
==== Front
Commun Math Phys
Commun Math Phys
Communications in Mathematical Physics
0010-3616
1432-0916
Springer Berlin Heidelberg Berlin/Heidelberg

5107
10.1007/s00220-024-05107-2
Article
Decompositions of Hyperbolic Kac–Moody Algebras with Respect to Imaginary Root Groups
http://orcid.org/0000-0002-8712-5644
Feingold Alex J. 1
Kleinschmidt Axel axel.kleinschmidt@aei.mpg.de

2
Nicolai Hermann 2
1 grid.189747.4 0000 0000 9554 2494 Department of Mathematics and Statistics, The State University of New York, Binghamton, NY 13902-6000 USA
2 https://ror.org/03sry2h30 grid.450243.4 0000 0001 0790 4262 Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany
Communicated by Y. Kawahigashi.

16 9 2024
16 9 2024
2024
405 10 2411 3 2024
23 7 2024
27 7 2024
© The Author(s) 2024
2024
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We propose a novel way to define imaginary root subgroups associated with (timelike) imaginary roots of hyperbolic Kac–Moody algebras. Using in an essential way the theory of unitary irreducible representation of covers of the group SO(2, 1), these imaginary root subgroups act on the complex Kac–Moody algebra viewed as a Hilbert space. We illustrate our new view on Kac–Moody groups by considering the example of a rank-two hyperbolic algebra that is related to the Fibonacci numbers. We also point out some open issues and new avenues for further research, and briefly discuss the potential relevance of the present results for physics and current attempts at unification.

http://dx.doi.org/10.13039/100019180 HORIZON EUROPE European Research Council 740209 Nicolai Hermann Max Planck Institute for Gravitational Physics (2)Open Access funding enabled and organized by Projekt DEAL.

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pmcIntroduction

The general theory of Kac–Moody (KM) Lie algebras [1, 2] has been recognized as a beautiful and natural generalization of the theory of finite-dimensional semi-simple Lie algebras over the complex numbers. Many important applications have been found for affine KM Lie algebras, which are characterized by positive semi-definite Cartan matrices, and their corresponding groups. This statement applies especially in physics, where the theory has found prominent applications in string theory and two-dimensional conformal field theory, but also in the context of axisymmetric stationary solutions of Einstein’s equations [3] and its generalizations. However, the situation with indefinite, and more specifically hyperbolic KM algebras, which come with indefinite Cartan matrices, is entirely different. These algebras are much more poorly understood on the mathematical side, which is mainly due to the presence of imaginary (time-like) roots whose associated root spaces exhibit exponential growth. As for physical applications there are tantalizing hints of their possible relevance to understanding the physics of the Big Bang in a quantum cosmological context [4–7] but it remains unclear how these symmetries are to be properly implemented and interpreted in a physical context. This is even more true with regard to the associated KM groups, where again the main problem resides with imaginary root space elements and their exponentiation.

In this paper we are interested in the hyperbolic KM Lie algebras and groups. So far, KM groups have been mostly defined by real root groups generated by exponentials of real root spaces whose adjoint action on the algebra is locally nilpotent. The only known definitions of imaginary root groups require completions of the algebra which allow infinite sums in only one direction (say for positive roots) [8, 9]. In this paper we study an alternative approach which works for one imaginary root at a time, and uses the extensive theory of unitary irreducible representations of (covers of) SO(2, 1). This approach has advantages and disadvantages, but we hope that the approach studied here can shed some new light on these remarkable algebraic structures which continue to challenge mathematicians and physicists.

We also believe that potential applications of our results to physics, unification and M-theory are very interesting. While there is now plenty of evidence that indefinite KM algebras are relevant in this context, we have very few tools for dealing with them, especially when it comes to the KM groups obtained by exponentiation of the corresponding KM algebras. Even for the KM Lie algebra a physical interpretation is so far established only for a finite subset of the real root generators and some very specific null roots associated to the elements of the spin connection [5, 10].1 Likewise the duality symmetries discussed so far only concern finite-dimensional regular subalgebras and their associated low level degrees of freedom. By contrast, the SO(2, 1) groups exhibited here reach ‘infinitely far’ into the space of imaginary root generators, beyond the low level elements for which a physical interpretation has been found. If a way could be found to imbue these groups with a physical meaning this would open entirely new windows on string unification, for instance providing new tools to study higher order corrections beyond perturbation theory. One important aspect of all proposals including KM symmetries in M-theory is the use of symmetric spaces based on the KM group and the usual physics approach is to include exponentials with all positive root generators, including imaginary root generators. A better understanding of such exponentials was one of the key motivations for this paper.

