Wick-rotations of pseudo-Riemannian Lie groups

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{1}, \cite{2}, \cite{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.


Introduction
This paper is motivated by the natural example of a Wick-rotation of a pseudo-Riemannian semi-simple Lie group (G, −κ) ⊂ (G C , −κ) 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: Remark 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 map g → −g induces 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.

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 C , if there is a real immersion G → G C (of Lie groups) where G C is viewed as a real Lie group, such that g is embedded as a real form of g C (the Lie algebra of G 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 C ), we shall say that the real form is an embedded real form. An embedded real form which also satisfies: G C = G · G 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: g ֒→ g C . We can find a complex connected Lie group: G C with Lie algebra g C . Thus using the exponential maps of these groups, we can find a smooth map (of real Lie groups): G → G C with differential i, thus G is an immersive real form of G C .
We shall abuse notation and write G ⊂ G C for an immersive real form.
Example 1. Consider the complex orthogonal group: O(4, C), then the map: g → I 3,1 gI 3,1 , is a conjugation map (i.e the differential is a conjugation map), where (I 3,1 ) ii = +1 for 1 ≤ i ≤ 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, C). Consider the universal covering Wick-rotations of pseudo-Riemannian Lie groups 3 group G := SL 2 (R) of SL 2 (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 (C).
We note that for a compact Lie group G, we can always complexify it to a complex Lie group: G C , such that G ⊂ G 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 ⊂ G C ⊃G be two immersive real forms which are Wick-related in (G C , g C ) for g C a left-invariant holomorphic metric. Then we shall say that the pseudo-Riemannian Lie group (G, g) is Wick-rotated to (G,g).
We call (G, g) ⊂ (G C , g C ) a real slice of Lie groups, and shall write (p, q) for the signature of g. If there is another real slice (G,g) ⊂ (G C , g C ) of Lie groups, then we shall refer to the signature ofg as (p,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 leftinvariant 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: is Wick-rotatable to (G,g) and they are both connected.
Proof. The proofs given in ( [5], Lemma 7.1 and 7.2) also hold for pseudo-Riemannian left-invariant metrics, with o(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.
Suppose we have a real slice of Lie groups: (G, g) ⊂ (G C , g C ). Let O(p, q) denote the real Lie group consisting of linear maps: g A − → g which preserves: for all x, y ∈ g, then O(p, q) is a real form of O(n, C) (isometries of g C ), by noting the anti-holomorphic involution (real structure): where σ is the conjugation map of g in g C . 4 Helleland We recall from [2], that if (E, g C ) is a holomorphic inner product space, and V,Ṽ 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: V,Ṽ , W , is said to be a compatible triple. Note that Example 2 is an example of a compatible triple: V := sl 2 (R),Ṽ := su(2), W := su (2) .
We shall call the eigenspace decomposition of a Cartan involution: θ, for the Cartan decomposition. Thus for a compact real slice: W ⊂ E, then by definition we see that θ = 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: θ j (j = 1, 2) then g → θ j gθ j is a global Cartan involution of O(p, q). Thus if gθ 1 g −1 = θ 2 for some g ∈ O(p, q), then writing g = k 2 e x , where k 2 commutes with θ 2 and x ∈ o(p, q), we obtain θ 1 = e x θ 2 e −x , and therefore θ 1 , θ 2 are conjugate by an element g ∈ O(p, q) 0 .
Suppose now we have a Wick-rotation of two real Lie groups: (G, g) ⊂ (G C , g C ) ⊃ (G,g). Let θ ∈ O(p, q) be a Cartan involution of the metric g, and let W denote the corresponding unique compact real slice associated with θ. Then by [3] it is possible to find a real form V ⊂ g C (as vector spaces) and an isomorphism: 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 ).
The following result is important to note: Lemma 3.6). The triple of real forms: o(p, q), o(p,q), o(n) , embedded into o(n, C) under a standard Wick-rotation is a compatible triple of Lie algebras.

Let now
tensor product space of the real Lie algebras, and let O(n, C) act by isometries on the complexification: V C =Ṽ 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 (G,g) be two Wick-rotatable Lie groups. Then two tensors v ∈ V andṽ ∈Ṽ are said to be Wick-rotatable at p := 1, if they lie in the same One also note if v =ṽ ∈ V ∩Ṽ, then from above we may take the map: φ C , such that v and φ −1 · v are Wick-rotatable, and we have the compatibility conditions satisfied. More , and thus results from [3] are applicable.
Wick-rotations of pseudo-Riemannian Lie groups 5 3. An invariant of Wick-rotation of Lie groups 3.1. The main theorem. Let (G, g) ⊂ (G C , g C ) ⊃ (G,g) be a Wick-rotation of Lie groups. Consider V to be the real vector space of bilinear forms on g, then V is a real form of V C (the set of complex bilinear forms on g C ). In particular the real Lie bracket [−, −] ∈ V w.r.t g, and is a restriction of the complex Lie bracket of g C . Note that the isometry action of O(p, q) on g extends to an action ρ on V , given simply by: for x, y ∈ g and g ∈ O(p, q). The metric induces a symmetric non-degenerate bilinear This Cartan involution also balances the action and thus the inner Cartan involutions w.r.t b are those conjugate to ρ(θ) by the action of O(p, q) (see [3]). We recall that a minimal vector v ∈ V is a vector satisfying ||g · v|| ≥ ||v|| for all g ∈ O(p, q), the set of minimal vectors of the action is denoted by M(O(p, q), V ). Analogously for the Lie algebra:g, we have the corresponding action of O(p,q) onṼ , 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: in such way that we may lend our selves to the results of [3].
The converse is identical. The theorem is proved.

