In this section we develop an abstract perspective on wedge domains in spacetimes, phrased completely in group theoretic terms. As wedge domains are supposed to correspond to standard subspaces in Hilbert spaces, we orient our approach on how standard subspaces are parametrized.
Let \({\mathrm{Stand}}(\mathcal{H })\) denote the set of standard subspaces of the complex Hilbert space \(\mathcal{H }\). In Sect. 4 we shall see that every standard subspace \(\mathtt{V}\) determines a pair \((\Delta _\mathtt{V}, J_\mathtt{V})\) of modular objects and that \(\mathtt{V}\) can be recovered from this pair by \(\mathtt{V}= {\mathrm{Fix}}(J_\mathtt{V}\Delta _\mathtt{V}^{1/2})\). This observation can be used to obtain a representation theoretic parametrization of \({\mathrm{Stand}}(\mathcal{H })\): each standard subspace \(\mathtt{V}\) specifies a continuous homomorphism
$$\begin{aligned} U^\mathtt{V}:\mathbb{R }^\times \rightarrow {\mathrm{AU}}(\mathcal{H })\quad \text{ by } \quad U^\mathtt{V}(e^t) := \Delta _\mathtt{V}^{-it/2\pi }, \quad U^\mathtt{V}(-1) := J_\mathtt{V}. \end{aligned}$$
(2.1)
We thus obtain a bijection between \({\mathrm{Stand}}(\mathcal{H })\) and the set \({\mathrm{Hom}}_{\mathrm{gr}}(\mathbb{R }^\times , {\mathrm{AU}}(\mathcal{H }))\) of continuous morphisms of graded topological groups.
The space \({\mathrm{Stand}}(\mathcal{H })\) carries three important features:
-
an order structure, defined by set inclusion
-
a duality operation \(\mathtt{V}\mapsto \mathtt{V}' = \{ \xi \in \mathcal{H }:(\forall v \in \mathtt{V})\, {\mathrm{Im}}\langle \xi , v \rangle = 0\}\)
-
the action of \({\mathrm{AU}}(\mathcal{H })\) as a symmetry group.
The order structure is hard to express in terms of the modular groups (see [Ne19b] for some first steps in this direction), but the duality operation corresponds to inversion
$$\begin{aligned} U^{\mathtt{V}'}(r) = U^{\mathtt{V}}(r^{-1}) \quad \text{ for } \quad r \in \mathbb{R }^\times , \end{aligned}$$
(2.2)
and the action of \({\mathrm{AU}}(\mathcal{H })\) translates into
$$\begin{aligned} U^{g\mathtt{V}}(r) = g U^{\mathtt{V}}(r^{\varepsilon (g)}) g^{-1} \quad \text{ for } \quad g \in {\mathrm{AU}}(\mathcal{H }), r \in \mathbb{R }^\times , \end{aligned}$$
(2.3)
where \(\varepsilon (g) = 1\) if g is unitary and \(\varepsilon (g) = -1\) otherwise. So unitary operators \(g \in \mathrm{U{}}(\mathcal{H })\) simply act by conjugation, but antiunitary operators also involve inversion. In particular, \(J_\mathtt{V}\mathtt{V}= \mathtt{V}'\) corresponds to
$$\begin{aligned} U^{\mathtt{V}'}(r) = J_\mathtt{V}U^{\mathtt{V}}(r^{-1}) J_\mathtt{V}= U^{\mathtt{V}}(r^{-1}) \quad \text{ for } \quad r \in \mathbb{R }^\times .\end{aligned}$$
We now develop the corresponding structures by replacing \({\mathrm{AU}}(\mathcal{H })\) by a finite dimensional graded Lie group.
Group theoretical setting
The basic ingredient of our approach is a finite dimensional graded Lie group \((G,\varepsilon _G)\), i.e., G is a Lie group and \(\varepsilon _G :G \rightarrow \{\pm 1\}\) a continuous homomorphism. We write
$$\begin{aligned} G^{{\uparrow }}= \varepsilon _G^{-1}(1) \quad \text{ and } \quad G^{{\downarrow }}= \varepsilon _G^{-1}(-1),\end{aligned}$$
so that \(G^{{\uparrow }}\trianglelefteq G\) is a normal subgroup of index 2 and \(G^{{\downarrow }}= G \setminus G^{{\uparrow }}\). We also fix a pointed closed convex cone \(C \subseteq \mathfrak{g }\) satisfying
$$\begin{aligned} {\mathrm{Ad}}(g) C = \varepsilon _G(g) C \quad \text{ for } \quad g \in G. \end{aligned}$$
(2.4)
As we shall see in the following, for graded Lie groups, it is more natural to work with the twisted adjoint action
$$\begin{aligned} {\mathrm{Ad}}^\varepsilon :G \rightarrow {\mathrm{Aut}}(\mathfrak{g }), \qquad {\mathrm{Ad}}^\varepsilon (g) := \varepsilon _G(g) {\mathrm{Ad}}(g), \end{aligned}$$
(2.5)
so that (2.4) actually means that C is invariant under the twisted adjoint action. The cone C will play a role in specifying an order structure. It is related to positive spectrum conditions on the level of unitary representations. We also allow \(C=\{0\}\). For instance, the Lie algebra \(\mathfrak{g }= {\mathfrak{so }}_{1,d}(\mathbb{R })\) of the Lorentz group \(G = \mathrm{O{}}_{1,d}(\mathbb{R })\), the isometry group of de Sitter space time \({\mathrm{dS}}^d\), contains no non-trivial invariant cone.
The space \({\mathrm{Hom}}_{\mathrm{gr}}(\mathbb{R }^\times ,G)\) and abstract wedges
In this section we define the fundamental objects we will need in the forthcoming discussion. We write \({\mathrm{Hom}}_{\mathrm{gr}}(\mathbb{R }^\times ,G)\) for the space of continuous morphisms of graded Lie groups \(\mathbb{R }^\times \rightarrow G\), where \(\mathbb{R }^\times \) is endowed with its canonical grading by \(\varepsilon (r) := {\mathrm{sgn}}(r)\). On this space G acts by
$$\begin{aligned} (g.\gamma )(r) := g \gamma (r^{\varepsilon _G(g)}) g^{-1}, \end{aligned}$$
(2.6)
where the twist is motivated by formula (2.2). Elements of \(G^{{\uparrow }}\) simply act by conjugation.
Since we are dealing with Lie groups, we also have the following simpler description of the space \({\mathrm{Hom}}_{\mathrm{gr}}(\mathbb{R }^\times ,G)\) by the set
$$\begin{aligned} \mathcal{G }:= \{ (x,\sigma )\in \mathfrak{g }\times G^{{\downarrow }}:\sigma ^2 = e, {\mathrm{Ad}}(\sigma )x = x\}.\end{aligned}$$
Proposition 2.1
The map
$$\begin{aligned} \Psi :{\mathrm{Hom}}_{\mathrm{gr}}(\mathbb{R }^\times ,G) \rightarrow \mathcal{G }, \quad \gamma \mapsto (\gamma '(1), \gamma (-1)) \end{aligned}$$
(2.7)
is a bijection. It is equivariant with respect to the action of G on \(\mathcal{G }\) by
$$\begin{aligned} g.(x,\sigma ) := ({\mathrm{Ad}}^\varepsilon (g)x, g\sigma g^{-1}). \end{aligned}$$
(2.8)
Note that center \(Z(G^{{\uparrow }})\) of \(G^{{\uparrow }}\) acts trivially on the Lie algebra but it may act non-trivially on involutions in \(G^{{\downarrow }}\).
Remark 2.2
For every involution \(\sigma \in G^{{\downarrow }}\), the involutive automorphism \(\sigma _G(g) := \sigma g \sigma \) defines the structure of a symmetric Lie group \((G^{{\uparrow }},\sigma _G)\), and \(G \cong G^{{\uparrow }}\rtimes \{{\mathrm{id}}, \sigma \}\), so that we can translate between G as a graded Lie group and the pair \((G^{{\uparrow }},\sigma _G)\), without loosing information.
To indicate the analogy of elements of \(\mathcal{G }\) with the wedge domains in QFT, we shall often denote the elements of \(\mathcal{G }\) by \(W = (x,\sigma )\).
Definition 2.3
(a) We assign to \(W = (x,\sigma ) \in \mathcal{G }\) the one-parameter group
$$\begin{aligned} \lambda _W :\mathbb{R }\rightarrow G^{{\uparrow }}\quad \text{ by } \quad \lambda _W(t) := \exp (t x) \end{aligned}$$
(2.9)
Then we have the graded homomorphism
$$\begin{aligned} \gamma _W :\mathbb{R }^\times \rightarrow G, \quad \gamma _W(e^t) := \lambda _W(t), \qquad \gamma _W(-1) := \sigma .\end{aligned}$$
Note that \(\Psi (\gamma _W) = W\) in terms of (2.7).
Definition 2.4
(a) We call an element x of the finite dimensional real Lie algebra \(\mathfrak{g }\) an Euler element if \({\mathrm{ad}}x\) is diagonalizable with \({\mathrm{Spec}}({\mathrm{ad}}x) \subseteq \{-1,0,1\}\), so that the eigenspace decomposition with respect to \({\mathrm{ad}}x\) defines a 3-grading of \(\mathfrak{g }\):
$$\begin{aligned} \mathfrak{g }= \mathfrak{g }_1(x) \oplus \mathfrak{g }_0(x) \oplus \mathfrak{g }_{-1}(x), \quad \text{ where } \quad \mathfrak{g }_\nu (x) = \ker ({\mathrm{ad}}x - \nu {\mathrm{id}}_\mathfrak{g })\end{aligned}$$
(see [BN04] for more details on Euler elements in more general Lie algebras). Then \(\sigma _x(y_j) = (-1)^j y_j\) for \(y_j \in \mathfrak{g }_j(x)\) defines an involutive automorphism of \(\mathfrak{g }\).
For an Euler element we write \(\mathcal{O }_x = {\mathrm{Inn}}(\mathfrak{g })x \subseteq \mathfrak{g }\) for the orbit of x under the group \({\mathrm{Inn}}(\mathfrak{g }) = \langle e^{{\mathrm{ad}}\mathfrak{g }} \rangle \) of inner automorphisms.Footnote 4 We say that x is symmetric if \(-x \in \mathcal{O }_x\).
We write \(\mathcal{E }(\mathfrak{g })\) for the set of non-zero Euler elements in \(\mathfrak{g }\) and \(\mathcal{E }_{\mathrm{sym}}(\mathfrak{g }) \subseteq \mathcal{E }(\mathfrak{g })\) for the subset of symmetric Euler elements.