In the theory of finite-dimensional semi-simple Lie algebras over the complex numbers, a great achievement was the Cartan–Killing classification of the simple Lie algebras in terms of an integral n×n Cartan matrix, A=[aij], which captures the geometry of the root system. Dynkin diagrams are very useful graphs which carry the same information as the Cartan matrix, but display it in a clearer way. Serre’s theorem gives generators, {ei,fi,hi∣1≤i≤n}, and relations1.1 [hi,ej]=aijej,[hi,fj]=-aijfj,[ei,fj]=δijhi,[hi,hj]=0,for1≤i,j≤n,(adei)1-aij(ej)=0=(adfi)1-aij(fj)for1≤i≠j≤n,

for finite-dimensional semi-simple Lie algebras from that Cartan matrix. Starting from a generalized Cartan matrix, A=[aij], Kac [13] and Moody [14] independently in 1968 defined a class of infinite-dimensional Lie algebras over C by generators and Serre relations (see [15]). Most of the results and applications of KM algebras have been for the affine KM algebras because they can be described explicitly as a central extension of a loop algebra of a finite-dimensional Lie algebra,1.2 g^=g⊗C[t,t-1]⊕Cc⊕Cd

(the “untwisted case”) where g is a finite-dimensional semi-simple Lie algebra over C, c is central and d is a derivation acting on the ring C[t,t-1] needed to extend the Cartan subalgebra because the affine Cartan matrix A=[aij] has det(A)=0. Lie brackets for these are explicitly given, in contrast with the indefinite KM algebras where the definition only gives a generators and relations description.

Among the indefinite KM algebras, the class of hyperbolic type has received the most attention, including some applications in theoretical physics to supergravity. The representation theories of the affine and hyperbolic types are also in stark contrast mainly because each affine KM algebra contains an infinite-dimensional Heisenberg Lie subalgebra,1.3 h^=h⊗C[t,t-1]⊕Cc

where h is the abelian Cartan subalgebra of g. The Fock-space representation of h^ on a space of polynomials in infinitely many variables as multiplication and partial differentiation operators plays a vital role in the vertex operator representations of g^. The rank 2 hyperbolic KM algebras do not contain any Heisenberg Lie subalgebra. For higher rank hyperbolic KM algebras which contain an affine KM subalgebra, one can decompose the hyperbolic algebra with respect to its affine subalgebra or some other kind of subalgebra whose representations can be understood.

There have been several choices studied for such a subalgebra in a hyperbolic g=g(A):

(1) A finite type KM algebra coming from a subset of the generators (a Dynkin sub-diagram),

(2) An affine type KM algebra coming from a subset of the generators,

(3) A subalgebra of fixed points under an automorphism of g,

(4) A subalgebra which is not obvious, e.g., not just from a Dynkin sub-diagram.

Option (1) has been used, for example, to study the hyperbolic algebra known as E10 by decomposing it with respect to a finite type A9 subalgebra [5, 16]. Similar decompositions have been performed with respect to the D9 and A8⊕A1 subalgebras in [17, 18]. In physical applications a real Lie algebra is preferred, usually the split real form, gR=g(A)R, which is just the real span of the generators and their Lie brackets. The split real form can also be understood under option (3) as the fixed points of the conjugate linear involutive automorphism that fixes the Chevalley generators, {ei,fi,hi∣1≤i≤n}.

Option (2) has been used, for example in [19], to study a particular rank 3 hyperbolic, F, also called AE3, which has an affine subalgebra of type A1(1), the simplest example of an affine KM algebra whose representation theory is well developed.