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 := −κ. Proof. The identity mapg 1 − →g is a Cartan involution of Lie algebras of the metric:g. Thus by Theorem 1 the corollary follows.
In view of Remark 1 with the signature change g → −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 (of Lie algebras) of −g. We can also easily supply the converse of the previous corollary, and thus obtain the equivalence result: Proof. (⇒) has been proved. Conversely suppose θ is a Cartan involution of Lie algebras of the metric g, and write g = t⊕p, for the Cartan decomposition. Then is is not difficult to show thatg := t ⊕ ip is a Lie algebra and is a real form of g C . Moreover the complex metric g C (−, −) restricts to an inner product ong by construction. Thus if we letG be the unique connected Lie subgroup of G C (the real Lie group) with Lie algebrag, 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 (G,g). Then (G 0 , g) and (G 0 ,g) are locally isometric Lie groups. In particular if moreover G andG are both simply connected then G andG are isometric Lie groups.
Proof. Choose a map g ∈ O(n, C) mapping g →g.

Wick-rotations of pseudo-Riemannian Lie groups 7
Now since h maps g to g and gh −1 fixes the complex Lie bracket. Then gh −1 ∈ O(n, C) is an automorphism of complex Lie algebras, and it maps g →g. Therefore since the metrics are left-invariant we can conclude that (G 0 , g) and (G 0 ,g) are locally isometric Lie groups as required.
In the case of a complex semi-simple Lie group: (G C , −κ), equipped with the Killing form, then any compact real form: u ⊂ g, gives rise to a real form: U ⊂ G 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 C are isomorphic Lie groups, and thus also isometric Lie groups. We may identify {e j } j with the standard basis of R 3 . Let g(−, −) be the standard Lorentzian pseudo-inner product on R 3 , i.e of signature (+, +, −). Thus (H 3 (R), −g) is a real slice (of Lie groups) of (H 3 (C), −g C ). Note that g(−, −) is not bi-invariant, since g([e 1 , e 2 ], e 3 ) = −1 = g(e 1 , [e 2 , e 3 ]) = 0. Define the linear map: θ ∈ End(h 3 (R)) by: Then it is easy to show that this is an involution of Lie algebras, and moreover θ is a Cartan involution of −g(−, −), thus by Corollary 2 it follows that H 3 (R) can be Wickrotated to a Riemannian Lie groupG. Note thatG is the real form of H 3 (C) consisting of matrices of the form:   1 ix iy 0 1 z 0 0 1   for x, y, z ∈ 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 (R) 2 ⊂ G C := SL 2 (C) 2 . Then we can equip G with a left-invariant metric g(−, −) of signature (3,3), by equipping one copy with −κ and the other copy with κ. The real forms up to isomorphism of sl 2 (C) 2 are: Letg be one of these real forms (except the last one), then we may Wick-rotate G to the corresponding real formsG of SL 2 (C) 2 of signature either: (3,3) or (1,5). In the case of Wick-rotating to (SU (2) 2 ,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 −g or +g respectively: Suppose the Cartan involution is ofg. Then if su(2) 2 = t⊕p, is the Cartan decomposition w.r.t θ, we have g 1 := t ⊕ ip ∼ = o (1,3). Indeed θ C is a Cartan involution of g 1 (i.e of 8 Helleland −κ), thus −κ has signature (3,3), hence it must be the case that g 1 ∼ = o (1,3). We recall that t ∼ = su (2), and that the Killing form of t is just: Now we note that t C ∼ = sl 2 (C) is simple and naturally t C ⊂ sl 2 (C) 2 simply because t ⊂ su(2) 2 is contained in a real form of sl 2 (C) 2 . Thus when restrictingg on t we must getg(X, Y ) = −λκ t (X, Y ) = −4λT r(XY ) for some λ > 0. Now if X := (x, y) ∈ t, then: Thus we conclude that y = 0, and therefore: This is impossible since then θ = 1⊕−1 which is not a Lie homomorphism. The argument for the signature case: (0, 6), is identical with the change:g → −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 (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).

