-invariant metric) can be Wick-rotated to a Riemannian Lie group.

We define a Cartan involution of a general Lie algebra, and prove a general version of E. Cartan’s result, namely the existence and conjugacy of Cartan involutions.´

©2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

This paper is motivated first of all by the study of Wick-rotations of pseudo-Riemannian manifolds defined in [3].

Given a pseudo-Riemannian manifold (M

,

^g) of signature (p

,

q), it is interesting know whether it can be Wick-rotated to another space (_M

˜ ,

g) (w.r.t. a fixed point

˜

p

∈

M

∩ ˜

M) of signaturep

˜ + ˜

q

=

p

+

q. In [2–4] the isometry action of the pseudo-orthogonal groupO(p

,

q) acting on tensors restricted topis explored. For instance it is proved that ifp

˜ =

0 (i.e.g

˜

is Riemannian) then there is a Cartan involution of the metric

θ ∈

O(p

,

^{q) (at}p) which fixes the Riemann tensorRunder the isometry action, i.e.

θ ·

R

=

R. Thus (M

,

^{g) is}Riemann purely electric (RPE) atp. More generally it is proved that for a space to bepurely electric(respectivelypurely magnetic) or (RPE) (respectivelyRiemann purely magnetic) is preserved under a Wick-rotation at a common fixed pointp.

A particular subclass of Wick-rotations which is of interest in its own right and deserves to be explored, is the class ofLie groups Gequipped with left-invariant metrics, so calledpseudo-Riemannian Lie groups. If we look at a semi-simple complex Lie groupG^Cequipped with the left-invariant Killing form:

− κ

, then there are natural examples of Wick-rotations to find at the identity point, simply because there exist real forms. Moreover by the theory of semi-simple Lie groups, one may always Wick-rotate a real form (G

, − κ

⁾

⊂

(G^C

, − κ

) to a Riemannian Lie group, simply because of the existence of a Cartan involution of the Lie algebrag. Thus motivated by this example, then for a general pseudo-Riemannian Lie group (G

,

^g), an interesting question one may ask:

Given a pseudo-Riemannian Lie group(G

,

g), when can it be Wick-rotated to a Riemannian Lie group(_G

˜ ,

g

˜

)?

Suppose (G

,

g) is Wick-rotated to a Riemannian Lie group (_G

˜ ,

g), then in view of the results given in [2–4], then the

˜

so called Wick-rotatable tensors restricted togmust be fixed by the isometry action (induced from the metric) of some (linear) Cartan involution

θ ∈

O(p

,

q) of the metric. This could for instance be the Riemann tensorR(as mentioned above), and is related to the fact thatRcan be embedded into the same complex orbit as_R

˜

(the Riemann tensor of (_G

˜ ,

g) restricted

˜

tog), i.e.

˜

O(p

+

q

,

C)

·

R

∋ ˜

R

.

E-mail address: [email protected].

https://doi.org/10.1016/j.geomphys.2020.103902

0393-0440/©2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/

licenses/by/4.0/).

However some 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 algebrag. Moreover in the semi-simple case (equipped with the left-invariant Killing form) such tensors are naturally fixed by the Cartan involutions of the Lie algebra:

θ ∈

Aut(g). For example the Levi-Civita connection is given by:

∇

_xy

=

¹

2

[

x

,

^y

]

, thus naturally

θ · ∇

_xy

= ∇

_xy.

The author of this paper therefore pondered about the existence of a Cartan involution:

θ ∈

Aut(g), for a general left-invariant metric (on a general Lie groupG) which can be Wick-rotated to a Riemannian Lie group_G.

˜

We prove an invariant for Wick-rotations of Lie groups, and give a complete answer to the question above, where we show that the answer is precisely related to the existence of a Cartan involution of the Lie algebra. Our main result of this paper isTheorem 3.1:

Theorem A. Suppose(G

,

^g)is a pseudo-Riemannian Lie group that can be Wick-rotated to another Lie group(_G

˜ ,

g

˜

). Then there exists a Cartan involution ofgif and only if there exists a Cartan involution ofg.

˜

We begin this paper by defining every notion we shall use throughout, and recall the definitions of Wick-rotations in [3]. Some new definitions are also given, in particular we define aWick-rotation of a Lie group, and aCartan involution of a general Lie algebra. We also state the results we use from [4], which makes the proofs easier to follow.

Remark 1.1. In this paper a Riemannian space shall always denote the signature: (

+ , + , . . . , +

), and a Lorentzian space shall denote the signature: (

+ , + , . . . , + , −

) and so on. The anti-isometry mapg

↦→ −

ginduces an isomorphism O(p

,

^q)

∼ =

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. Moreover using a right-invariant metric instead of a left-invariant metric does not change the results of this paper.

Conventions: Throughout this paper

κ

shall denote the Killing form of a Lie algebra. A product of vector spacesV

×

V shall often be denoted by justV². A complex Lie group shall always be denoted by the symbol:G^C.

2. Preliminaries

2.1. Real forms and left-invariant metrics

In this paper a real Lie groupGshall be said to be animmersive real formof a complex Lie groupG^C, if there is a real immersionG

→

G^C(of Lie groups) whereG^Cis viewed as a real Lie group, such thatgis embedded as a real form of g^C(the Lie algebra ofG^C). If the immersion is also injective then we shall callGavirtual real form. A virtual real formG which is also an embedding (i.e. the image ofGis closed inG^C), we shall say that the real form is anembedded real form.

An embedded real form which also satisfies:G^C

=

G

·

G^C₀ (abstract group product) shall be said to be areal 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 ifGis a connected semi-simple Lie group, then it is a fact thatGis a virtual real form if and only ifGis linear.

One shall note that given any 1-connected real Lie groupG, then we can complexify the Lie algebra via an inclusioni:

g

↪ →

g^C. We can find a complex 1-connected Lie group:G^Cwith Lie algebrag^C. One can find a smooth map (of real Lie groups):G

→

G^Cwith differentiali, thusGis an immersive real form ofG^C.