(b) An element \((x,\sigma ) \in \mathcal{G }\) is called an Euler couple or Euler wedge if
$$\begin{aligned} {\mathrm{Ad}}(\sigma )=e^{\pi i {\mathrm{ad}}x}.\end{aligned}$$
(2.10)
Then \(\sigma \) is called an Euler involution and \(\sigma =\sigma _x\) as introduced before. We write \(\mathcal{G }_E \subseteq \mathcal{G }\) for the subset of Euler couples and note that the relation \(e^{\pi i {\mathrm{ad}}x} = e^{-\pi i {\mathrm{ad}}x}\) implies that the subset \(\mathcal{G }_E\) is invariant under the G-action.
For an Euler element \(x \in \mathcal{E }(\mathfrak{g })\), the relation (2.10) only determines \(\sigma \) up to an element \(z \in G^{{\uparrow }}\cap \ker ({\mathrm{Ad}})\) for which \((\sigma z)^2 = e\), i.e., \(\sigma z \sigma = z^{-1}\). Note that, if \(G^{{\uparrow }}\) is connected, then \(G^{{\uparrow }}\cap \ker ({\mathrm{Ad}}) = Z(G^{{\uparrow }})\) is the center of \(G^{{\uparrow }}\). The couples \((x,\sigma )\) that we have seen in the physics literature are all Euler couples (cf. [NÓ17, Ex. 5.15]). This ensures many properties, such as the proper relation between spin and statistics, see for instance [GL95].
Definition 2.5
(a) (Duality operation) For \(W = (x,\sigma ) \in \mathcal{G }\), we define \(W' := (-x,\sigma )\). Under \(\Psi \), this operation corresponds to inverting the homomorphism \(\mathbb{R }^\times \rightarrow G\) pointwise. Note that \((W')' = W\) and \((gW)' = gW'\) for \(g \in G\) by (2.8).
(b) (Order structure on \(\mathcal{G }\)) We now define an order structure on \(\mathcal{G }\) that depends on the invariant cone C from (2.4). We associate to \(W = (x,\sigma ) \in \mathcal{G }\)
-
the Lie wedge
$$\begin{aligned} L_W := L(x,\sigma ) := C_+(W) \oplus \underbrace{(\mathfrak{g }^{\sigma }\cap \ker ({\mathrm{ad}}x))}_{\mathfrak{g }_W :=} \oplus C_-(W), \end{aligned}$$
where
$$\begin{aligned} C_\pm (W) = \pm C \cap \mathfrak{g }^{-\sigma } \cap \ker ({\mathrm{ad}}x \mp \mathbf{1 }) \quad \text{ and } \quad \mathfrak{g }^{\pm \sigma } := \{ y \in \mathfrak{g }:{\mathrm{Ad}}(\sigma )(y) = \pm y \}. \end{aligned}$$
-
\(\mathfrak{g }(W) := L_W - L_W\), the Lie algebra generated by \(L_W\).
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the semigroup associated to the triple \((C,x,\sigma )\):
$$\begin{aligned} \mathcal{S }_W := \exp (C_+(W)) G^{{\uparrow }}_{W} \exp (C_-(W)) = G^{{\uparrow }}_{W} \exp \big (C_+(W) + C_-(W)\big ),\end{aligned}$$
where
$$\begin{aligned} G^{{\uparrow }}_{W} = \{g \in G^{{\uparrow }}:g.W = W \} = \{ g \in G^{{\uparrow }}:\sigma _G(g) = g, {\mathrm{Ad}}(g)x = x\} \end{aligned}$$
is the stabilizer of \(W = (x,\sigma )\) in \(G^{{\uparrow }}\) (cf. [Ne19b, Thm. 3.4]).Footnote 5
-
the subgroups \(G^{{\uparrow }}(W) := \langle \exp \mathfrak{g }(W) \rangle G^{{\uparrow }}_{W}\) and \(G(W) := G^{{\uparrow }}(W) \{e,{\sigma }\}\) with Lie algebra \(\mathfrak{g }(W)\).
As the unit group of \(\mathcal{S }_W\) is given by \(\mathcal{S }_W \cap \mathcal{S }_W^{-1} = G^{{\uparrow }}_{W}\) ([Ne19b, Thm. III.4]), the semigroup \(\mathcal{S }_W\) defines a \(G^{{\uparrow }}\)-invariant partial order on the orbit \(G^{{\uparrow }}.W \subseteq \mathcal{G }\) by
$$\begin{aligned} g_1.W \le g_2.W \quad :\Longleftrightarrow \quad g_2^{-1}g_1 \in \mathcal{S }_W. \end{aligned}$$
(2.11)
In particular, \(g.W \le W\) is equivalent to \(g \in \mathcal{S }_W\).
We have the following relations among these objects:
Lemma 2.6
For every \(W=(x_W,\sigma _W) \in \mathcal{G }\), \(g \in G\), and \(t \in \mathbb{R }\), the following assertions hold:
-
(i)
\(\lambda _W(t) W = W, \lambda _W(t) W' = W'\) and \(\sigma _W.W = W'.\)
-
(ii)
\(\sigma _{W'} = \sigma _W\) and \(\lambda _{W'}(t) = \lambda _W(-t)\).
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(iii)
\(\sigma _W\) commutes with \(\lambda _W(\mathbb{R })\).
-
(iv)
\(L_{W'} = - L_W\) and \(\mathcal{S }_{W'} = \mathcal{S }_W^{-1}\).
-
(v)
\(C_\pm (g.W) = {\mathrm{Ad}}(g) C_{\pm \varepsilon _G(g)}(W)\), \(L_{g.W} = {\mathrm{Ad}}(g)L_W\), and \(\mathcal{S }_{g.W} = g \mathcal{S }_W g^{-1}\).
-
(vi)
For \(W_1, W_2 \in \mathcal{G }\), the relation \(W_1 \le W_2\) in \(\mathcal{G }\) implies \(g.W_1 \le g.W_2\).
Proof
(i) For \(W = (x,\sigma )\in \mathcal{G }\), the first two relations follow from the fact that \(\exp (\mathbb{R }x)\) commutes with x and \(\sigma \). The second follows from \(\sigma _W. W = \sigma .(x,\sigma ) = (-{\mathrm{Ad}}(\sigma ) x, \sigma ) = (-x, \sigma ) = W'.\)
(ii) is clear from the definition of \(W'\).
(iii) follows from (i).
(iv) follows from \(C_\pm (W') = - C_{\mp }(W)\).
(v) The assertion is clear for \(g \in G^{{\uparrow }}\). For \(g \in G^{{\downarrow }}\), we have \(g\sigma \in G^{{\uparrow }}\), so that
$$\begin{aligned} C_\pm (g.W)&= C_\pm (g\sigma .W') = {\mathrm{Ad}}(g\sigma )C_{\pm }(W') = -{\mathrm{Ad}}(g\sigma )C_{\mp }(W) = {\mathrm{Ad}}(g)C_{\mp }(W)\\&= {\mathrm{Ad}}(g)C_{\pm \varepsilon _G(g)}(W). \end{aligned}$$
This implies in particular that \(L_{g.W} = {\mathrm{Ad}}(g)L_W\). From \(G^{{\uparrow }}_{g.W} = g G^{{\uparrow }}_{W} g^{-1}\), we thus obtain \(\mathcal{S }_{g.W} = g \mathcal{S }_W g^{-1}\).
(vi) If \(W_1 \le W_2\), then \(W_1 = s.W_2\) for \(s \in \mathcal{S }_{W_2}\). Then \(g.W_1 = gs.W_2 = gsg^{-1}.(g.W_2)\) with \(gsg^{-1} \in g \mathcal{S }_{W_2} g^{-1} = \mathcal{S }_{g.W_2}\) implies \(g.W_1 \le g.W_2\). \(\square \)
In this discussion we started with a Lie group. We remark that one can also start with a Lie algebra as follows: Consider a quadruple \((\mathfrak{g }, \sigma _\mathfrak{g }, h, C)\) of a Lie algebra \(\mathfrak{g }\), an involutive automorphism \(\sigma _\mathfrak{g }\) of \(\mathfrak{g }\), fixing the Euler element h and a pointed closed convex invariant cone \(C \subseteq \mathfrak{g }\) with \(\sigma _\mathfrak{g }(C) = - C\). Then \(\sigma _\mathfrak{g }\) integrates to an automorphism \(\sigma _G\) of the 1-connected Lie group \(G^{{\uparrow }}\) with Lie algebra \(\mathfrak{g }\), so that we obtain all the data required above with \(G := G^{{\uparrow }}\rtimes \{{\mathrm{id}}_G,\sigma _G\}\).
For two such quadruples \((\mathfrak{g }_j, \tau _{\mathfrak{g },j}, h_j, C_j)_{j =1,2}\), a homomorphism \(\varphi :\mathfrak{g }_1 \rightarrow \mathfrak{g }_2\) of Lie algebras is compatible with this structure if
$$\begin{aligned} \varphi \circ \tau _{\mathfrak{g },1} = \tau _{\mathfrak{g },2} \circ \varphi ,\qquad \varphi (h_1) =h_2 \quad \text{ and } \quad \varphi (C_1) \subseteq C_2.\end{aligned}$$
We thus obtain a category whose objects are the quadruples \((\mathfrak{g }, \tau _\mathfrak{g }, h, C)\) and its morphisms are the compatible homomorphisms.
A similar category can be defined on the group level, but there are some subtle ambiguities concerning the possible extensions of the group structure from \(G^{{\uparrow }}\) to G.
Remark 2.7
(Twisted extensions of \(G^{{\uparrow }}\) to G) We start with a graded group G for which \(G^{{\downarrow }}\) contains an involution \(\sigma \), so that \(G \cong G^{{{\uparrow }}} \rtimes \{e,\sigma \}\), where \(\sigma \) acts on \(G^{{\uparrow }}\) by the automorphism \(\sigma _G(g) := \sigma g \sigma \). This defines a split group extension
$$\begin{aligned} G^{{\uparrow }}\rightarrow G \rightarrow \mathbb{Z }_2 \end{aligned}$$
and we are now asking for other group extensions
$$\begin{aligned} G^{{\uparrow }}\rightarrow {\widehat{G}} \rightarrow \mathbb{Z }_2 \end{aligned}$$
for which the elements in \({\widehat{G}}^{{\downarrow }}\) define the same element in the group \({\mathrm{Out}}(G^{{\uparrow }}) = {\mathrm{Aut}}(G^{{\uparrow }})/{\mathrm{Inn}}(G^{{\uparrow }})\) of outer automorphisms of \(G^{{\uparrow }}\). These extensions are parametrized by the group
$$\begin{aligned} Z(G^{{\uparrow }})^+ := \{ z \in Z(G^{{\uparrow }}) :\sigma _G(z) = z\},\end{aligned}$$
by assigning to \(z \in Z(G^{{\uparrow }})^+\) the group structure on \(G^{{\uparrow }}\times \{1,-1\}\) given by
$$\begin{aligned} (g,1)(g',\varepsilon ') = (gg',\varepsilon '), \quad (e,-1)(g',1) = (\sigma _G(g'),-1) \quad \text{ and } \quad (e,-1)^2 = (z,1).\nonumber \\ \end{aligned}$$
(2.12)
We write \({\widehat{G}}_z\) for the corresponding Lie group. Basically, this means that the element \({\widehat{\sigma }} := (e,-1)\) has the same commutation relations with \(G^{{\uparrow }}\) but its square is z instead of e:
$$\begin{aligned} {\widehat{\sigma }} g {\widehat{\sigma }}^{-1} = \sigma _G(g) \quad \text{ for } \quad g\in G \quad \text{ and } \quad {\widehat{\sigma }}^2 = z. \end{aligned}$$
(2.13)
For two elements \(z,z' \in Z(G^{{\uparrow }})^+\), the corresponding extensions are equivalent if and only if
$$\begin{aligned} z^{-1}z' \in B := \{ w \sigma _G(w) :w \in Z(G^{{\uparrow }})\}. \end{aligned}$$
(2.14)
This follows from [HN12, Thm 18.1.13], combined with [HN12, Ex. 18.3.5(b)].