Option (3) includes the split real form mentioned above, as well as the “compact” real form, K(g), which is a real Lie subalgebra of fixed points under the Cartan–Chevalley involution ω(ei)=-fi, ω(fi)=-ei and ω(hi)=-hi on the complex KM algebra. The intersection of the split and the compact real form is of interest to physicists [20–25], who have studied finite-dimensional representations of the infinite-dimensional involutive subalgebra K(gR) generated by {ki=ei-fi∣1≤i≤n} and satisfying the Berman relations [26]. When a finite type algebra has a Dynkin diagram with an automorphism (symmetry), twisted affine KM algebras result from the fixed point subalgebra of g^.

Option (4) can be applied using the results of [27] on subalgebras of hyperbolic KM algebras. They found inside F all the rank 2 hyperbolics whose Cartan matrix is symmetric. The simplest example is the rank 2 “Fibonacci” hyperbolic Fib [28] whose 2×2 Cartan matrix has a12=-3=a21. A study was made in [29] of the decomposition of F with respect to Fib that showed some interesting Fib-modules occur, including some integrable modules which are neither highest nor lowest weight modules, and not the adjoint module. We will later use Fib as one of the simplest examples of a hyperbolic KM algebra to illustrate how it and two of its irreducible highest weight representations (see Figs. 3 and 4) might be decomposed in a new way using a three-dimensional imaginary subalgebra determined by a choice of an imaginary root vector in some imaginary root space of a hyperbolic KM algebra. For comparison we will also discuss how a choice of a real root vector in a real root space gives a decomposition into finite-dimensional sl(2,C)-modules. The use of “real” versus “imaginary” for kinds of roots should not be confused with the choice of field R versus C for the scalars of the Lie algebra and its representations.

Included in option (1) is the obvious choice of an sl(2,C) subalgebra corresponding to a simple root, αi, for a fixed i, that is, the subalgebra gi=sl(2,C)i with basis {ei,fi,hi}. The Serre relations defining g imply that it decomposes with respect to gi into an infinite number of finite-dimensional gi-modules. We could have taken any real root, α, whose root space gα must be one-dimensional with basis vector E(α), and found an opposite root vector F(α) in g-α, such that with H(α)=[E(α),F(α)] a subalgebra sl(2)α is defined. But since any real root is by definition in the Weyl group orbit of the simple roots, it is sufficient to study just the decompositions with respect to the subalgebras sl(2)i. In a later section we will discuss in some detail how this decomposition works for the rank 2 hyperbolic algebra Fib, see Sect. 4. See Fig. 1 for a graphical display of some positive roots of Fib along with their root multiplicities.Fig. 1 Some positive roots of Fib with root multiplicities. First published in [27]

Our construction raises several interesting questions and opens new avenues for further research. One of them concerns the issue of ‘combining’ different SO(2,1)α groups and their interplay for different imaginary roots α. Unlike for real root subgroups, there are no (Steinberg-type) relations that could be exploited towards the evaluation of products of elements of different imaginary root groups due to the lack of local nilpotency.2 Although each action would involve distinct Hilbert spaces, the repeated action of such operations is well-defined, because the unitary action guarantees that norms are preserved by the repeated group action.

What we would like to stress here that for the definition of imaginary root subgroups some notion of completion is definitely required. The one we employ here relies on the Hilbert space completion of the vector space of the KM algebra viewed as an SO(2,1)α module. To what extent the KM algebra structure is compatible with this completion remains an open question, but the results of [25] suggest the thus completed space may no longer be a KM algebra, in the sense that the commutator of two elements of the completion is no longer an element of the Hilbert space. This could mean that the Hilbert space norm used here which is induced by the standard bilinear form may not be the appropriate tool to define a completion of the KM algebra, and that one may have to resort to different notions of completion (we note that there are many topologies on infinite dimensional vector spaces that might be used here). This is also exemplified by comparing between Kac–Moody commutators and tensor products in the general theory of unitary representations of SO(2, 1) as described for instance in [31–34]. More specifically, take the two principal series that arise in the adjoint of Fib derived in Sect. 4. They are both unitary but their commutator contains, among other things, the non-unitary adjoint of so(2,1). This is in tension with the tensor product results given in [31–34], according to which the product of two unitary representations is again unitary. A possible explanation of this tension is that the norm of a commutator in the Kac–Moody algebra is not equal to the product of the norms of its two elements which underlies the tensor product construction in the general theory of [31–34].