We shall abuse notation and writeG

⊂

G^Cfor an immersive real form.

Example 2.1. Consider the complex orthogonal group:O(4

,

C), then the map:g

↦→

I_3,1gI_3,1, is a conjugation map (i.e. the differential is a conjugation map), where (I_3,1)_ii

= +

1 for 1

≤

i

≤

2

,

^(I3,1)₃₃

= −

1 and zero otherwise. The fix points of this map are justO(1

,

3), which is an example of a real form ofO(4

,

C). Consider the universal covering groupG

:=

SL˜₂(R) ofSL₂(R), then it is a fact thatGis not a virtual real form of any complex Lie group. HoweverGis an immersive real form ofSL₂(C).

LetGbe a real Lie group, then a left-invariant metricg onGis a pseudo-Riemannian metric satisfying:

g_gh(L_gh^∗(x_h)

,

^Lgh∗(y_h))

=

g_h(x_h

,

^yh)

, ∀

g

,

^h

∈

G

, ∀

x_h

,

^yh

∈

T_hG

,

whereL_g^∗ is the push-forward of the translation map:G

− →

^Lg G:h

↦→

gh. Instead of writingg_e(

− , −

) for the metric at the identity point, we simply write justg(

− , −

). A bi-invariant metricgon a real Lie groupGis a left-invariant metric which is also right-invariant i.e.L_g above is replaced withR_g

:

h

↦→

hg.

On a real vector spaceV a symmetric non-degenerate bilinear formg shall be referred to as apseudo-inner product, and an inner product in the case of positive definite. A pair (V

,

^g) shall be referred to as a pseudo-inner product space (respectively inner product space). If we have a Lie algebragwith a pseudo-inner productg which satisfies:

g(

[

x

,

^y

] ,

^z)

=

g(x

, [

y

,

^z

]

)

,

^x

,

^y

,

^z

∈

g

,

theng shall be calledinvariant. Such a pair: (g

,

g) is called aquadratic Lie algebra. For example the pair:

(

sl₂(R)

, − κ

) is a quadratic Lie algebra, however the 3-dimensional Heisenberg Lie algebra:h₃(R), is never a quadratic Lie algebra. We

recall that an idealI_◁gis callednon-degenerateifg

=

I

⊕

I^⊥w.r.t. the invariant formg. In the case thatgis a reductive Lie algebra, then all ideals are in fact non-degenerate.

Aholomorphic inner product g^Con a complex vector spaceV^Cshall be a symmetric non-degenerate complex bilinear form. The definitions of left-invariance and so on above are analogous in the case of a complex Lie group equipped with a holomorphic metric.

Definition 2.1. A real Lie groupG equipped with a left-invariant metric g, denoted (G

,

g) shall be called a pseudo- Riemannian Lie group. Ifgis also a Riemannian metric then the pair (G

,

^g) shall be called aRiemannian Lie group. A complex Lie groupG^Cequipped with a left-invariant holomorphic metric, shall be called aholomorphic Riemannian Lie group(or a complex Riemannian Lie group).

Definition 2.2. Let (G

,

^g1) and (H

,

^g2) be two pseudo-Riemannian Lie groups. ThenGis said to be isometric toHif there exists a Lie group isomorphism:G

− →

^F H, such thatF∗

:

g

→

h is an isomorphism of pseudo-inner product spaces:

(g

,

^g1)

∼ =

(h

,

^g2). The spaces are said to be locally isometric if there exists a local homomorphismG

⊃

U

− →

^F V

⊂

Hsuch thatF∗is an isomorphism of pseudo-inner product spaces: (g

,

^g1)

∼ =

(h

,

^g2).

The left-invariant metrics on a real Lie groupGare in bijections with the pseudo-inner products on the Lie algebrag.

So we shall always work with a pseudo-inner productg on the Lie algebra and induce a left-invariant metric on the Lie group by:

g_h(x_h

,

^yh)

:=

g (

L_h⁻1

∗ (x_h)

,

^L_h⁻_∗^1(yh)

)

,

^xh

,

^yh

∈

T_hG

.

We note that for a compact Lie groupG, we can always complexify it to a complex Lie group:G^C, such thatG

⊂

G^C is a real form, by using the universal complexification group. In particular starting from a compact Lie group with a left-invariant metric we naturally have a candidate for a holomorphic Riemannian Lie group such that G

⊂

G^C is a real form. Recall that the universal complexification group of a real Lie group G, is a pair: (G^C

, η

^{), where}

η

^{is a real} Lie homomorphism:G

→

G^C, satisfying the universal property (see for instance [5]). For example the pseudo-orthogonal groups:O(p

,

q) has universal complexification groupO(p

+

q

,

C).

2.2. Wick-rotations of pseudo-Riemannian manifolds

We recall some of the definitions of Wick-rotations given in [3], and define a Wick-rotation of a pseudo-Riemannian Lie group.

Definition 2.3. Given a holomorphic inner product space (E

,

^{gC). Then ifV}

⊂

E is a real linear subspace for which g

:=

g^C⏐

⏐V is non-degenerate and real valued, i.e.,g(X

,

^Y)

∈

_R

, ∀

X

,

^Y

∈

V, we will callV areal slice.

Remark 2.1. In this paper we always assumeV

⊂

(E

,

^gC) has the same real dimension as the complex dimension of E. ThusV is also a real form of E, i.e. there is a conjugation mapE

− →

^σ E with fix pointsV. We shall simply refer to V

⊂

(E

,

^gC) as a real form in such a case, to mean both a real slice and a real form.

Thus in the definition (V

,

^g

:=

g^C⏐

⏐V) is a pseudo-inner product space, and if (p

,

q) denotes the signature ofg, then the isometry groupO(p

,

^{q) of (V}

,

g) is a real Lie group and is a real form ofO(p

+

q

,

C) (the isometries of (E

,

^gC)). Indeed if

σ

is the conjugation map ofVinEthen note the involutionF of real Lie groups:

g

↦→ σ

^g

σ ,

^g

∈

O(p

+

q

,

C)

.