(a) For \(G = \mathrm{O{}}_{n}(\mathbb{R })\), \(n > 3\), and \(G^{{\uparrow }}= {\mathrm{SO}}_{n}(\mathbb{R })\), the situation depends on the parity of n. If n is odd, then \(Z(G^{{\uparrow }}) = \{e\}\) and no twists exist. If n is even, then \(Z(G^{{\uparrow }}) = \{\pm \mathbf{1 }\} = Z(G)\). Therefore \(Z(G^{{\uparrow }})^+ = \{ \pm \mathbf{1 }\} \not = B = \{e\}\). We therefore have one twisted group \({\widehat{G}} = {\mathrm{SO}}_n(\mathbb{R }) \{e,{\widehat{\sigma \}}}\), where \(\sigma \in \mathrm{O{}}_n(\mathbb{R })\) corresponds to a hyperplane reflection, and \({\widehat{\sigma }}^2 = - \mathbf{1 }\) in \({\widehat{G}}\).
(b) The same phenomenon occurs for Spin groups. Let \(G := {\mathrm{Pin}}_n(\mathbb{R }) \cong {\mathrm{Spin}}_n(\mathbb{R }) \rtimes \{e,\sigma \}\), where \(\sigma \) corresponds to a hyperplane reflection. If n is odd, then \(Z({\mathrm{Spin}}_n(\mathbb{R })) = \{ e,z\}\) contains two elements, and we have a twisted group
$$\begin{aligned} {\widehat{G}} = {\mathrm{Spin}}_n(\mathbb{R }) \{ e, {\widehat{\sigma }} \} \quad \text{ with } \quad {\widehat{\sigma }}^2 = z \end{aligned}$$
(cf. [HN12, Rem. B.3.25]). If n is even, then the situation is more complicated because the center of \({\mathrm{Spin}}_n(\mathbb{R })\) has order 4.
(c) For \(G = {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}} \rtimes \{e,\sigma \}\), where \(\sigma \) corresponds to a reflection \(\sigma (x) = -x\) on \(\mathbb{R }^\infty \cong \mathbb{S }^1\), we have \(Z(G^{{\uparrow }}) \cong \mathbb{Z }\) and \(\sigma _G(z) = z^{-1}\) for \(z \in Z(G^{{\uparrow }})\). Hence \(Z(G^{{\uparrow }})^+ = \{e\}\), so that there are no twists.
(d) If \(G = {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}^{(2n)} \rtimes \{e,\sigma \}\), where \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}^{(2n)}\) is the covering of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) of even order, then \(Z(G^{{\uparrow }}) \cong \mathbb{Z }_{2n}\) and \(\sigma _G(z) = z^{-1}\) for \(z \in Z(G^{{\uparrow }})\). Therefore \(Z(G^{{\uparrow }})^+ = \{ e, \gamma \}\), where \(\gamma \) is the unique non-trivial involution in \(Z(G^{{\uparrow }})\) and \(B = \{e\}\). Hence there exists a non-trivial twist \({\widehat{G}} = G^{{\uparrow }}\{ e,{\widehat{\sigma \}}}\) with \({\widehat{\sigma }}^2 = \gamma \).
(e) As we shall see in Example 2.13 below, it may happen that, for the twisted groups \({\widehat{G}}_z\), the coset \({\widehat{G}}_z^{{\downarrow }}\) contains no involutions. In this example \(G^{{\uparrow }}= {\mathrm{SL}}_2(\mathbb{R })\) and \(G = G^{{\uparrow }}\{e,\gamma \}\) with \(\gamma ^2 = - \mathbf{1 }\).
In general, elements in \({\widehat{G}}_z^{{\downarrow }}\) are of the form \(g{\widehat{\sigma }}\) with \(g \in G^{{\uparrow }}\), and then
$$\begin{aligned} (g{\widehat{\sigma }})^2 = g {\widehat{\sigma }} g {\widehat{\sigma }} = g \sigma _G(g) {\widehat{\sigma }}^2 = g \sigma _G(g) z. \end{aligned}$$
(2.15)
Hence \({\widehat{G}}_z^{{\downarrow }}\) contains an involution if and only if
$$\begin{aligned} z \in \{ \sigma _G(g)^{-1} g^{-1} :g \in G^{{\uparrow }}\} = \{ g \sigma _G(g) :g \in G^{{\uparrow }}\}.\end{aligned}$$
If \(z = g \sigma _G(g)\) for some \(g \in G^{{\uparrow }}\), then conjugating with g implies that g and \(\sigma _G(g)\) commute.
The discussion in Example 2.13 shows that (2.15) is not satisfied for \(z = -\mathbf{1 }\) and the Euler involution of \(G^{{\uparrow }}= {\mathrm{SL}}_2(\mathbb{R })\). For any odd degree covering \({\mathrm{SL}}_2(\mathbb{R })^{(2k+1)} \rightarrow {\mathrm{SL}}_2(\mathbb{R })\), the central involution is mapped onto \(-\mathbf{1 }\), so that this observation carries over to odd coverings of \({\mathrm{SL}}_2(\mathbb{R })\).
The situation changes if we consider \(G^{{\uparrow }}= {\mathrm{SL}}_2(\mathbb{C })\) instead. Then \(g := i \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix}\) satisfies \(g^2 = -\mathbf{1 }\), so that the group \({\widehat{G}} = G^{{\uparrow }}\{ \mathbf{1 },{\widehat{\sigma \}}}\) with \({\widehat{\sigma }}^2 = - \mathbf{1 }\) contains the non-trivial involution \(g{\widehat{\sigma }} \in {\widehat{G}}^{{\downarrow }}\). As this involution is central, \({\widehat{G}} \cong {\widehat{G}}^{{\uparrow }}\times \mathbb{Z }_2\) is a direct product.
The abstract wedge space, some fundamental examples
Definition 2.8
(The abstract wedge space). From here on, we always assume that \(\mathcal{G }\not =\emptyset \), i.e., that \(G^{{\downarrow }}\) contains an involution \(\sigma \). Then
$$\begin{aligned} G \cong G^{{\uparrow }}\rtimes \{{\mathrm{id}},\sigma \} \end{aligned}$$
(cf. Remark 2.2). For a fixed couple \(W_0 = (h,\sigma ) \in \mathcal{G }\), the orbits
$$\begin{aligned} \mathcal{W }_+ (W_0):= G^{{\uparrow }}.W_0 \subseteq \mathcal{G }\quad \text{ and } \quad \mathcal{W }(W_0) := G.W_0 \subseteq \mathcal{G }\end{aligned}$$
are called the positive and the full wedge space containing \(W_0\).
Remark 2.9
(a) As \(\sigma .W_0 = (-h,\sigma ) = W_0'\), we have \(\mathcal{W }(W_0) = \mathcal{W }_+(W_0) \cup \mathcal{W }_+(W_0')\), and \(\mathcal{W }(W_0)\) coincides with \(\mathcal{W }_+(W_0)\) if and only if \(W_0' = (-h,\sigma ) \in \mathcal{W }_+(W_0)\). This is equivalent to the existence of an element \(g \in G^{{\uparrow }}\) with \(g.W_0 = W_0'\), i.e., \(g \in (G^{{\uparrow }})^\sigma \) with \({\mathrm{Ad}}(g)h = - h\).
(b) If \(W_0\) is an Euler couple, then \(\mathcal{W }(W_0)\) is a family of Euler couples, and we shall see below that in this case we have \(\mathcal{W }(W_0) = \mathcal{W }_+(W_0)\) in many important cases.
We collect some fundamental examples, starting from the low dimensional cases, that we shall refer to throughout the paper.
Examples 2.10
(a) The smallest example is the abelian group \(G = \mathbb{R }\times \{\pm 1 \}\), where \(G^{{\uparrow }}= \mathbb{R }\), \(C = \{0\}\) and \(L = \mathfrak{g }\). For \(W_0 = (h,\sigma )\) with \(h = 1\) and \(\sigma = (0,1)\), we then have the one-point set \(\mathcal{W }_+ = \{ (h,\sigma )\}\), and \(\mathcal{W }= \{ (h,\sigma ), (-h,\sigma )\}\).
(b) The affine group \(G :={\mathrm{Aff}}(\mathbb{R })\cong \mathbb{R }\rtimes \mathbb{R }^\times \) of the real line is two-dimensional. Its elements are denoted (b, a), and they act by \((b,a)x = ax + b\) on the real line. The identity component \(G^{{\uparrow }}= \mathbb{R }\rtimes \mathbb{R }^\times _+\) acts by orientation preserving maps, and \(G^{{\downarrow }}\) consists of reflections \(r_p(x) = 2p-x\), \(p \in \mathbb{R }\).
Let \(\zeta (t) = (t,1)\) and \(\delta (t) = (0,e^t)\) be the translation and dilation one-parameter groups, respectively. We write \(\lambda = (0,1) \in \mathfrak{g }= \mathbb{R }\rtimes \mathbb{R }\) for the infinitesimal generator of \(\delta \), which is an Euler element. Therefore \(W := (\lambda , r_0)\) is an Euler couple.