Decompositions of Hyperbolic Kac–Moody Algebras

In this section we will discuss the main idea of the paper, how the choice of an imaginary root vector (multi-bracket) gives an imaginary three-dimensional subalgebra whose split real form is isomorphic to so(2,1). We use the representation theory of sl(2,R)≅so(2,1) on well-known series of unitary modules to decompose any hyperbolic Kac–Moody algebra or representation in such a way that the action of the group SO(2, 1) (or its covers) is given explicitly on a Hilbert space completion of each irreducible summand. This approach defines imaginary root groups in a different way from other methods that use a completion of the Kac–Moody group in only one “direction”, see e.g. [9] and other references on page 268 of that book or [8].

Kac–Moody algebras and involutions

Let g=g(A) be a Kac–Moody (KM) algebra over C with a non-degenerate symmetric r×r Cartan matrix A=[aij]. The generalization of this work to symmetrizable Cartan matrices should be straightforward. Our main focus will be on hyperbolic KM algebras where it is well-known that A is Lorentzian with signature (1,r-1) and the maximal rank r is 10 [1], but these introductory remarks are valid in greater generality.

We recall that g has a root space decomposition2.1 g=h⊕⨁α∈Δgα

where h is the r-dimensional Cartan subalgebra (CSA) that acts semi-simply by the adjoint action adh(x)=[h,x] on g, and gα denotes the eigenspaces under this action with eigenvalue given by roots α∈Δ of the form α:h→C such that the eigenspace is non-trivial. Roots α are divided into real roots (characterized by positive norm squared) and imaginary roots. The latter can be further subdivided into lightlike (with vanishing norm squared) and timelike roots (with negative norm squared).

The Cartan–Chevalley involution is the C-antilinear automorphism of g (ω(zx)=z¯ω(x) for z∈C and x∈g) defined by2.2 ω(ei)=-fi,ω(fi)=-ei,ω(hi)=-hi

and extended to the whole KM algebra by means of ω([x,y])=[ω(x),ω(y)]. In particular, for any multi-bracket we have2.3 ω(ei1⋯in)=(-1)nfi1⋯in

where we use the notation ei1…in:=[ei1,[ei2,⋯,[ein-1,ein]...]] and similarly for fi1⋯in. The standard bilinear form is defined by2.4 ⟨ei|fj⟩=δij,⟨hi|hj⟩=aij

and ⟨[x,y]|z⟩=⟨x|[y,z]⟩. Then Theorem 11.7 of [1] shows that the Hermitian form (complex-conjugate linear in the second argument)2.5 (x,y):=-〈x|ω(y)〉

is positive definite on the whole (complex) KM algebra except on its Cartan subalgebra, where it has precisely one negative eigenvalue. For any operator O on g, its Hermitian conjugate O† is defined by (O(x),y)=(x,O†(y)) for any x,y∈g.

With respect to this Hermitian form, for any element z∈g, the adjoint operator adz, defined by adz(x)=[z,x] for any x∈g, satisfies2.6 (adz)†=ad-ω(z).

To see this we check2.7 (adz(x),y)=([z,x],y)=-〈[z,x]|ω(y)〉=+〈x|[z,ω(y)]〉=〈x|ωω(z),y〉=(x,[-ω(z),y])=(x,ad-ω(z)(y)).

In particular, adz is self-conjugate if and only if z=-ω(z).

Subalgebras so(2,1)α associated with roots α

The following works for any indefinite KM algebra g whose Cartan matrix has indefinite signature, not only for hyperbolic algebras.

Let α=αi1+⋯+αin be a positive root belonging to some multi-commutator ei1⋯in, where each αij is a simple root and the eij are the Chevalley generators. The order of indices is significant if α is not a real root. Define2.8 E(α):=ei1⋯in∈gα,F(α):=-ω(E(α))=-(-1)nfi1⋯in∈g-α

so that2.9 N:=⟨E(α)|F(α)⟩=(E(α),E(α))>0

In principle we should use the multi-index label (i1,…,in) instead of just α to distinguish the independent elements of the root space gα, but we suppress this for simplicity of notation.3 The fact that α is the root of E(α) means that2.10 [hj,E(α)]=α(hj)E(α).