The differential of this map is a conjugation map, andO(p

,

q) is the fix points ofF, i.e. is a real form. Such a mapFis often called areal structure.

Definition 2.4. Given a complex (holomorphic) manifold M^Cwith complex (holomorphic) Riemannian metricg^C. If a submanifoldM

⊂

M^Cfor any pointp

∈

M we have thatT_pMis a real slice of (T_pM^C

,

^gC) (in the sense ofDefinition 2.3), we will callMa real slice of (M^C

,

^gC).

This definition implies that the induced metric fromM^C is real valued onM. M is therefore a pseudo-Riemannian manifold.

Definition 2.5 (Wick-related Spaces).Two pseudo-Riemannian manifoldsM and _M

˜

are said to be Wick-relatedif there exists a holomorphic Riemannian manifold (M^C

,

^gC) such thatMand_M

˜

are embedded as real slices ofM^C.

Definition 2.6(Wick-rotation).If two Wick-related spaces (of the same real dimension) intersect at a pointpinM^C, then we will use the termWick-rotation: the manifoldMcan be Wick-rotated to the manifold_M

˜

(with respect to the pointp).

We now define a Wick-rotation of a pseudo-Riemannian Lie group:

Definition 2.7(Wick-rotation of a Pseudo-Riemannian Lie Group).LetG

⊂

G^C

⊃ ˜

Gbe two immersive real forms which are Wick-related in (G^C

,

^{gC) forgC}a left-invariant holomorphic metric. Then we shall say that the pseudo-Riemannian Lie group (G

,

^g) is Wick-rotated to (_G

˜ ,

g

˜

).

Thus from the definition: (G

,

^g)

⊂

(G^C

,

^gC) is a real slice of Lie groups, and shall write (p

,

q) for the signature ofg.

If there is another real slice (_G

˜ ,

g)

˜ ⊂

(G^C

,

^gC) of Lie groups, then we shall refer to the signature ofg

˜

as (p

˜ ,

q). We shall

˜

often just say a Wick-rotations of Lie groups. Note that two Lie groups which are Wick-related are also Wick-rotated at the identity pointp

:=

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 2.1. Suppose(G

,

^g)is Wick-rotatable to(_G

˜ ,

g

˜

)and they are both connected. Then g(

− , −

)is bi-invariant if and only ifg

˜

(

− , −

)is bi-invariant.

Proof. The proofs given in ([9], Lemma 7.1 and 7.2) also hold for pseudo-Riemannian left-invariant metrics, witho(n) replaced with o(p

,

q). Moreover if the metric g(

− , −

) is bi-invariant, then because ad(g)

⊂

o(p

,

^q)

⊂

o(n

,

C), and ad(g)^C

=

ad(g^C) it follows that the holomorphic metric must also be bi-invariant, thus alsog

˜

(

− , −

). 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 2.2. LetSL₂(R)

⊂

SL₂(C)

⊃

SU(2) be the natural inclusions. Then they are real forms, and Wick-rotated w.r.t.

to the holomorphic Killing form

κ

^onsl2(C). Note that (SL₂(R)

, κ

) is Lorentzian and (SU(2)

, κ

) has signature: (

− , − , −

).

We also define:

Definition 2.8. LetV

⊂

(E

,

^gC) be a real slice. We say an involutionV

− →

^θ V

∈

O(p

,

^{q), is a}Cartan involutionofg

:=

g^C⏐

⏐V, ifg_θ(

· , ·

)

:=

g^C⏐

⏐V(

· , θ

⁽

·

)), is an inner product onV. If

θ =

1 thenVis said to be acompact real slice, or in the case thatV is also a real form, thenVshall be said to be acompact real form.

Note the resemblance (in the definition) with a compact real form of a complex semi-simple Lie algebra and its Killing form. In the case of Lie algebras: (g

,

^g)

⊂

(g^C

,

^gC), then a Cartan involution

θ

^ofgis not necessarily a homomorphism of Lie algebras, since we do not know it they exist. We do not even know if there exists a compact real form which is also a Lie subalgebra of (g^C

,

^gC). But we know ifgis semi-simple, andg^C

= − κ

, then there exists a Cartan involution

θ

^which is also homomorphism of the Lie algebra.

But more generally we shall define:

Definition 2.9. Letg

⊂

(g^C

,

^gC) be a real form. A Cartan involution

θ

^ofgis a Cartan involution ofg

:=

g_|^C

g(

− , −

) which is also a homomorphism of Lie algebras.

Thus a Cartan involution ofg is only a linear Cartan involution of the pseudo-inner productg, but a Cartan involution ofgis a Cartan involution ofg which is also a homomorphism of Lie algebras. Currently at this point we only know that Cartan involutions ofgexist whengis abelian orgis semi-simple equipped with the Killing form:

− κ

. One shall note that there are examples where they do not exist, indeed by changing the sign to:

κ

, then it is straighforward to show that there are no Cartan involutions ofg.

Definition 2.10. Two real formsVand˜VofEare said to be compatible if their conjugation maps commute, i.e.

[ σ , σ ˜ ] =

0.

Often we shall refer to a pair (V

,

V) as a compatible pair, to mean that the spaces are compatible.

˜

We recall from [3], that if (E

,

^gC) is a holomorphic inner product space, andV

,

_V

˜

_andW are real forms such thatW is a compact real form (i.e. of Euclidean signature), then if they are pairwise compatible, the triple:

(

V

,

_V

˜ ,

^W)

, is said to be acompatible triple. Note thatExample 2.2is an example of a compatible triple:

(

V

:=

sl₂(R)

,

_V

˜ :=

su(2)

,

^W

:=

su(2) )

.

We shall call the eigenspace decomposition of a Cartan involution:

θ

, for the Cartan decomposition.