The cone \(C = \mathbb{R }_+ \times \{0\} \subseteq \mathfrak{g }\) satisfies the invariance condition (2.4) and the corresponding semigroup \(\mathcal{S }_W\) is
$$\begin{aligned} \mathcal{S }_W = [0,\infty ) \rtimes \mathbb{R }^\times _+ = \{ g = (b,a) :g.0 = b \ge 0, {a>0}\} = \{ g \in G^{{\uparrow }}:g\mathbb{R }_+ \subseteq \mathbb{R }_+\}. \end{aligned}$$
Therefore the map
$$\begin{aligned} \mathcal{W }_+(W) \ni {g.(\lambda , r_0) }\mapsto g{(0,+\infty )} \end{aligned}$$
defines an order preserving bijection between the abstract wedge space \(\mathcal{W }_+(W) \subseteq \mathcal{G }\) and the set \(\mathcal{I }_+(\mathbb{R }) = \{ (t,\infty ) :t \in \mathbb{R }\}\) of lower bounded open intervals in \(\mathbb{R }\). Accordingly, we may write \(W_{(t,\infty )}=(\Lambda _{(t,\infty )},r_t) := \zeta (t) W = ({\mathrm{Ad}}(\zeta (t))\lambda , r_t)\) for \(t \in \mathbb{R }\). Acting with reflections, we also obtain the couples
$$\begin{aligned} W_{(-\infty ,t)}:=(\Lambda _{(-\infty ,t)},r_t) = r_t.W_{(t,\infty )} = (-{\mathrm{Ad}}(\zeta (t))\lambda , r_t)\end{aligned}$$
corresponding to past pointing half-lines \((-\infty ,t)\subset \mathbb{R }\). We thus obtain a bijection between the full wedge space \(\mathcal{W }(W)\) and the set \(\mathcal{I }(\mathbb{R })\) of open semibounded intervals in \(\mathbb{R }\). We shall denote with \(\delta _I\) the one-parameter group of dilations with generator \(\lambda _I\) corresponding to the half line I.
The set \(\mathcal{E }(\mathfrak{g }) = {\mathrm{Ad}}(G^{{\uparrow }})\{\pm \lambda \}\) of non-zero Euler elements in \(\mathfrak{g }\) consists of two \(G^{{\uparrow }}\)-orbits and, for each non-zero Euler element \(\pm {\mathrm{Ad}}(\zeta (t))\lambda \in \mathcal{E }(\mathfrak{g })\), the reflection \(r_t\) is the unique partner for which \((\pm {\mathrm{Ad}}(\zeta (t))\lambda , r_t) \in \mathcal{G }\). Accordingly, Euler couples in \(\mathcal{G }\) are in one-to-one correspondence with semi-infinite open intervals in \(\mathbb{R }\).
(c) The Möbius group \(G :={\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2 := {\mathrm{PGL}}_2(\mathbb{R })\cong {\mathrm{GL}}_2(\mathbb{R })/\mathbb{R }^\times \) acts on the compactification \(\overline{\mathbb{R }}= \mathbb{R }\cup \{\infty \}\) of the real line by
$$\begin{aligned} g.x := \frac{a x + b}{cx + d} \quad \text{ on } \quad \overline{\mathbb{R }}:= \mathbb{R }\cup \{\infty \}, \qquad \text{ for } \quad g = \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix}\in {\mathrm{GL}}_2(\mathbb{R }).\end{aligned}$$
We write \(G^{{\uparrow }}= {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\cong {\mathrm{PSL}}_2(\mathbb{R })\) for the subgroup of orientation preserving maps. The Cayley transform
$$\begin{aligned} C:{\overline{\mathbb {R}}} \rightarrow \mathbb{S }^1 := \{z\in \mathbb {C}: |z|=1\}, \quad C(x) := \frac{i-x}{i+x}, \qquad C(\infty ) := -1, \end{aligned}$$
is a homeomorphism, identifying \(\overline{\mathbb{R }}\) with the circle. Its inverse is the stereographic map
$$\begin{aligned} C^{-1}:\mathbb{S }^1 \rightarrow \overline{\mathbb{R }}, \quad C^{-1}(z) = i\frac{1-z}{1+z}.\end{aligned}$$
It maps the upper semicircle \(\{ z \in \mathbb{S }^1 :{\mathrm{Im}}z > 0\}\) to the positive half line \((0,+\infty )\). The Cayley transform intertwines the action of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) on \(\overline{\mathbb{R }}\) with the action of \({\mathrm {PSU}}_{1,1}(\mathbb{C }) ={\mathrm{SU}}_{1,1}(\mathbb{C })/\{\pm \mathbf{1 }\}\), given by
$$\begin{aligned} \begin{pmatrix} \alpha &{} \beta \\ \overline{\beta }&{} \overline{\alpha } \end{pmatrix}.z := \frac{\alpha z + \beta }{\overline{\beta }z + \overline{\alpha }} \quad \text{ for } \quad z \in \mathbb{S }^1, \begin{pmatrix} \alpha &{} \beta \\ \overline{\beta }&{} \overline{\alpha } \end{pmatrix} \in {\mathrm{SU}}_{1,1}(\mathbb{C }).\end{aligned}$$
The three-dimensional Lie group \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) is generated by the following one-parameter subgroups:
-
Rotations: \(\rho (\theta )(x) = \frac{\cos (\theta /2)x + \sin (\theta /2)}{-\sin (\theta /2)x + \cos (\theta /2)}\) for \(\theta \in \mathbb{R }\); note that \(C(\rho (\theta )x) = e^{i\theta } C(x)\).
-
Dilations: \(\delta (t)(x)= e^t x\) for \(t \in \mathbb{R }\).
-
Translation: \(\zeta (t) x = x+t\) for \(t \in \mathbb{R }\).
In the circle picture \(\delta \) and \(\zeta \) will be denoted by \(\delta _{\cap }\) and \(\zeta _\cap \), referring to the upper semicircle with endpoints \(\{-1,1\} = C(\{0,\infty \})\). Note that \(-1\) is the unique fixed point of \(\zeta _\cap \) and one of the two fixed points \(\{\pm 1\}\) of \(\delta _\cap \). On the circle, \(\rho (\pi )\) maps 1 to \(-1\) and exchanges the upper and the lower semicircle. Accordingly, \(\zeta _\cup =\rho (\pi )\zeta \rho (\pi ) \) is the subgroup of conjugated translations fixing the point \(1\in \mathbb{S }^1\).
We write \(\mathbf{K } = \rho (\mathbb{R })\), \(\mathbf{A } = \delta (\mathbb{R })\), \(\mathbf{N }^+ = \zeta (\mathbb{R })\) and \(\mathbf{N }^- = \zeta _\cup (\mathbb{R })\) for the corresponding one-dimensional subgroups of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\), and \(\mathbf{P }^+ = \mathbf{A }\mathbf{N }^+ = {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_\infty \), \(\mathbf{P }^- := \mathbf{A }\mathbf{N }^- = {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_0\) for the stabilizer groups of \(\infty \) and 0 in \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\). We observe that \(\overline{\mathbb{R }}\cong {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}/\mathbf{P }^-\) and that the circle group \(K= {\mathrm{PSO}}_2(\mathbb{R })\) acts simply transitively on \(\overline{\mathbb{R }}\).
On the compactified line, the point reflection \(\tau (x) = -x\) in 0 acts on the Lie algebra by
$$\begin{aligned} {\mathrm{Ad}}(\tau )\begin{pmatrix} a &{} \quad b \\ c &{} \quad -a \end{pmatrix} = \begin{pmatrix} -1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix} \begin{pmatrix} a &{} \quad b \\ c &{} \quad -a \end{pmatrix} \begin{pmatrix} -1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix} = \begin{pmatrix} a &{} \quad -b \\ -c &{} \quad -a \end{pmatrix}. \end{aligned}$$
(2.16)
Note that \(\tau \in G^{{\downarrow }}\).
The infinitesimal generator \(h := \begin{pmatrix} \frac{1}{2} &{} 0 \\ 0 &{} -\frac{1}{2} \end{pmatrix}\) of \(\delta \) is an Euler element and \(W := (h,\tau )\) is an Euler couple. Since \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2 \cong {\mathrm{PGL}}_2(\mathbb{R }) \cong {\mathrm{Aut}}({\mathfrak{sl }}_2(\mathbb{R }))\), for any Euler couple \((x,\tau )\), the involution \(\tau \) is determined by the requirement that it acts on \(\mathfrak{g }={\mathfrak{sl }}_2(\mathbb{R })\) by \(e^{\pi i {\mathrm{ad}}x}\). We conclude that the action of \(G^{{\uparrow }}= {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) on the set of Euler couples is transitive, i.e., \(\mathcal{G }_E = G^{{\uparrow }}.(h,\tau )\).
To see the geometric side of Euler couples, let us call a non-dense, non-empty open connected subset \(I \subseteq \mathbb{S }^1\) an interval and write \(\mathcal{I }(\mathbb{S }^1)\) for the set of intervals in \(\mathbb{S }^1\). It is easy to see that \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) acts transitively on \(\mathcal{I }(\mathbb{S }^1)\). To determine the stabilizer of an interval, we consider the upper half circle, which corresponds to the half line \((0,\infty ) \subseteq \overline{\mathbb{R }}\). Each element \(g \in {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) mapping \((0,\infty )\) onto itself fixes 0 and \(\infty \). Since it is completely determined by the image of a third point, it is of the form \(\delta (t)\) if \(g.1 = e^t\). Therefore the stabilizer of \((0,\infty )\) in \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) is the subgroup \(\delta (\mathbb{R })\), which coincides with the stabilizer of h under the adjoint action. This already shows that \(\mathcal{W }_+(W)\) and \(\mathcal{I }(\mathbb{S }^1)\) are isomorphic homogeneous spaces of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\). In particular, we can associate to an interval \(I = g(0,\infty )\) the reflection \(\tau _I=g \tau g^{-1}\) and the one-parameter group \(\delta _I := g \delta g^{-1}\). Note that \(\tau _I\) is an orientation reversing involution mapping I to the complementary open interval \(I'\). We write \(x_I := {\mathrm{Ad}}(g)h\) for the infinitesimal generator of \(\delta _I\), so that the assignment \(I \mapsto x_I\) defines an equivariant bijection \(\mathcal{I }(\mathbb{S }^1) \rightarrow \mathcal{E }(\mathfrak{g })\). The anticlockwise orientation of \(\mathbb{S }^1\), which can also be considered as a causal structure, is used here to pick the sign of \(x_I\) in such a way that the flow \(\delta _I\) is counter clockwise (future pointing) on I. Accordingly, \(x_{I'} = -x_I\) corresponds to the complementary interval \(I'\).