Writing α=∑j=1rnjαj and defining H(α)=∑j=1rnjhj we also have2.11 [H(α),E(α)]=α2E(α)[H(α),F(α)]=-α2F(α)[E(α),F(α)]=NH(α).

with α2=∑niaijnj and where the last equation uses the invariance of the standard bilinear form ⟨[H(α),E(α)]|F(α)⟩=⟨H(α)|[E(α),F(α)]⟩ with the normalization ⟨hi|hj⟩=aij.

Now we have to distinguish two cases. When α is a real root (α2:=α·α>0), we define2.12 J3=(α2)-1H(α),J+=(Nα2)-1/2E(α),J-=(Nα2)-1/2F(α),

The commutation relations are2.13 J+,J-=+J3,J3,J±=±J±.

These are elements of the KM algebra, so we understand them as operators under the adjoint action. The hermiticity properties of these operators are inherited from the bilinear form, that is, with respect to the Cartan–Chevalley involution ω the generators satisfy2.14 ω(J±)=-J∓,ω(J3)=-J3

whence we have2.15 J±†=J∓,J3†=J3.

One easily checks that these elements have positive norm with respect to the Hermitian form (2.5). The generators J± and J3 together with the commutation relations (2.13) and the hermiticity properties (2.15) therefore represent the real Lie algebra so(3). For real roots, the (adjoint) action of J± on the KM algebra g(A) generates finite-dimensional representation spaces because the multiple addition of a real root α to any root β will satisfy (β+kα)2>2 for sufficiently large k. The associated groups obtained by exponentiating these Lie algebra elements are referred to as real root groups, where the exponentiation can be performed over R or C, or any other field of characteristic zero. These real root groups generate the minimal Kac–Moody group associated with the Cartan matrix [8, 9, 35].

The second case to be considered concerns imaginary roots, for which α2≤0. For lightlike imaginary roots α, for which α2=0, one obtains a Heisenberg algebra from (2.8) that corresponds to a contraction of sl(2). However, our main interest here is the case of timelike imaginary roots, for which α2<0. In that case we can define a subalgebra so(2,1)α of the KM algebra g(A) for any element of a timelike imaginary root space gα.

Instead of (2.12), the relevant definition reads now for timelike roots2.16 J3=(α2)-1H(α),J+=(-Nα2)-1/2E(α),J-=(-Nα2)-1/2F(α).

It is straightforward to see that these operators satisfy the bracket relations of an so(2,1) Lie algebra, that is,2.17 J+,J-=-J3,J3,J±=±J±,

which differs by a crucial minus sign from (2.13) in the first commutator, while the hermiticity properties (2.15) are maintained. The latter point is essential, since otherwise the minus sign could simply be redefined away, for instance by rescaling J+→-J+, but this redefinition would violate the hermiticity properties (2.15). The normalization (2.16) implies that for α2<02.18 ||J+||2=||J-||2=-(α2)-1>0,||J3||2=(α2)-1<0

so these norms shrink to zero as α2→-∞.

The difference between the real Lie algebra so(2,1)α for timelike roots compared to so(3) for real roots becomes apparent when writing these algebras in terms of standard Lorentz or rotation algebras as reviewed in Appendix A. The change of basis from the standard basis to (2.13) or (2.17) involves complex coefficients in such a way that the hermiticity properties of the algebras are different in unitary representations. This will also be important when considering the implications for the Kac–Moody group in Sect. 3. Since the definition of the generators in (2.16) depends on the root α, we will keep this dependence in the notation for the algebra so(2,1)α. As a real Lie algebra we have the isomorphism so(2,1)α≅sl(2,R).