Remark 2.2. By the uniqueness of a signature associated to a pseudo-inner productg then all Cartan involutions ofg are conjugate inO(p

,

q). In fact given two Cartan involutions:

θ

j(j

=

1

,

^{2) theng}

↦→ θ

jg

θ

jis a global Cartan involution of O(p

,

q). Thus ifg

θ

1g⁻¹

= θ

2for someg

∈

O(p

,

q), then writingg

=

k₂e^x, wherek₂commutes with

θ

2andx

∈

o(p

,

^{q), we} obtain

θ

1

=

e^x

θ

2e^−x, and therefore

θ

1

, θ

2are conjugate by an elementg

∈

O(p

,

^q)0.

Suppose now we have a Wick-rotation of two real Lie groups: (G

,

^g)

⊂

(G^C

,

^gC)

⊃

(_G

˜ ,

g). Let

˜ θ ∈

O(p

,

^{q) be a} Cartan involution of the metricg, and letW denote the corresponding unique compact real form associated with

θ

^{, i.e.}

W

:=

V+

⊕

iV−, whereg

=

V+

⊕

V−is the Cartan decomposition. Then by [4] it is possible to find a real form_V

˜ ⊂

g^C(as vector spaces) and a linear isomorphism:_V

˜ − → ˜

^φ gsuch that

φ

^C

∈

O(n

,

C), and (g

,

_V

˜ ,

^W) is a compatible triple. So consider the triple:(

o(p

,

^q)

,

^o(p

˜ ,

q)

˜ ,

^o(n))

, of Lie algebras of the isometry groups associated with the compatible triple (g

,

_V

˜ ,

^W).

Then the following straightforward result is important to note:

Lemma 2.1([3], Lemma 3.6). The triple of real forms:

(

o(p

,

^q)

,

^o(_p

˜ ,

_q)

˜ ,

^o(n))

, embedded intoo(n

,

C)is a compatible triple of Lie algebras.

Thus we note that up to an isometryg

∈

O(n

,

C) we may assume our two Lie algebrasgand

˜

g(viewed as a vector space) form a compatible triple with a compact real formW

⊂

(g^C

,

^gC).

2.3. Real GIT on compatible representations

In this section we recall some definitions and results of [4] that we shall use. We consider certain type of groups here. When considering a real form:G

⊂

G^C, thenG^Cshall be of type linearly complex reductive, andGshould either be linearly real reductive, or in the case whereG^C

⊂

GL(V^C) is defined overR, the real points:G

:=

GL(V)

∩

G^C. This is the assumptions in the paper [4]. Thus we may for instance use the pseudo-orthogonal groupO(p

,

^q)

⊂

O(n

,

C) defined as the isometry group of some pseudo-inner product space: (V

,

^g)

⊂

(V^C

,

^gC). A compact real form ofG^Cshall always be denoted byU.

LetG

⊂

GL(V) be such a group. A Cartan involution

θ

^ofgis now a Cartan involution in the sense of a reductive Lie algebra. Recall that this means that

θ

is the restriction of a Cartan involution ofgl(V). In view ofDefinition 2.9,

θ

^{is a} Cartan involution of (g

,

^{g), withg}

= λκ ⊕

B(

λ <

^{0), where}

κ

is the Killing form on

[

g

,

^g

]

andBa pseudo-inner product onz(g). We refer to for example [4] or [11] where such Cartan involutions are considered in more detail. A global Cartan involutionΘ^withdΘ

= θ

^ofGalways exists for such groups. For example the class of linear semisimple Lie groups of finitely many connected components (fcc) are one such class.

Definition 2.11. LetG

⊂

G^C

⊃ ˜

Gbe two real Lie subgroups of a complex Lie group such that the real Lie algebras are real forms ofg^C. Then we sayGand_G

˜

_{arecompatible}if the Lie algebras are compatible.

Definition 2.12. LetG

⊂

G^C

⊃ ˜

GandU

⊂

G^Cbe real Lie subgroups of a complex Lie group such that the real Lie algebras are real forms ofg^C. Moreover assumeUis compact. Then we say

( G

,

_G

˜ ,

^U)

is acompatible tripleif the Lie algebras are pairwise compatible.

If we useLemma 2.1, in the context of Wick-rotations (see the previous section), then the triple of isometry groups:

(

O(p

,

^q)

,

^O(p

˜ ,

q)

˜ ,

^O(n))

form a compatible triple when the pseudo-inner product spaces they are isometries of, form a compatible triple.

Definition 2.13([11]).Let G ^ρ

G

− →

V GL(V) be a real representation, then

ρ

V^Gis said to be abalanced representationif there exist an involutionV

− →

^θ V, and a global Cartan involution:G

− →

^Θ Gsuch that:

(

∀

g

∈

G)(

ρ

V^G(Θ^(g))

= θ ◦ ρ

V^G(g)

◦ θ

)

.

Thus if we have an involution

θ

^ofVbalancing our action, then w.r.t. the global Cartan involutionΘ^ofGwith Cartan decomposition:G

=

Ke^p, there exists a pseudo-inner productg(

− , −

) onVsuch that

θ

is a Cartan involution ofg(

− , −

), and the inner productg_θ(

− , −

)

:=

g(

− , θ

⁽

−

)) isK-invariant. LetM(G

,

V) denote the minimal vectors of our action, i.e.

those

v ∈

Vsatisfying:

∥

g

· v ∥ ≥ ∥ v ∥

for allg

∈

G, where

∥ v ∥

²

:=

g_θ(

v, v

^{). Then ifV}

=

V+

⊕

V−is the Cartan decomposition, we naturally haveV+

∪

V−

⊂

M(G

,

V). The Cartan involutions ofg(

− , −

) which are conjugate by the action ofGto

θ

^are defined as theinner Cartan involutionsofg(

− , −

).

A complex action:

ρ

^{CofGCacting onVC}is said to be a complexified action of a real action

ρ

_V^Gif

ρ

^C(g)(

v

⁾

= ρ

^(g)(

v

⁾ for allg

∈

Gand

v ∈

V.

Definition 2.14. LetG

⊂

G^C

⊃ ˜

Gbe real forms, andG ^ρ

G

− →

V GL(V) and_G

˜

^ρ

G˜ V˜

− →

GL(V) be real representations of Lie groups.