To identify the natural order on the abstract wedge space \(\mathcal{G }_E = \mathcal{W }_+(W)\), we consider for \(X = \begin{pmatrix} a &{} b \\ c &{} -a \end{pmatrix} \in \mathfrak{g }= {\mathfrak{sl }}_2(\mathbb{R })\) the corresponding fundamental vector field
$$\begin{aligned} {V_X}(x) = \frac{d}{dt}\Big |_{t = 0} \exp (tX).x = (a - d)x + b - c x^2 = b + 2 ax - c x^2.\end{aligned}$$
This shows that
$$\begin{aligned} C := \{ X \in \mathfrak{g }:{V_X} \ge 0\} = \Big \{ X= \begin{pmatrix} a &{} \quad b \\ c &{} \quad -a \end{pmatrix} :b \ge 0, c \le 0, a^2 \le -bc\Big \} \end{aligned}$$
(2.17)
is a pointed generating invariant cone in \(\mathfrak{g }\). The Lie wedge specified by the triple \((h,\tau , C)\) is
$$\begin{aligned} L_W = L(h,\tau ,C) = \underbrace{\mathbb{R }_+ \begin{pmatrix} 0 &{} \quad 1 \\ 0 &{} \quad 0 \end{pmatrix}}_{C_+} \oplus \mathbb{R }h \oplus \underbrace{\mathbb{R }_+ \begin{pmatrix} 0 &{} \quad 0 \\ 1 &{} \quad 0 \end{pmatrix}}_{C_-} = \Big \{ \begin{pmatrix} a &{} \quad b \\ c &{} \quad -a \end{pmatrix} :a \in \mathbb{R }, b\ge 0, c\ge 0\Big \}.\end{aligned}$$
We further have \(G(W) = G^{{\uparrow }}\), and the associated semigroup is
$$\begin{aligned} \mathcal{S }_W = \exp (C_+) \exp (\mathbb{R }h) \exp (C_-) = \{ g \in G^{{\uparrow }}:g(0,\infty ) \subseteq (0,\infty ) \}.\end{aligned}$$
Therefore the map
$$\begin{aligned} \mathcal{G }_E = \mathcal{W }_+(W) = \mathcal{W }(W) \rightarrow \mathcal{I }(\mathbb{S }^1), \quad g.W \mapsto g (0,\infty ) \end{aligned}$$
(2.18)
defines an order preserving bijection between the abstract wedge space \(\mathcal{W }(W)\) and the ordered set \(\mathcal{I }(\mathbb{S }^1)\).
(d) We now consider the universal covering of the Möbius group \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\). Concretely, we put \(G := {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}} \rtimes \{\mathbf{1 },{\widetilde{\tau }}\}\), where \({\widetilde{\tau }}\) acts on \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) by integrating \({\mathrm{Ad}}(\tau )\) from (2.16) to an automorphism of \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\). The group G is a graded Lie group and \(G^{{\uparrow }}:= {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) is its identity component. We have a covering homomorphism \(q_G: G \rightarrow {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2\) whose kernel \(Z({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}) \cong \mathbb{Z }\) is discrete cyclic. We write \({\widetilde{\rho }}\), \({\widetilde{\delta }}\), \({\widetilde{\zeta }}\) and \({\widetilde{\zeta }}_\cup \) for the canonical lifts of the one-parameter groups \(\rho \), \(\delta \), \(\zeta \), \(\zeta _\cup \) of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\), \({\widetilde{\mathbf{P }}}^+ := {\widetilde{\delta }}(\mathbb{R }) {\widetilde{\zeta }}(\mathbb{R })\), and \({\widetilde{\mathbf{P }}}^- := {\widetilde{\delta }}(\mathbb{R }) {\widetilde{\zeta }}_\cup (\mathbb{R })\).
The action of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) on \(\mathbb{S }^1\) lifts canonically to an action of the connected group \(G^{{\uparrow }}= {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) on the universal covering \(\widetilde{\mathbb{S }^1} \cong \mathbb{R }\), where we fix the covering map \(q_{\mathbb{S }^1} :\mathbb{R }\rightarrow \widetilde{\overline{\mathbb{R }}}\), defined by \(q_{\mathbb{S }^1}(\theta ) = {\widetilde{\rho }}(\theta ).0\), which corresponds to the map \(\theta \mapsto e^{i\theta } = C({\widetilde{\rho }}(\theta ).0)\) in the circle picture. We may thus identify \(\widetilde{\mathbb{S }^1}\) with the homogeneous space \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}/{\widetilde{\mathbf{P }}}^-\cong \mathbb{R }\). As conjugation with \({\widetilde{\tau }}\) on \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) preserves the subgroup \({\widetilde{\mathbf{P }}}^-\), it also acts on \(\widetilde{\mathbb{S }^1}\). From (2.16) it follows that it simply acts by the point reflection \({\widetilde{\tau }}.x = - x\) in the base point 0. We also note that \(Z := \ker (q_G) = {\widetilde{\rho }}(2\pi \mathbb{Z })\) is the group of deck transformations of the covering \(q_{\mathbb{S }^1}\), which acts by
$$\begin{aligned} {\widetilde{\rho }}(2\pi n).x = x + 2 \pi n \quad \text{ for } \quad n \in \mathbb{Z }. \end{aligned}$$
(2.19)
We call a non-empty interval \(I \subseteq \mathbb{R }\) admissible if its length is strictly smaller than \(2\pi \) and write \(\mathcal{I }(\mathbb{R })\) for the set of admissible intervals. An interval \(I \subseteq \mathbb{R }\) is admissible if and only if there exists an interval \(\underline{I} \in \mathcal{I }(\mathbb{S }^1)\) such that I is a connected component of \(q_{\mathbb{S }^1}^{-1}(\underline{I})\). The group Z acts transitively on the set of these connected components. As \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) acts transitively on \(\mathcal{I }(\mathbb{S }^1)\), it follows that the group \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) acts transitively on the set \(\mathcal{I }(\mathbb{R })\), and that composition with \(q_{\mathbb{S }^1}\) yields an equivariant covering map
$$\begin{aligned} \mathcal{I }(\mathbb{R }) \cong {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}/{\widetilde{\delta }}(\mathbb{R }) \rightarrow \mathcal{I }(\mathbb{S }^1) \cong {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}/\delta (\mathbb{R }), \quad I \mapsto q_{\mathbb{S }^1}(I). \end{aligned}$$
(2.20)
We further have:
-
The group \({\widetilde{\mathbf{P }}}^+ = {\widetilde{\delta }}(\mathbb{R }) {\widetilde{\zeta }}(\mathbb{R })\) fixes the points \(\{(2k+1)\pi :k\in \mathbb{Z }\}\).
-
For \(I \in \mathcal{I }(\mathbb{S }^1)\), let \({\widetilde{\delta }}_I\) be the lift of the one-parameter group \(\delta _I\). Then \({\widetilde{\delta }}_I\) preserves every interval in the preimage \(q_{\mathbb{S }^1}^{-1}(I)\).
-
The inverse images of \(\tau \in {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2\) in \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_2\) are the elements \({\widetilde{\tau }}_n := {\widetilde{\rho }}(2\pi n){\widetilde{\tau }}\), \(n \in \mathbb{Z }\). These are involutions, acting by
$$\begin{aligned} {\widetilde{\tau }}_n(x)= 2\pi n -x \quad \text{ for } \quad x\in \mathbb{R }\end{aligned}$$
(2.21)
which is a point reflection in the point \(\pi n\). All pairs \((h, {\widetilde{\tau }}_n)\) are Euler couples in \(\mathcal{G }({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_2)\), and from the discussion of the set of Euler couples \(\mathcal{G }_E({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2)\) under (c), we know that the involutions \({\widetilde{\tau }}_n\) exhaust all possibilities for supplementing h to an Euler couple.
There is an interesting difference to the situation for \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2\), where \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) acts transitively on the set \(\mathcal{G }_E({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2)\) of Euler couples. To see what happens for \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_2\), recall that the stabilizer of the element \((h,\tau )\in \mathcal{G }_E({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}_2)\) in \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) is the subgroup \(\delta (\mathbb{R })\). Its inverse image is the group
$$\begin{aligned} {\widetilde{\delta }}(\mathbb{R }) {\widetilde{\rho }}(2\pi \mathbb{Z }) \cong \mathbb{R }\times \mathbb{Z }.\end{aligned}$$
An element \(g \in {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) fixes \((h,{\widetilde{\tau }}_n)\) if and only if \({\mathrm{Ad}}(g)h = h\) and \(g {\widetilde{\tau }}_n g^{-1} = {\widetilde{\tau }}_n\). The first condition is equivalent to g being of the form
$$\begin{aligned} g = {\widetilde{\delta }}(t) {\widetilde{\rho }}(2\pi k) \quad \text{ for } \text{ some } \quad t\in \mathbb{R }, k \in \mathbb{Z }.\end{aligned}$$
The second condition is equivalent to \({\widetilde{\tau }} g {\widetilde{\tau }} = {\widetilde{\tau }}_n g {\widetilde{\tau }}_n = g\), which takes the form
$$\begin{aligned} {\widetilde{\delta }}(t) {\widetilde{\rho }}(-2\pi k) = {\widetilde{\delta }}(t) {\widetilde{\rho }}(2\pi k),\end{aligned}$$
and this is equivalent to \(k = 0\). We conclude that the stabilizer of \((h,{\widetilde{\tau }}_n)\) is
$$\begin{aligned} {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_{(h,{\widetilde{\tau }}_n)} = {\widetilde{\delta }}(\mathbb{R }). \end{aligned}$$
(2.22)
We also note that
$$\begin{aligned} {\widetilde{\rho }}(\pi k).(h,{\widetilde{\tau }}_n) = ( (-1)^k h, {\widetilde{\rho }}(\pi k){\widetilde{\tau }}_n{\widetilde{\rho }}(-\pi k)) = ((-1)^k h, {\widetilde{\rho }}(2\pi k){\widetilde{\tau }}_n) = ((-1)^k h, {\widetilde{\tau }}_{n+k}).\end{aligned}$$
We conclude that the group \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) does not act transitively on the set \(\mathcal{G }_E\) of Euler couples. It has two orbits:
$$\begin{aligned} \mathcal{G }_E({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_2) = G^{{\uparrow }}.W_0 {\dot{\cup }} G^{{\uparrow }}.{W_1} = \mathcal{W }_{+}(W_0) {\dot{\cup }} \mathcal{W }_{+}({W_1}) \quad \text{ for } \quad W_0 := (h,{\widetilde{\tau }}_0), W_1 := (h,{\widetilde{\tau }}_1).\nonumber \\ \end{aligned}$$
(2.23)
We also refer to Example 2.14 for a discussion of this issue from a different perspective.