Before continuing we note that there is another way to define so(2,1) subalgebras of g(A) in the case of hyperbolic algebras that does not make use of timelike imaginary roots, but rather appropriate linear combinations of real roots. Distinguished among these is the principal so(2,1) subalgebra introduced in [36]. Generalizations of this construction are studied in [37]. The principal so(2,1) subalgebra can be constructed using the inverse Cartan matrix A-1=[bij]=[Λi·Λj], where Λi, 1≤i≤r, are the fundamental weights. The entries of A-1 satisfy bij≤0 since all fundamental weights are null or time-like for hyperbolic KM algebras, so their scalar products are non-positive. We recall that we assume the Cartan matrix to be symmetric for simplicity. If we define2.19 ri=-∑jbij>0

then the generators2.20 J3=-∑irihi=∑i,jbijhi,J+=∑iriei,J-=∑irifi

again satisfy the commutation relations (2.17) and the hermiticity properties (2.15).

Decomposing g(A) under the action of so(2,1)α

The subalgebra so(2,1)α⊂g can be used to decompose the adjoint representation (or any other representation) of g under its action. Since so(2,1)α≅sl(2,R) we will be dealing with representations of sl(2,R). In view of the hermiticity properties (2.15) these representations will typically be unitary representations of sl(2,R) so we review the relevant infinite-dimensional representation spaces, called principal series and discrete series representations, in Appendix B. From now on we take g to be a complex hyperbolic KM algebra.

To analyse the decomposition of the adjoint g under the algebra so(2,1)α generated by (2.11) let us consider an arbitrary imaginary root β and any element E(β)∈gβ of its associated root space; then2.21 [J3,E(β)]=νE(β),whereν=α·βα2∈Q.

For β a positive timelike imaginary root we have α·β<0 and therefore the prefactor on the right-hand side is positive. In general, the rational number ν is not an integer. While this does not matter much for the representations of the Lie algebra sl(2,R), this matters for the group: the exponential operator e2πirJ3 is not periodic modulo 2π if the J3 eigenvalue ν not an integer. In other words, the group obtained by exponentiation of so(2,1)α is not SO(2, 1) but a covering of it. We note that the parameter ν can become arbitrarily small. There are infinitely many covers since the fundamental group π1(SO(2,1))≅Z and this agrees with the fact that any denominator can occur in ν as α varies. The most well-known cover is Spin(2, 1) corresponding to the double cover (with the metaplectic Weil representation of SL(2,R)) but more complicated situations are possible. In physical terms, such representations are anyonic representations of covers of SO(2, 1) [38–43].4 There are such representations both for the so-called principal series, the discrete series and the complementary series.

Returning to the decomposition of g under so(2,1)α for positive timelike α, we note that, with respect to the bilinear form (2.4), the orthogonal complement in the Cartan subalgebra h of J3 from (2.16) consists of singlets. In particular, choosing a basis of (r-1) CSA generators H(vi) with α·vi=0 (with space-like vi), we have [J±,H(vi)]=0; all these states have positive norm because the vi are spacelike.

For other representations let us pick any positive root β, and apply J± to any element E(β)∈gβ. The successive application of the lowering operator J- will result in a chain of maps2.22 …⟶J-gβ⟶J-gβ-α⟶J-gβ-2α⟶J-gβ-3α⟶J-…

along an infinite string of subspaces2.23 ⨁k∈Zgβ+kα⊂g(A)

Likewise, the application of J+ moves in the opposite direction:2.24 …⟵J+gβ+3α⟵J+gβ+2α⟵J+gβ+α⟵J+gβ⟵J+…

For the root string {β+kα∣k∈Z} we must distinguish two main cases:there exists a minimal k0<0 such that β+k0α is not a root;

the elements β+kα are roots for all k∈Z (the chain may or may not contain a real root)