˜

SupposeG^{C ρ}

C

− →

GL(V^C) is a complexified action of both

ρ

^G_V ^and

ρ

_V^G˜^˜. Then we say that

ρ

_V^{G iscompatiblewith}

ρ

_V^G˜^˜, if the following two criteria are fulfilled:

(1) Gand_G

˜

are compatible real forms ofG^C.

(2) V and_V

˜

are compatible real forms ofV^C.

Definition 2.15. Let

ρ

_V^G

, ρ

_V^G˜˜ and

ρ

_W^U be pairwise compatible representations, whereU

⊂

G^C, is a compact real form. Then the triple:

(

ρ

_V^G

, ρ

_V^G˜^˜

, ρ

_W^U)

is said to be acompatible triple.

If we have such a compatible triple, then all the real actions in the triple are balanced, and we can choose pseudo-inner productsg(

− , −

) andg(

˜ − , −

) onV and_V

˜

respectively, in such a way that they restrict from the same Hermitian form onV^C. Moreover if

τ

denotes the conjugation map ofWinV^Cthen it restricts to Cartan involutions:

θ

^{(ofg) and}

θ ˜

^(ofg

˜

).

The Cartan involutions also balance the real actions respectively. In particular the inner productsg_θ andg

˜

θ˜ both restrict from theU-invariant Hermitian inner productH(

− , τ

⁽

−

)). The minimal vectors satisfy:

M(G

,

^V)

⊂

M(G^C

,

^VC)

⊃

M(_G

˜ ,

_V

˜

₎

,

^W

⊂

M(G^C

,

^VC)

.

Denote the Cartan decompositions byV

=

V+

⊕

V−and_V

˜ = ˜

V+

⊕ ˜

V−respectively.

Definition 2.16. Let(

ρ

V^G

, ρ

_V^G˜˜

)

be a compatible pair. Suppose

v ∈

V and

v ˜ ∈ ˜

V are such that

v ˜ ∈

G^C

v

, then we shall say thatG

v

^{iscompatiblewith}_G

˜ v ˜

^{. We writeG}

v ∼ ˜

G

v ˜

^.

It is important to note the following result:

Theorem 2.1([4]). Let

(

ρ

_V^G

, ρ

_V^G˜^˜

, ρ

_W^U)

be a compatible triple. Suppose

v ∈

V and

v ˜ ∈ ˜

V are such that: _G

˜ v ˜ ∼

G

v

^{. Then} G

v ∩

V+

̸= ∅

(respectively G

v ∩

V−

̸= ∅

) if and only if_G

˜ v ˜ ∩ ˜

V+

̸= ∅

(respectively_G

˜ v ˜ ∩ ˜

V−

̸= ∅

).

Observe that if there exists

v

⁺

∈

G

v

^{, then}

θ

⁽

v

⁺⁾

= v

^{+, i.e. ifg}

∈

Gis such thatg

· v = v

⁺, then there is an inner Cartan involution

θ

^′ofg(

− , −

) such that

θ

^′(

v

⁾

= v

^usingg.

We shall also state the following important result:

Theorem 2.2([4]). Let(

ρ

_V^G

, ρ

_W^U)be a compatible pair. Let

v ∈

V , then the following statements are equivalent:

A There exists

w ∈

W such that U

w ∼

G

v

^.

B There exists an inner Cartan involution V

− →

^θ V such that

θ

⁽

v

⁾

= v

^. C There exists

w ∈

W such that U

w ∩

G

v ̸= ∅

.

In fact if there is a

w ∈

W and

v ∈

V such thatU

w ∼

G

v

^then:

∅ ̸=

U

w ∩

G

v =

G

v ∩

M(G

,

^V)

=

K

v,

whereK

=

U

∩

G.

A worked out example of compatible representations is given in the next section in the context of Wick-rotations of Lie groups.

2.4. The isometry action on bilinear maps into the Lie algebra

In this section we shall consider the action that we are going to use to prove our main result of this paper. We shall explain in detail that under a Wick-rotation, the isometry groups of the pseudo-inner product spaces induces compatible representations (see Defn. Section2.3).

Suppose we have a Wick-rotation of pseudo-Riemannian Lie groups:

(G

,

^g)

⊂

(G^C

,

^gC)

⊃

(_G

˜ ,

g

˜

). As we have seen we can choose a mapg

∈

O(n

,

C) such that we obtain a compatible triple:

(g

,

_V

˜ ,

^{W), with}_V

˜ :=

g(g). We shall denote

˜

g

˜

also for the pseudo-inner product on_V

˜

restricted fromg^C. We can choose a pseudo-orthonormal basis:

{

e₁

, . . . ,

^ep

, . . . ,

^en

}

(ofg) and similarly

{˜

e₁

, . . . ,

e

˜

_p˜

, . . . ,

e

˜

_n

}

(ofg), such that

˜

W is the real span of both the sets:Y

:= {

e₁

, . . . ,

^ep

,

^iep+₁

. . .

^ien

}

and_Y

˜ := {˜

e₁

, . . . ,

e

˜

_p˜

,

ⁱe

˜

_p˜+1

, . . . ,

ⁱ

˜

e_n

}

. Denote the corresponding Cartan involutions by

θ

^{(ofg) and}

θ ˜

^(ofg). Note that

˜

Y and_Y

˜

are both an orthonormal basis ofg^C.

Consider the complex isometry action of O(n

,

C) ong^C byg

·

x

:=

g(x). This action restricts to the real isometry actions ofO(p

,

^{q) ongandO(}p

˜ ,

q) on

˜

Vrespectively. LetVand_V

˜

denote the real vector spaces of bilinear maps:g²

→

g (respectively_V

˜

²

→ ˜

V). ThusV

⊂

V^C

⊃ ˜

Vare real forms, whereV^Cis the complex vector space of complex bilinear maps:

(g^C)²

→

g^C. The complex isometry action naturally extends to a complex action ofO(n

,

C) onb

∈

V^C, by (g

·

b)(x

,

^y)

:=

g

(

b(g⁻¹(x)

,

^g−1(y)))

,

^x

,

^y

∈

g^C

,

^g

∈

O(n

,

C)

.