-
The subgroup \({\widetilde{\delta }}(\mathbb{R })\) preserves every interval which is a non-trivial orbit of \({\widetilde{\delta }}(\mathbb{R })\), acting on \(\mathbb{R }\). If, conversely, \(g \in {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}\) preserves such an interval, then its image in \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) is contained in \(\delta (\mathbb{R })\), so that
$$\begin{aligned} g = {\widetilde{\delta }}(t) {\widetilde{\rho }}(2\pi k)\quad \text{ for } \text{ some } \quad t\in \mathbb{R }, k \in \mathbb{Z }.\end{aligned}$$
As every open orbit of \({\widetilde{\delta }}(\mathbb{R })\) is an interval of length \(\pi \), the element g can only preserve such an orbit if \(k = 0\). This shows that \({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_{(h,{\widetilde{\tau }}_n)}\) also is the stabilizer group of any open \({\widetilde{\delta }}(\mathbb{R })\)-orbit in \(\mathbb{R }\). We conclude that, for the Euler couple \(W_0 = (h, {\widetilde{\tau }}_0)\), the map
$$\begin{aligned} {\Phi } :\mathcal{W }_+(W_0) \rightarrow \mathcal{I }(\mathbb{R }), \quad g.(h,{\widetilde{\tau }}_0) \mapsto g (0,\pi ) \end{aligned}$$
(2.24)
defines a \(G^{{\uparrow }}\)-equivariant bijection between the abstract wedge space \(\mathcal{W }_+(W_0) \subseteq \mathcal{G }\) and the set \(\mathcal{I }(\mathbb{R })\) of admissible intervals in \(\mathbb{R }\). Since the full group G acts on the space \(\mathcal{I }(\mathbb{R })\) of intervals, \(\Phi \) can be used to transport this action to a G-action on the space \(\mathcal{W }_+(W_0)\), extending the action of the subgroup \(G^{{\uparrow }}\). Since \(\tau _0(0,\pi )=(-\pi ,0)=\rho (-\pi )(0,\pi ),\) we have
$$\begin{aligned} \Phi ^{-1}(\tau _0(0,\pi ))=\Phi ^{-1}(\rho (-\pi )(0,\pi )) =\rho (-\pi ).\Phi ((0,\pi ))^{-1}=(-h,\rho (-2\pi )\tau _0), \end{aligned}$$
so that \(\tau _0.W_0:=(-h,\rho (-2\pi )\tau _0)\). By \(G^{{\uparrow }}\)-equivariance of the map \(\Phi \), we conclude that the action of \(G^{{\downarrow }}\) on \(\mathcal{W }_+(W_0)\) is given by
$$\begin{aligned} g *_{\rho (-2\pi )} (x,\sigma ) := ({\mathrm{Ad}}^\varepsilon (g), \rho (-2\pi ) g \sigma g^{-1}) \quad \text{ for } \text{ every } \quad g \in G^{{\downarrow }}.\end{aligned}$$
(2.25)
Here we use that \({\widetilde{\rho }}(-2\pi ) \in Z(G^{{\uparrow }})\). Note that we have chosen \((0,\pi )\) to be the image of \(W_0\) through \(\Phi \). Further possible actions come from the identifications
$$\begin{aligned} {\Phi _n} :\mathcal{W }_+(W_n) \rightarrow \mathcal{I }(\mathbb{R }), \quad g.(h,{\widetilde{\tau }}_n) \mapsto g (0,\pi ) \quad \text{ with } \quad W_n=(h,\tau _n), \end{aligned}$$
(2.26)
and one can likewise see that
$$\begin{aligned} g *_{\alpha _n} (x,\sigma ) := ({\mathrm{Ad}}^\varepsilon (g), \alpha _n g \sigma g^{-1}) \quad \text{ for } \quad g \in G^{{\downarrow }}\quad \text{ and } \quad \alpha _n = {\widetilde{\rho }}((2n-1)2\pi ) \in Z(G^{{\uparrow }}), \end{aligned}$$
extends the action of \(G^{{\uparrow }}\) on \(\mathcal{W }_+(W_n)\) to G and \(\Phi =\Phi _0\) for \(n=0\) (see also (2.37) and Section 2.4.2 for this kind of action).
(e) Let \(q :{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}^{(n)} \rightarrow {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) be the n-fold covering group of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\) and \(\rho ^{(n)}, \delta ^{(n)}, \zeta ^{(n)}\) and \(\zeta _{\cup }^{(n)}\) be the lifts of the corresponding one-parameter groups of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\). We further put \(\mathbf{P }^{-,(n)} := \delta ^{(n)}(\mathbb{R }) \zeta _{\cup }^{(n)}(\mathbb{R })\), so that we obtain an n-fold covering
$$\begin{aligned} q_n :\mathbb{S }^1_n := {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}^{(n)}/\mathbf{P }^{-,(n)} \rightarrow \mathbb{S }^1= {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}/\mathbf{P }^{-}, \quad g \mathbf{P }^{-,(n)} \mapsto q(g) \mathbf{P }^{-} \end{aligned}$$
of the circle, and the action of the one-parameter group \(\rho ^{(n)}\) induces a diffeomorphism
$$\begin{aligned} \mathbb{R }/2\pi n \mathbb{Z }\rightarrow \mathbb{S }^1_n, \quad [t] \mapsto \rho ^{(n)}(t).0 \end{aligned}$$
The set of wedges can be described analogously to the case (d), but there is a difference depending on the parity of n. If n is even, the group \(G^{{\uparrow }}\) has two orbits in the set \(\mathcal{G }_E\) of Euler couples, but if n is odd, there is only one. Indeed, for \(n = 2k\), the element \(\rho ^{(n)}(2\pi k)\) acts as an involution on \(\mathbb{S }^1_n\). So it fixes all Euler couples \((h,{\widetilde{\tau }}_n)\), even if it does NOT fix any proper interval in \(\mathbb{S }^1_n\) (see also Example 2.14).
(f) The example arising most prominently in physics is the proper Poincaré group
$$\begin{aligned} G := \mathcal{P }_+ := \mathbb{R }^{1,d} \rtimes {\mathrm{SO}}_{1,d}(\mathbb{R }), \qquad G^{{\uparrow }}:= \mathcal{P }_+^{{\uparrow }}:= \mathbb{R }^{1,d} \rtimes {\mathrm{SO}}_{1,d}(\mathbb{R })^{{\uparrow }}.\end{aligned}$$
It acts on \(1+d\)-dimensional Minkowski space \(\mathbb{R }^{1,d}\) as an isometry group of the Lorentzian metric given by \((x,y)= x_0y_0-{\mathbf{x}}{\mathbf{y}}\) for \(x = (x_0, {\mathbf{x}}) \in \mathbb{R }^{1,d}\). Writing
$$\begin{aligned} V_+ := \{ (x_0, {\mathbf{x}}) \in \mathbb{R }^{1,d} :x_0> 0, x_0^2 > {\mathbf{x}}^2\}\end{aligned}$$
for the open future light cone, the grading on G is specified by time reversal, i.e., \(gV_+ = \varepsilon (x,g) V_+\). In particular \(C := \overline{V_+}\) is a pointed closed convex cone satisfying (2.4). For \(d > 1\), this is, up to sign, the only non-zero pointed invariant cone in the Lie algebra \(\mathfrak{g }\).
The generator \(k_{1} \in {\mathfrak{so }}_{1,d}(\mathbb{R })\) of the Lorentz boost on the \((x_0,x_1)\)-plane
$$\begin{aligned} k_1(x_0,x_1,x_2, \ldots , x_{d}) = (x_1, x_0, x_2, \ldots , x_{d})\end{aligned}$$
is an Euler element. It combines with the spacetime reflection \(j_1(x) =(-x_0,-x_1,x_2,\ldots ,x_d)\) to the Euler couple \((k_1, j_1)\). We associate to \((k_1, j_1)\) the spacetime region
$$\begin{aligned} W_1=\{x\in \mathbb {R}^{1+d}: |x_0|<x_1\}, \end{aligned}$$
the standard right wedge, and note that \(W_1\) is invariant under \(\exp (\mathbb{R }k_{1})\). It turns out that the semigroup \(\mathcal{S }_{( k_1,j_1)}\) associated to the couple \(( k_1,j_1)\) in Definition 2.5 satisfies
$$\begin{aligned} \mathcal{S }_{( k_1, j_1)} = \{ g \in G :gW_1 \subseteq W_1\} =: \mathcal{S }_{W_1} \end{aligned}$$
(2.27)
(see [NÓ17, Lemma 4.12]). From (2.27) it follows that the map
$$\begin{aligned} \mathcal{W }_+ = \mathcal{W }= G^{{\uparrow }}.(k_1, j_1) \ni g.( k_1, j_1) \mapsto gW_1 \end{aligned}$$
(2.28)
defines an order preserving bijection between the abstract wedge space \(\mathcal{W }\subseteq \mathcal{G }\) and the set of wedge domains in Minkowski space \(\mathbb{R }^{1+d}\). For an abstract wedge \(W = ( k_W, j_W) \in \mathcal{W }\), the Euler element \(k_W\) is the corresponding boost generator. For an axial wedge \(W_i:=\{x\in \mathbb {R}^{1+d}:|x_0|<x_i\}\), \(i =1,\ldots , n\), the corresponding Euler couple will be denoted \((k_i,j_i)\).
Nets of wedges, isotony, central locality and covering groups
In the following sections we will focus on the description of relative positions of wedges, in particular wedge inclusions and the locality principle.
Wedge inclusion
Firstly consider this wedge inclusion configuration called half-sided modular inclusion:
Definition 2.11
Let \(W_0 = (x,\sigma ) \in \mathcal{G }\) and \(y \in \pm C\) with \([x,y] = \pm y\). Then \(\exp (y) \in \mathcal{S }_{W_0}\) (Definition 2.5(b)), so that
$$\begin{aligned} W_1 := \exp (y).W_0 \le W_0.\end{aligned}$$
We then call \(W_1 \le W_0\) a ±half-sided modular inclusion.
The next lemma shows that any wedge inclusion can be described in terms of positive and negative half-sided modular inclusions.
Lemma 2.12
If \(W_1 \le W_3\) in \(\mathcal{G }\), then there exists an element \(W_2 \in \mathcal{G }\) with \(W_1 \le W_2 \le W_3\) for which the inclusion \(W_1 \le W_2\) is \(+\)half-sided modular and the inclusion \(W_2 \le W_3\) is −half-sided modular.
Proof
That \(W_1 \le W_3\) means that \(W_1 = s W_3\) for some
$$\begin{aligned} s \in \mathcal{S }_{W_3} = \exp (C_-(W_3)) \exp (C_+(W_3)) G^{{\uparrow }}_{W_3}. \end{aligned}$$
Accordingly, we write \(s = g_- g_+ g_0\) and observe that \(W_1 = g_- g_+ W_3\) because \(g_0 W_3 = W_3\). Put \(W_2 := g_- W_3\). Then \(W_2 \le W_3\) and \(g_+ W_3 \le W_3\) implies \(W_1 = g_- g_+ W_3 \le g_- W_3 = W_2\).
Further, the inclusion \(W_2 \le W_3\) is −half-sided modular because \(g_- \in \exp (C_-(W_3))\). Likewise the inclusion \(g_+ W_3 \le W_3\) is \(+\)half-sided modular, and therefore \(W_1 \le W_2\) is also \(+\)half-sided modular. \(\square \)
Central locality
For a wedge \(W = (x,\sigma )\), the dual wedge \(W' = (-x,\sigma )\) need not be contained in the orbit \(\mathcal{W }_+ = G^{{\uparrow }}.W\). If, however, \(G^{{\uparrow }}\) has a non-trivial central subgroup Z such that, modulo Z, the complement \(W'\) is contained in \(\mathcal{W }_+\), then we use central elements \(\alpha \in Z\) to define “twisted complements” \(W^{'\alpha }\) which are contained in \(\mathcal{W }_+\), and this in turn leads to a twisted action of the full group G on \(\mathcal{W }_+\). We also obtain on \(\mathcal{W }_+\) a complementation map \(W \mapsto W^{'\alpha }\).