Conjugacy of Cartan involutions of a left-invariant metric.
Given a pseudo-Riemannian Lie group (G, g), with two Cartan involutions θ j (j = 1, 2) of Lie algebras of the metric g, one may wonder if they are conjugate in Aut(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 (−κ). Theorem 2. Suppose (G, g) ⊂ (G C , g C ) is a pseudo-Riemannian Lie group. Assume there exist two Cartan involutions: θ 1 , θ 2 of the metric, which are homomorphisms of Lie algebras. Then θ 1 is conjugate to θ 2 in Aut(g) 0 ∩ O(p, q) 0 .
Proof. Write g = t 1 ⊕ p 1 = t 2 ⊕ p 2 for the Cartan decompositions w.r.t θ 1 and θ 2 respectively. Denote also: t j ⊕ ip j := u j (j = 1, 2) for the real forms of g C . There exist Wick-rotations of G to connected virtual real forms: U j ⊂ G C with Lie algebras u j which are Riemannian (by Corollary 3). If σ denotes the conjugation map w.r.t g, and τ j denotes the conjugation map of u j , then we have θ C j = στ j . Now since θ 1 is conjugate to θ 2 in O(p, q) 0 , as linear maps, say by φ then φ C sends u 1 → u 2 . Thus following the notation in the proof of Proposition 2 we may take g ∈ O(p, q) 0 . The Lie brackets of the real Lie algebras are restrictions of the same complex Lie bracket: [−, −] C , on g C . Denote [−, −] for the Lie bracket on g. Thus if we letṽ := g −1 · [−, −] C , thenṽ is a bilinear form on u 1 , and: by the compatibility conditions of the pair: (g, u 1 ). Here Wick-rotations of pseudo-Riemannian Lie groups 9 is the maximal compact subgroup associated with the fixed Cartan involution of O(p, q): g → θ 1 gθ 1 . Thus the element h from the proof of Proposition 2, leaves invariant g also. It follows that [σ, gh −1 ] = 0, i.e for some gh −1 ∈ Aut(g) 0 ∩ 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): r.t our basis. We conclude that there exist a unique Cartan involution of −g(−, −), namely θ.

3.4.
Wick-rotating to a compact semi-simple Riemannian Lie group. Let (G, g) ⊂ (G C , g C ) be a real slice, and g C be simple. If g is bi-invariant, then g is proportional to the Killing form. Assume w.l.o.g that g = −λκ(−, −) (λ > 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 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) ⊂ (G C , g 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 θ of the metric which is also a Cartan involution of g (i.e of −κ).
Proof. (⇒). If (G, g) is Wick-rotated to a Riemannian Lie group, then by Corollary 2, we can choose a Cartan involution (of Lie algebras) θ of the metric. Denote g = t ⊕ p for the Cartan decomposition. Then following the proof of Corollary 3, then we can find a Riemannian Lie groupG with Lie algebra:g := t ⊕ ip, which is Wick-rotated to G in (G C , g C ). By Proposition 2,g is compact, since we can Wick-rotate G to a Riemannian compact Lie group (by assumption). But since θ is a Cartan involution of g if and only ifg is compact, then the direction is proved. (⇐). Suppose θ is a Cartan involution (of Lie algebras) of g(−, −) which is also a Cartan involution of −κ. Thus if u := t ⊕ ip is the compact real form of g C associated with θ, then there exist a compact real form U ⊂ G C with Lie algebra u. The proposition follows.
Thus since we may lift a local Cartan involution: g θ − → g, to a global Cartan involution: G Θ − → G, then in view of the previous proposition, there is a Θ which is an isometry of (G, g), i.e Θ ∈ Isom(G). Observe also that if there exist a real slice of Lie groups of G C which is compact Riemannian, then the possible signatures (p, q) w.r.t g C is a subset of the possible signatures of −κ (of g C ).
Note in general that if (G, g) has a Cartan involution θ g (of Lie algebras) of the metric g, then (by the theory of semi-simple Lie algebras) there exist a Cartan involution of g (i.e of −κ) such that [θ g , θ] = 0.
If (G, g) is locally isometric to λκ for some λ ∈ R, then we can Wick-rotate to aG ⊂ G C with Lie algebra o(1, 3). However o(1, 3) on g C is not a real slice. We thus conclude that (G, g) is not locally isometric to (G, λκ) for any λ ∈ R.

3.5.
Wick-rotating a Lorentzian signature. If we assume our metric on our Lie group G is Lorentzian or of signature (+, −, · · · , −), 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 g can be written as a semi-direct sum g = s ⋉ h, where h is the radical of g and s ⊂ g is either trivial or a semi-simple subalgebra of g called the Levi-factor.
It is clear that g C = s C ⋉ h C , and ifg is another real form of g C , then writing a Levi-decomposition:g =s ⋉h, thenh is a real form of h C . To see thats is a real form of s C , we note that there exist a k ≥ 1 such that s C = [g C (k) , g C (k) ] ⊃ [g (k) ,g (k) ] =s.
In view of the existence of an involution of Lorentzian decomposition we can say the following: Proposition 4. Let (G, g) ⊂ (G C , g C ) be a real slice of Lie groups. Then the following statements hold: (1) Suppose g(−, −) has Lorentzian signature. If (G, g) can be Wick-rotated to a Riemannian Lie group (G,g) then s = 0 or h = 0. Moreover ifs is a Levi-factor ofg, thens ∼ = s. (2) Suppose g(−, −) has signature (+, −, · · · , −). If (G, g) can be Wick-rotated to a Riemannian Lie group, then either s = 0 or s ∼ = sl 2 (R).