Note that the action again restricts to action of the real isometry groups onVand_V

˜

respectively. Denote the real actions by

ρ

^and

ρ ˜

respectively. The Cartan involution

θ ∈

O(p

,

q) (respectively

θ ˜ ∈

O(p

˜ ,

q)) naturally extends to an involution

˜

ofV (respectively_V

˜

), by the action:

ρ

⁽

θ

) (respectively

ρ ˜

⁽

θ ˜

)). The holomorphic inner productg^Cextends naturally to a holomorphic inner product:g^C, by defining:

g^C(b₁

,

^b2)

:=

n

∑

j

g^C (

b₁(y_j

,

^yj)

,

^b2(y_j

,

^yj) )

.

Observe that if we change basis w.r.t. to_Y

˜

instead then we obtain the same holomorphic inner product. Indeed this follows since we can findg

∈

O(n

,

C) sendingY

↦→ ˜

Y. It is easy to check thatV

⊂

(V^C

,

^gC)

⊃ ˜

V are real slices. Similarly if we defineWto be all bilinear maps:W²

→

W, then by constructionWis a compact real form of (V^C

,

^gC). Observe that the three real forms form a compatible triple inV^C. Therefore the actions form a compatible triple (see Section2.3). There is a natural choice ofO(n)-invariant Hermitian inner product onV^C, namely:H

:=

g^C(

· ,

^T(

·

)), whereT is the conjugation map ofW. This Hermitian inner product restricts to inner products onV

,

_V

˜

_andW. Observe that the inner Cartan involutions of

ρ

(respectively

ρ ˜

) are those conjugate to

ρ

⁽

θ

) (respectively

ρ ˜

⁽

θ ˜

^)).

2.5. Wick-rotatable tensors of pseudo-Riemannian manifolds

For a Wick-rotation of Lie groups it is worth noting that the action in the previous section is just an example of a tensor action ofO(n

,

C) on a general tensor space of finite form:

V^C

:=

⨁

k,m

((⨂^k

i=₁

g^C) ⨂(⨂^m

i=₁

(g^C)^∗ ))

,

induced from the isometry action of the holomorphic metricg^C. Analogously we define:

V

:=

⨁

k,m

((⨂^k

i=₁

g) ⨂(⨂^m

i=₁

g^∗

))

,

_V

˜ :=

⨁

k,m

((⨂^k

i=₁

˜

g) ⨂(⨂^m

i=₁

˜

g^∗

))

.

The real isometry groups:O(p

,

q) (respectivelyO(p

˜ ,

q)) restrict to acting on

˜

V(respectively_V

˜

_).

More generally for a Wick-rotation of pseudo-Riemannian manifolds:

(M

,

^g)

⊂

(M^C

,

^gC)

⊃

(_M

˜ ,

g

˜

)

,

at a common pointp

∈

M

∩ ˜

M, then by replacinggwithT_pM(respectivelyg

˜

withT_{pM), and}

˜

_gCwithT_pM^C, we obtain the induced tensor action on real forms:V

⊂

V^C

⊃ ˜

V.

One shall note that the metrics, Cartan involutions all extend naturally to these spaces via the tangent spaces. Moreover ifg

∈

O(n

,

C) is such thatT_pMandg(T_pM) form a compatible triple with a compact real form

˜

W

⊂

T_pM^C, then naturally alsoVandg

· ˜

Vform a compatible triple withW

:=

⨁

k,m

((⨂k i=₁W

)⨂(

⨂m i=₁W^∗

)) .

For example the induced action ofO(n

,

C) onEnd(T_pM^C) given by conjugation:g

·

f

:=

gfg⁻¹is just the tensor action:

g

·

(

v

1

⊗ v

2)

:=

g(

v

1)

⊗

g(

v

2), for anO(n

,

C)-module isomorphism:End(T_pM^C)

∼ =

T_pM^C

⊗

T_pM^C. For a more detailed explanation of this example, and on the tensor action in general we refer to [4].

Consider the action in the previous section for instance, then one should observe that the complex Lie bracket

v := [− , −]

ofg^Cis a vector inV, but also there is ag

∈

O(n

,

C) such that

v ˜ :=

g

· v ∈

g

· ˜

V, i.e.

v

^and

v ˜

^{lie in the} same complex orbit:O(n

,

C)

v ∋ ˜ v

, in such a way thatO(p

,

^q)

v ∼

O(p

˜ ,

q)

˜ v ˜

are compatible real orbits.

Thus it useful to define for general tensors

v ∈

Vand

v ˜ ∈ ˜

V:

Definition 2.17([4]).Let (M

,

^{g) and (}_M

˜ ,

_g

˜

) be two Wick-rotatable pseudo-Riemannian manifolds at a common pointp.

Then two tensors

v ∈

Vand

v ˜ ∈ ˜

Vare said to beWick-rotatableatp, if they lie in the sameO(n

,

C)-orbit, i.e.

O(n

,

C)

v =

O(n

,

C)

v. ˜

One should note the subset of Wick-rotatable tensors consisting of those in the intersection:

v ∈

V

∩ ˜

V. Then there is a mapg

∈

O(n

,

C), such that

v

^{and g}

· v ∈

g

· ˜

V lie in the same complex orbit such thatO(p

,

^q)

v ∼

O(_p

˜ ,

_q)

˜ v ˜

^are compatible real orbits. More generally if

v

^and

v ˜

are Wick-rotatable i.e. by definitionO(n

,

C)

v =

O(n

,

C)

v ˜

, then also O(n

,

C)

v =

O(n

,

C)g

· ˜ v

. The main point is to be able to embed the vectors into the same complex orbit, such that we may apply the results of Section2.3.

Let (M

,

^g) be a pseudo-Riemannian manifold of signature (p

,

^{q), and}

θ ∈

O(p

,

q) be a Cartan involution ofg(

− , −

).

Consider the isometry tensor action ofO(p

,

^{q) onVas above:}

O(p

,

^{q) ρ}

O(p,^q)

−−−→

V GL(V)

.