Let \(Z \subseteq Z(G^{{\uparrow }})\) be a closed normal subgroup of G, and \(q :G \rightarrow {\underline{G}}:= G/Z\) be the corresponding surjective morphism of graded Lie groups with kernel Z. If Z is discrete, then q is a covering map. The morphism of graded Lie groups q induces a natural map
$$\begin{aligned} q_\mathcal{G }:\mathcal{G }( G) \rightarrow \underline{\mathcal{G }}:= \{ (x,\underline{\sigma })\in \mathfrak{g }\times {\underline{G}}^{{\downarrow }}:\underline{\sigma }^2 = e, {\mathrm{Ad}}_\mathfrak{g }(\underline{\sigma })x = x\}, \quad (x,\sigma ) \mapsto (x, q(\sigma )),\nonumber \\ \end{aligned}$$
(2.29)
where \({\mathrm{Ad}}_\mathfrak{g }:{\underline{G}}\rightarrow {\mathrm{Aut}}(\mathfrak{g })\) denotes the factorized adjoint action which exists because \(Z = \ker (q)\) acts trivially on \(\mathfrak{g }\). It restricts to a map
$$\begin{aligned} \mathcal{G }_E(G) \rightarrow \underline{\mathcal{G }}_E := \{ (x,\underline{\sigma })\in \mathcal{E }(\mathfrak{g }) \times {\underline{G}}^{{\downarrow }}:\underline{\sigma }^2 = e, {\mathrm{Ad}}_\mathfrak{g }(\underline{\sigma }) = e^{\pi i {\mathrm{ad}}x}\}. \end{aligned}$$
(2.30)
As the following example shows, neither of these maps is always surjective. The main obstruction is that, although the differential \(\mathbf{L{}}(q):\mathbf{L{}}(G) \rightarrow \mathbf{L{}}({\underline{G}})\) is surjective, there may be involutions \(\tau \in {\underline{G}}^{{\downarrow }}\) for which no involution \(\sigma \in G^{{\downarrow }}\) with \(q(\sigma ) = \tau \) exists. This phenomenon is tightly related to the twisted groups \({\widehat{G}}_z\) discussed in Remark 2.7 because these twists disappear for \(z \in Z\) in \({\widehat{G}}/Z \cong G/Z\).
Example 2.13
We consider the graded Lie group
$$\begin{aligned} G := {\mathrm{SL}}_2(\mathbb{R }) \{ \mathbf{1 },\gamma \} \subseteq {\mathrm{SL}}_2(\mathbb{C }), \quad \text{ where } \quad \gamma := \begin{pmatrix} i &{} \quad 0 \\ 0 &{} \quad -i \end{pmatrix} \quad \text{ satisfies } \quad \gamma ^2 = - \mathbf{1 }. \end{aligned}$$
It has two connected component and \(G^{{\uparrow }}= {\mathrm{SL}}_2(\mathbb{R })\).Footnote 6 The subgroup \(Z := \{\pm \mathbf{1 }\}\) is central and the quotient map \(q :G \rightarrow {\underline{G}}:= G/Z\) is a 2-fold covering. The Euler element \(x := \frac{1}{2}\begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad -1 \end{pmatrix} \in \mathfrak{g }= {\mathfrak{sl }}_2(\mathbb{R })\) combines with the involution \(q(\gamma ) \in {\underline{G}}^{{\downarrow }}\) to the Euler couple \((x,q(\gamma )) \in \underline{\mathcal{G }}\). However, the set \(\mathcal{G }(G)\) is empty because \(G^{{\downarrow }}\) contains no involution. In fact, for \(g = \begin{pmatrix} a &{} \quad b \\ c &{} \quad d \end{pmatrix} \in {\mathrm{SL}}_2(\mathbb{R })\), the condition that \(g\gamma \) is an involution is equivalent to
$$\begin{aligned} \begin{pmatrix} -a &{} \quad b \\ c &{} \quad -d \end{pmatrix} = \gamma g \gamma = g^{-1} = \begin{pmatrix} d &{} \quad -b \\ -c &{} \quad a \end{pmatrix}. \end{aligned}$$
This is equivalent to \(a = -d\) and \(b = c = 0\), contradicting that \(1 = \det (g) = -a^2\). We conclude in particular that the maps \(\mathcal{G }(G) \rightarrow \underline{\mathcal{G }}\) and \(\mathcal{G }_E(G) \rightarrow \mathcal{G }_E({\underline{G}})\) are not surjective.
We now discuss \(G^{{\uparrow }}\)-orbits in \(\mathcal{G }(G)\). In the examples we have in mind, the central subgroup Z is discrete.
Involution lifts and central wedge orbit. Each element \(\sigma \in G^{{\downarrow }}\) acts in the same way on the abelian normal subgroup Z by the involution
$$\begin{aligned} \sigma _ Z :Z \rightarrow Z, \quad \gamma \mapsto \gamma ^\sigma := \sigma \gamma \sigma \end{aligned}$$
which restricts to an involution \(\sigma _ Z \in {\mathrm{Aut}}( Z)\) because Z is central in \(G^{{\uparrow }}\) and a normal subgroup of G. In the following we shall need the subgroups
$$\begin{aligned} Z^- := \{ \gamma \in Z:\gamma ^\sigma = \gamma ^{-1}\} \supseteq Z_1 := \{\gamma ^\sigma \gamma ^{-1} :\gamma \in Z\}. \end{aligned}$$
(2.31)
For \(\gamma \in Z^-\), the element \(\gamma ^2 = (\gamma ^\sigma \gamma ^{-1})^{-1}\) is contained in \( Z_1\), so that the quotient group \( Z^-/ Z_1\) is an elementary abelian 2-group, i.e., isomorphic to \(\mathbb{Z }_2^{(B)}\) for some index set B.
For an involution \(\sigma \in G^{{\downarrow }}\) and \(\beta \in Z(G^{{\uparrow }})\), the element \(\beta \sigma \in G^{{\downarrow }}\) is an involution if and only if \(\beta \in Z^-\). Therefore
$$\begin{aligned} \alpha *(x,\sigma ):= (x,\alpha \sigma ) \end{aligned}$$
(2.32)
defines an action of \( Z^-\) on \(\mathcal{G }(G)\), commuting with the conjugation action of \(G^{{\uparrow }}\) and satisfying
$$\begin{aligned} g.(\alpha * (x,\sigma )) = \alpha ^{-1} * (g.(x,\sigma )) \quad \text{ for } \quad g \in G^{{\downarrow }}, \alpha \in Z^-. \end{aligned}$$
(2.33)
For \(W = (x,\sigma ) \in \mathcal{G }( G)\), the fiber over \({\underline{W}}:= (x,q(\sigma ))\) is thus given by
$$\begin{aligned} Z^-* W:= \{ (x,\alpha \sigma ) :\alpha \in Z^- \}. \end{aligned}$$
(2.34)
The subgroup \( Z\subseteq G^{{\uparrow }}\) acts by conjugation on the fiber \(Z^-*W\):
$$\begin{aligned} \gamma .(x,\sigma ) = (x, \gamma \sigma \gamma ^{-1}) = (x, \gamma (\gamma ^\sigma )^{-1} \sigma ),\end{aligned}$$
so that the quotient group \( Z^-/ Z_1\) parametrizes the Z-conjugation orbits in the fiber \( Z^-*W\).Footnote 7 Here is an example.
Example 2.14
(a) If \( Z \cong \mathbb{Z }\) and \(n^\sigma = -n\), then \( Z^- = \mathbb{Z }\) and \( Z_1 = 2 \mathbb{Z }\), so that \( Z^-/ Z_1 \cong \mathbb{Z }/2\mathbb{Z }\).
(b) If \( Z = \mathbb{Z }_n\) and \(\overline{n}^\sigma = -\overline{n}\), then \( Z^- = \mathbb{Z }_n\) and \( Z_1 = 2 \mathbb{Z }_n\), so that
$$\begin{aligned} Z^-/ Z_1\cong {\left\{ \begin{array}{ll} \mathbb{Z }/2\mathbb{Z }&{}\text { if } n \ \text { is } \text { even} \\ \{0\} &{}\text { if } n \ \text { is } \text { odd.} \end{array}\right. } \end{aligned}$$
Wedge \(G^{{\uparrow }}\)-orbits. Let \(W = (x,\sigma ) \in \mathcal{G }_E(G)\) and \(\underline{W} = (x,q(\sigma )) \in \underline{\mathcal{G }}\). In general the group \( G^{{\uparrow }}\) does not act transitively on the inverse image of the orbit \({\underline{\mathcal{W }}}_+ := {\underline{G}}^{{\uparrow }}.{\underline{W}}\subseteq \underline{\mathcal{G }}\) under \(q_\mathcal{G }\). We now describe how this set decomposes into orbits. By the transitivity of the \({\underline{G}}^{{\uparrow }}\)-action on \({\underline{\mathcal{W }}}_{+}\), it suffices to consider the orbits of the stabilizer
$$\begin{aligned} G^{{\uparrow }}_{{\underline{W}}}=\{g\in G^{{\uparrow }}: q(g).{\underline{W}}={\underline{W}}\} \end{aligned}$$
on the fiber \(Z^- * W\). That \(g \in G^{{\uparrow }}\) fixes \({\underline{W}}\) implies in particular that \(g\sigma g^{-1}\sigma = g (g^\sigma )^{-1} \in Z\). This leads to a homomorphism
$$\begin{aligned} \partial :G^{{\uparrow }}_{\underline{W}}\rightarrow Z^-, \quad g \mapsto g (g^\sigma )^{-1} \quad \text{ with } \quad g.(x,\sigma ) = ({\mathrm{Ad}}(g)x, g\sigma g^{-1}) = (x, \partial (g)\sigma ) .\nonumber \\ \end{aligned}$$
(2.35)
As \( Z \subseteq G^{{\uparrow }}_{{\underline{W}}}\), the image \( Z_2 := \partial ( G^{{\uparrow }}_{{\underline{W}}})\) is a subgroup containing \( Z_1\).