Then

θ

naturally extends to an involutionΘ

:= ρ

V^O(p,q)(

θ

^{) onV}, and the metric naturally induces a pseudo-inner product:

g(

− , −

) onVsuch thatΘis a Cartan involution. Let nowR

∈

Vbe the Riemann tensor ofMatpforVsome tensor space.

For exampleRmay be considered as a multilinear form intoT_pM:T_pM³

→

T_pM, where the action is given by:

(g

·

R)(x

,

^y

,

^z)

:=

g (

R(g⁻¹(x)

,

^g−1(y)

,

^g−1(z)))

,

^x

,

^y

,

^z

∈

T_pM

,

^g

∈

O(p

,

^q)

.

Another approach is to considerRas a map inEnd(o(p

,

^q))

⊂

End

(

End(T_pM) )

at the pointp, where the action is given by:

(g

·

R)(X)

:=

gR(g⁻¹Xg)g⁻¹

,

^X

∈

o(p

,

^q)

,

^g

∈

O(p

,

^q)

.

The Riemann tensor atpis viewed in this way for instance in [2]. One may show that these two actions are equivalent up to anO(p

,

q)-module isomorphism, by identifying the spaces with the tensor space:T_pM

⊗

T_pM

⊗

T_pM

⊗

T_pM.

We also recall the following definition:

Definition 2.18. If there exists a Cartan involutionΘ ^{such that}Θ^(R)

=

R (respectivelyΘ^(R)

= −

R), then the space (M

,

^{g) atp}is called Riemann purely electric RPE (respectively Riemann purely magnetic (RPM)). If there is such aΘ^for the Weyl tensor atp, then (M

,

^{g) atp}is called purely electric (PE) (respectively purely magnetic (PM)).

Any Riemannian space (M

,

^g) is RPE at any pointp

∈

M, since the identity map

θ :=

1_TpMis a Cartan involution of the metricg at any point, thus the Cartan involution extended to tensors:Vis also the identity map, i.e.Θ^(R)

=

R.

The Levi-Civita connection

∇

of a real slice of a holomorphic Riemannian manifold (M

,

^g)

⊂

(M^C

,

^{gC) atp}

∈

M, restricts from the complex Levi-Civita connection:

∇

^Catpof the complex manifoldM^C. Thus the real Riemann curvature tensor R(ofM) atprestricts from the complex Riemann curvature tensorR^CofM^C(atp). Moreover ifric_g denotes the real Ricci curvature:T_pM²

→

_R, defined by:

ric_g(x

,

^y)

:=

Tr (

z

↦→

R(z

,

^y)(x))

,

then using a real basis ofT_pMalso forT_pM^Cwe see that restricting the complex Ricci curvature:ric_gC onM^CtoT_pMwe getric_g. Similarly the real Ricci operator:

Ric_g

∈

End(T_pM)

,

^gp(Ric_g(x)

,

^y)

=

ric_g(x

,

^y)

,

restricts form the complex Ricci curvature operator ofM^C(atp).

This means that in terms of Wick-rotations of pseudo-Riemannian manifolds at a common pointp: (M

,

^g)

⊂

(M^C

,

^gC)

⊃

(_M

˜ ,

g

˜

), we see that the pairs of tensors:

(

∇ , ∇ ˜

)

,

^(R

,

_R)

˜ ,

^(ricg

,

^ricg^˜)

,

^(Ricg

,

^Ric˜g)

,

are examples of Wick-rotatable tensors (atp) in the intersectionV

∩ ˜

V. The induced isometry action ofO(n

,

C) on these tensors (induced from the isometry action of the metric) can be naturally seen as the actions:

(g

· ∇

)(x

,

^y)

:=

g(

∇

_g−1xg⁻¹y)

,

^(g

·

R)(x

,

^y

,

^z)

:=

g (

R(g⁻¹x

,

^g−1y

,

^g−1z)) and

(g

·

ric_g)(x

,

^y)

:=

ric_g(gx

,

^gy)

,

^(g

·

Ric_g)(x)

:=

(g

◦

Ric_g

◦

g⁻¹)(x)

.

An immediate new result is the following:

Theorem 2.3. Let(M

,

^g)

⊂

(M^C

,

^gC)

⊃

(_M

˜ ,

g)

˜

be a Wick-rotation at a common point p

∈

M

∩ ˜

M. Assume(_M

˜ ,

g)

˜

is a Riemannian space. Then the following statements hold:

(1) There exists a Cartan involution

θ

of g such that

∇

_θ(x)

θ

^(y)

= θ

⁽

∇

_xy)for all x

,

^y

∈

T_pM.

(2) There exists a Cartan involution

θ

of g such that ric_g

(

θ

^(x)

, θ

^(y))

=

ric_g(x

,

^{y)for all x}

,

^y

∈

T_pM.

(3) There exists a Cartan involution

θ

of g such that

[ θ,

^Ricg

] =

0.

(4) There exists a Cartan involution

θ

of g such that R

(

θ

^(x)

, θ

^(y))(

θ

^(z))

= θ

(

R(x

,

^y)(z))

for all x

,

^y

,

^z

∈

T_pM. Thus(M

,

^g) is (RPE) at p.

Proof. It is enough to spell out the proof for the first case, as the other cases are identical. Let

v := ∇ ∈

Vand

v ˜ := ˜ ∇ ∈ ˜

V, and consider the isometry tensor action as above. The vectors

v

^and

v ˜

are Wick-rotatable, thus up to a mapg

∈

O(n

,

C) we can assume the real actions are compatible, and that

v

^and

v ˜

lie in the same complex orbit, such that the real orbits:

O(p

,

^q)

v ∼

O(p

˜ ,

q) are compatible. The result now follows from

˜

Theorem 2.2, sinceO(p

˜ ,

q)

˜ =

O(n) is a compact real form ofO(n

,

C). □

One shall note that Case (4) of the theorem is proved in [2]. We shall strengthenTheorem 2.3for Wick-rotations of pseudo-Riemannian Lie groups in the last section of the paper, by proving that a Cartan involution ofg may be chosen to be a homomorphism of Lie algebras.

3. An invariant of Wick-rotation of Lie groups

In this section we shall prove the main theorem of the paper, which is an invariance result based on the existence of a Cartan involution of the Lie algebras (Definition 2.9).

Let (G

,

^g)

⊂

(G^C

,

^gC)

⊃

(_G

˜ ,

g

˜

) be a Wick-rotation of Lie groups. Consider the action in Section2.4and following the notation there, then by our preparations, the main result is now easily deducible:

Theorem 3.1. Suppose(G

,

^g)is a pseudo-Riemannian Lie group that can be Wick-rotated to another Lie group(_G

˜ ,

g). Then

˜

there exists a Cartan involution ofgif and only if there exists a Cartan involution ofg.

˜

Proof. Consider the group action and the notation as in Section2.4. Thus if

v := [− , −]

is the Lie bracket ofg^Cthen

v ∈

Vand restricts to the Lie bracket ofg. We can choose

˜

g

∈

O(n

,

C) such thatg

· v ∈ ˜

V, i.e.

v

^and

v ˜ :=

g

· v

^{lie in} the same complex orbit, thusO(p

,

^q)

v ∼

O(p

˜ ,

q)

˜ v ˜

are compatible real orbits. Suppose

θ

is a Cartan involution ofg, and denoteV

=

V+

⊕

V−(respectively_V

˜ = ˜

V+

⊕ ˜

V−) for the Cartan decomposition w.r.t. to

ρ

⁽

θ

) (respectively

ρ ˜

⁽

θ ˜

^{)). Then} the action of

θ

^on

v

^fixes

v

^{, i.e.}

ρ

⁽

θ

⁾⁽

v

⁾

:= θ · v = v

^{, thus}

v ∈

V+. Hence the real orbit:O(p

,

^q)

v

^{intersectsV+. But then} byTheorem 2.1, it follows that there exists also

v ˜

^′

∈ ˜

V+

∩

O(p

˜ ,

q)

˜ v ˜

. Therefore chooseh

∈

O(p

˜ ,

q) such that

˜

h

· ˜ v = ˜ v

^′. By conjugating

ρ ˜

⁽

θ ˜

^{) byh}we obtain a Cartan involution

θ ˜

^′of g

˜

such that

θ ˜

^′

· ˜ v = ˜ v

. Finally since_V

˜ :=

g(g) for some

˜

g

∈

O(n

,

C) then the Cartan involutiong⁻¹

θ ˜

^′gfixes

v

, i.e. is a Cartan involution ofg

˜

and a homomorphism of Lie algebras.

The converse is symmetric. The theorem is proved. □ We find it useful to define for future exploration:

Definition 3.1. A property of a pseudo-Riemannian Lie group (G

,

g) is said to beWick-rotatableif it is an invariant under a Wick rotation of Lie groups.

Corollary 3.1. The existence of a Cartan involution ofgis Wick-rotatable.

Other Wick-rotatable properties include (see for example [1] on complexification of real Lie algebras): being semi- simple, abelian, nilpotent, solvable, reductive. Note that being simple, is not Wick-rotatable, indeed as an example consider the Lie groupO(1

,

3) with the left-invariant metric being the Killing form. Theno(1

,

3) is simple, but we may Wick-rotate O(1

,

^{3) toO(2}

,

2) which is semi-simple but not simple, aso(2

,

²⁾

∼ =

sl₂(R)²(two copies).

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 left-invariant Killing form:g

:= − κ

^.

Corollary 3.2. Suppose(G

,

^g)

⊂

(G^C

,

^gC)is a real slice of Lie groups. Then(G

,

^g)can be Wick-rotated to a Riemannian Lie group(_G

˜ ,

_g

˜

₎if and only if there exists a Cartan involution ofg.

Proof. (

⇒

). The identity mapg

˜ − → ˜

¹ gis a Cartan involution ofg. Thus by

˜

Theorem 3.1the direction follows. Conversely suppose

θ

is a Cartan involution ofg, and writeg

=

k

⊕

p, for the Cartan decomposition. Then is not difficult to show that

˜

g

:=

k

⊕

ipis a Lie algebra and is a real form ofg^C. Moreover the complex metricg^C(

− , −

) restricts to an inner product ong

˜

by construction. Thus if we let_G

˜

be the unique connected Lie subgroup ofG^C(the real Lie group) with Lie algebrag,

˜

then the corollary follows. □

In view ofRemark 1.1with the signature changeg

↦→ −

g, if (G

,

g) can be Wick-rotated to a signature (

− , − , . . . , −

), then (G

, −

g) can be Wick-rotated to a Riemannian space, thus there would exist a Cartan involution ofgw.r.t.

−

g. We note in the Corollary that w.r.t. the existing Cartan involution, then the Wick-rotated Riemannian Lie group may be chosen to be a virtual real form. Moreover note that since a Wick-rotation is a local condition then on Lie algebra level we have proved:

Corollary 3.3. Let(g^C

,

^gC)be a holomorphic inner product space, whereg^Cis a complex Lie algebra. Letg

⊂

g^Cbe a real form which is a real slice. Assume there exists a compact real formu

⊂

g^Cwhich is also a real Lie subalgebra. Let

σ

^{be the} conjugation map ofg. Then there exists an automorphism

φ ∈

Aut(g^C)

∩

O(n

,

C)such that:

σ

⁽

φ

^(u))

⊂ φ

^(u).

Note in the corollary that if

τ

denotes the conjugation map of the compact real form

φ

^(u)

⊂

(g^C

,

^gC), then the map

θ

^C

:= στ

restricts to a Cartan involution

θ

^ofg.

Thus we have proved a general version ofE. Cartan’s result: ([1], 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: (g

,

g) as above, indeed following the notation of the proof, it is not obvious thatN

:= στ ∈

O(n

,

C)

∩

Aut(g^C).

One shall note that 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 such a case we have the following (again one should compare this to semi-simple compact real forms w.r.t.

− κ

^):