Example 2.15
(An example where \(Z_1 \not = Z_2\).) We consider the group \(G = {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}} \rtimes \{\mathbf{1 },{\widetilde{\tau }}\}\) from Example 2.10(d) and the canonical homomorphism
$$\begin{aligned} q :G \rightarrow \underline{G} := {\mathrm{SL}}_2(\mathbb{R }) \rtimes \{\mathbf{1 },\sigma \}, \quad \sigma := \begin{pmatrix} -1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\end{aligned}$$
whose kernel is the central subgroup \(Z := 2 Z(G^{{\uparrow }}) \subseteq Z(G^{{\uparrow }}) \cong \mathbb{Z }\) of index two. Now \(W = (h,{\widetilde{\tau }}) \in \mathcal{G }(G)\) is an Euler couple mapped to \(\underline{W} = (h,\sigma ) \in \underline{\mathcal{G }}\). As \(z^{{\widetilde{\tau }}} = z^{-1}\) for every \(z \in Z\), we have \(Z = Z^-\) and \(Z_1 = 2Z\) is a subgroup of index 2. To calculate \(Z_2\), we observe that
$$\begin{aligned} {\underline{G}}^{{\uparrow }}_{\underline{W}} = {\underline{G}}_h = \exp (\mathbb{R }h) \{ \pm \mathbf{1 }\} \quad \text{ and } \quad G^{{\uparrow }}_{\underline{W}} = \exp (\mathbb{R }h) Z(G^{{\uparrow }}).\end{aligned}$$
We conclude that
$$\begin{aligned} Z_2 = \partial \big (G^{{\uparrow }}_{\overline{W}}\big ) = \partial (Z(G^{{\uparrow }})) = 2 Z(G^{{\uparrow }}) = Z^- \not = Z_1.\end{aligned}$$
The situation changes if we consider \(Z=Z(G^{{\uparrow }})\) and the center-free group \({\underline{G}}= {\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\rtimes \{\mathbf{1 },\tau \}\) instead. Then \(Z = Z^- = Z(G^{{\uparrow }})\) and \(Z_1 = Z_2 = 2 Z\).
As the \(G^{{\uparrow }}\) orbits in \(q_\mathcal{G }^{-1}(\underline{G}^{{\uparrow }}.\underline{W}) = q_\mathcal{G }^{-1}(\underline{\mathcal{W }}_+)\) correspond to the \(G^{{\uparrow }}_{\underline{W}}\)-orbits in the fiber \(q_\mathcal{G }^{-1}(\underline{W}) = Z^- * W\), we obtain the following lemma.
Lemma 2.16
The quotient group \(Z^-/Z_2\) parametrizes the set of \(G^{{\uparrow }}\)-orbits in \(q_\mathcal{G }^{-1}({\underline{\mathcal{W }}}_{+})\).
\(\alpha \)-twisted complement. The following definition generalizes the notion of complementary wedge given in Definition 2.5 (a).
Definition 2.17
For \(\alpha \in { Z^-}\), we define the \(\alpha \)-twisted complement of \(W = (x,\sigma ) \in \mathcal{G }( G)\) by
$$\begin{aligned} (x,\sigma )^{'\alpha } := {(-x,\alpha \sigma )}.\end{aligned}$$
We will refer to couples of the form \(W^{'\alpha }\) as complementary wedges. We consider \(W^{'\alpha }\) as a “complement” of W because \(q_\mathcal{G }\) maps \(W^{'\alpha }\) to \(W'\) (see item (a) below).
Lemma 2.18
For each \(\alpha \in Z^-\), the \(\alpha \)-twisted complementation \(W \mapsto W^{'\alpha }\) satisfies:
-
(a)
For \(\alpha \in Z^-\), \(W^{'\alpha }\) is mapped by \(q_\mathcal{G }\) onto the complement \(W' = (-x, q(\sigma ))\) of \({\underline{W}}= (x,q(\sigma ))\).
-
(b)
The \(\alpha \)-twisted complementation is not involutive if \(\alpha ^2 \not =e\).
-
(c)
The map \({}^{'\alpha } :\mathcal{G }( G) \rightarrow \mathcal{G }( G), (x,\sigma ) \mapsto (-x, \alpha \sigma )\) is \( G^{{\uparrow }}\)-equivariant.
-
(d)
In terms of the action (2.32) of \(Z^-\) on \(\mathcal{G }(G)\), we have
$$\begin{aligned} W^{'\alpha } = \alpha * W' \quad \text{ for } \quad W \in \mathcal{G }(G), \alpha \in Z^-. \end{aligned}$$
(2.36)
-
(e)
The prescription
$$\begin{aligned} g *_\alpha (x,\sigma ) :={\left\{ \begin{array}{ll} g.(x,\sigma ) &{} \text { for } g \in G^{{\uparrow }}\\ g.(\alpha ^{-1} * (x,\sigma )) = \alpha * (g.(x,\sigma )) &{} \text { for } g \in G^{{\downarrow }}. \end{array}\right. } \end{aligned}$$
(2.37)
defines an action of G on \(\mathcal{G }(G)\). This action satisfies
$$\begin{aligned} W^{'\alpha } = \sigma *_\alpha W \quad \text{ for } \quad W = (x,\sigma ) \in \mathcal{G }(G), \alpha \in Z^-. \end{aligned}$$
(2.38)
If \(W^{'\alpha }\in G^{{\uparrow }}.W\), then \(\mathcal{W }_+ = G^{{\uparrow }}.W\) is invariant under the full group G with respect to the \(\alpha \)-twisted action.
-
(f)
There exists an \(\alpha \in Z^-\) with \(W^{'\alpha } \in \mathcal{W }_+\) if and only if \(\underline{W}' := (-x,q(\sigma )) \in G^{{\uparrow }}.\underline{W}\). If this is the case, then \(W^{'\beta } \in \mathcal{W }_+\) for \(\beta \in Z^-\) if and only if \(\beta ^{-1}\alpha \in Z_2\). In this case, the twisted actions of \(g \in G^{{\downarrow }}\) are related by \(g*_\beta = (\beta \alpha ^{-1}) * g *_\alpha \).
Proof
(a) and (b) are easy to see.
(c) follows from \(\alpha \in Z( G^{{\uparrow }})\) and the \(G^{{\uparrow }}\)-equivariance of the complementation map.
(d) is immediate from the definition of \(\alpha * W\).
(e) That the prescription defines an action follows easily from the fact that \(g_1 *_\alpha (g_2 *_\alpha W)= (g_1 g_2).W\) for \(g_1, g_2 \in G^{{\downarrow }}\) (cf. (2.33))). The relation (2.38) follows from \(\sigma .W = \sigma .(x,\sigma ) = (-x,\sigma )\). For the last statement, we note that by (2.38), the relation \(W^{'\alpha } \in \mathcal{W }_+\) implies
$$\begin{aligned} G *_\alpha \mathcal{W }_+ = \mathcal{W }_+ \cup \sigma *_\alpha \mathcal{W }_+ = \mathcal{W }_+ \cup G^{{\uparrow }}.W^{'\alpha } = \mathcal{W }_+ \cup G^{{\uparrow }}\mathcal{W }_+ = \mathcal{W }_+.\end{aligned}$$
(f) As \(q_\mathcal{G }(\mathcal{W }_+) = \underline{\mathcal{W }}_+ = G^{{\uparrow }}.\underline{W}\) and \(q_\mathcal{G }(W^{'\alpha }) = \underline{W}'\), the inclusion \(W^{'\alpha } \in \mathcal{W }_+\) implies that \(\underline{W}' \in \underline{\mathcal{W }}_+\). If, conversely, \(\underline{W}' \in \underline{\mathcal{W }}_+\), then there exists a \(g \in G^{{\uparrow }}\) with
$$\begin{aligned} (-x, q(\sigma )) = g.(x,q(\sigma )) = ({\mathrm{Ad}}(g)x, q(g\sigma g^{-1})), \end{aligned}$$
so that \(\alpha := g\sigma g^{-1} \sigma \in \ker (q) = Z\) satisfies
$$\begin{aligned} \mathcal{W }_+ \ni g.W = g.(x,\sigma ) = (-x, g\sigma g^{-1}) = (-x, \alpha \sigma ) = \alpha * W' = W^{'\alpha }.\end{aligned}$$
Now suppose that \(W^{'\alpha } = \alpha * W' \in \mathcal{W }_+\). Then \(W^{'\beta } = \beta * W' \in \mathcal{W }_+\) is equivalent to \(\beta \alpha ^{-1}* W^{'\alpha } = W^{'\beta } \in \mathcal{W }_+\), and this is equivalent to \(\beta ^{-1}\alpha * \mathcal{W }_+ = \mathcal{W }_+\). Next we observe that the relation \(\beta \alpha ^{-1}* W \in \mathcal{W }_+\) is equivalent to the existence of some \(g \in G^{{\uparrow }}_{\underline{W}}\) with \(g.W = (x,\beta ^{-1}\alpha \sigma )\), which means that \(\beta \alpha ^{-1} \in Z_2 = \partial (G^{{\uparrow }}_{\underline{W}})\). \(\square \)
Example 2.19
We show that for \(G = {\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}} \rtimes \{\mathbf{1 },{\widetilde{\tau }}\}\) as in Example 2.10(d), we have to use twisted complements to obtain a \(G^{{\uparrow }}\)-orbit in \(\mathcal{G }_E(G)\) invariant under complementation. We have already seen that \(\mathcal{G }_E({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}})\) contains two \(G^{{\uparrow }}\)-orbits, represented by the couples \(W_0 = (h,{\widetilde{\tau }})\) and \(W_1 = (h,{\widetilde{\tau }}_1)\). The complement \(W_0' = (-h, {\widetilde{\tau }})\) satisfies
$$\begin{aligned} {\widetilde{\rho }}(\pi )W_0' = (h, {\widetilde{\rho }}(\pi ){\widetilde{\tau }}{\widetilde{\rho }}(-\pi )) = (h, {\widetilde{\rho }}(2\pi ){\widetilde{\tau }}) = (h, {\widetilde{\tau }}_1) = W_1,\end{aligned}$$
so that complementation exchanges the two \(G^{{\uparrow }}\)-orbits in \(\mathcal{G }_E({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}})\). On the other hand, for the action \(*_\alpha \) defined in (2.37), the full group G preserves both \(G^{{\uparrow }}\)-orbits.
Since \({\mathrm{Ad}}(\rho (-\pi ))h = -h\), the element \(g := {\widetilde{\rho }}(-\pi )\) can be used to define a suitable \(\alpha \)-twisted conjugation as follows. We note that
$$\begin{aligned}\alpha := g (g^{{\widetilde{\tau }}})^{-1} = {\widetilde{\rho }}(-\pi ) {\widetilde{\rho }}(-\pi ) = {\widetilde{\rho }}(-2\pi ) \end{aligned}$$
is a generator of \(Z := Z({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}) = Z^-\). We now have
$$\begin{aligned} W_0^{'\alpha } = (-h, \alpha {\widetilde{\tau }}) = {\widetilde{\rho }}(-\pi ).(h, {\widetilde{\tau }}) = {\widetilde{\rho }}(-\pi ).W_0 \in G^{{\uparrow }}.W_0.\end{aligned}$$
Thus \(\mathcal{G }_E({\widetilde{{\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}}}_2)\) consists of two \(G^{{\uparrow }}\)-orbits, none of which is invariant under complementation, but both are invariant under \(\alpha \)-complementation. An analogous computation leads to the same picture for even coverings of \({\textsf {M}}\ddot{\textsf {o}}{\textsf {b}}\), in particular for the fermionic case.