In the first case the chain terminates and all elements of the root spaces along the chain belong to discrete representations (idem for the negative side). In the second case we may encounter continuous representations. However, also in that case there will occur (many!) discrete representations in the subspace (2.23) of g. This is because the root multiplicities, defined as mult(β)=dimgβ, vary with β, and increase exponentially with the height of the root. Namely, the subspaces gβ+kα are in general of different dimensions, with multiplicities increasing in the leftward direction for positive roots (as long as β+kα is positive), and likewise in the rightward direction for negative roots. This implies that for positive β+kα each root space in the descending chain (2.22) has a large kernel whose elements are annihilated by the action of J-. Consequently, for each root space, every element of the kernel is a lowest weight vector of a discrete series representation that extends to the left and is generated by the successive application of J+, with the value of m given by formula (2.21). Because α·β/α2>0 for all positive β the unitarity condition (B.4) discussed in the appendix is satisfied. This number is in general fractional and thus we are dealing with an anyonic discrete series representation of a cover of SO(2,1)α. This covering has at most (-α2) sheets, but since (-α2) can become arbitrarily large we eventually reach all positive rationals. A natural framework for considering all possibilities of imaginary timelike roots together is therefore the universal cover.

If the chain of roots β+kα does not terminate we are left with principal series representations after the elimination of the discrete series representations. In the examples we have studied so far, we noticed that no complementary series representations occurred in the decompositions and this is discussed further in Sect. 4.4. In the following, we shall assume that they are always absent for simplicity although their presence would not qualitatively change our structural analysis.

We do have a proof that there can only be a finite number of principal series representations since any principal series has a string of weights β+kα with k∈Z. This string must therefore intersect the region in h∗ bounded by two planes orthogonal to the timelike α that are separated by α. The intersection of this region with Δ∪{0} only contains finitely many elements and therefore only finitely many Lie algebra generators can be part of principal series representations, showing that these are finite in number. The determination of the Casimir for a principal series representation must be done ‘by hand’, as there appears to be no general formula for evaluating the requisite multi-commutators. In Sect. 4 we will therefore present a few exemplary calculations involving the so-called “Fibonacci” algebra, Fib, which is the simplest example of a (strictly) hyperbolic KM algebra.

In total, we obtain therefore the following decomposition of the vector space of the complex KM algebra g as modules for so(2,1)α:2.25 g=(so(2,1)α⊗C)⊕Cr-1⊕⨁s∈Prα(g)MultsprPs⊕⨁s>0Mults+Ds+⊕⨁s>0Mults-Ds-

The first term is the complexified adjoint of so(2,1)α, the second term represents the singlets in the CSA, the third term is a sum over the set Prα(g) of principal series representations which occur in this decomposition with multiplicity Multspr. The last two terms are the infinite number of lowest and highest weight discrete series representations, each labelled by parameter s, and the number of copies for each value of s given by the multiplicity Mults±. Since we are dealing with covers of SO(2, 1), only restriction on the value of s for discrete series representations is being a positive real number. For the group SO(2,1)≅PSL(2,R), we would require a positive integer. The various multiplicities which occur in this decomposition are computable in any particular example from a knowledge of the root multiplicities of g, but we have no general formula for them. It follows from the Cartan–Chevalley involution that there is a bijection between the lowest and highest weight representations that arise, so we get Mults+=Mults-. The representation spaces are normed and carry a unitary action of so(2,1)α, hence they have completions, P^s and D^s±, which are complex Hilbert spaces. While this gives a Hilbert space completion of the vector space, g, for each choice of α, we do not claim that the Lie algebra structure of g extends to the completion. We have used the notation introduced in Appendix B. In Sect. 3 this will be important since we know that we can define the action of a group (a cover of SO(2, 1)) on these spaces.

We may also decompose any highest weight representation, V=V(Λ), of g with respect to so(2,1)α:2.26 V(Λ)=⨁s∈Discr(V)Mults-Ds-

where Discr(V) is the set of values of parameter s which occur in the decomposition, and Mults- is the multiplicity giving the number of copies which occur of the irreducible highest weight discrete series representation Ds-. A similar formula holds for lowest weight representations of g where only lowest weight discrete series representations Ds+ occur. In Sect. 4 we will give some terms in the decompositions of two highest weight representations of Fib. In order to get representations of covers of SO(2,1)α, these discrete series representations must be completed with respect to a norm to Hilbert spaces, D^s±, so there is a Hilbert space completion, V^(Λ), of V(Λ). Now the question is whether this completion, which depends on the choice of α, is a g-module.

To close this section, we give the decomposition of g under the principal so(2,1) subalgebra (2.20) that was already studied in [36]. There it was shown to take the following form2.27 g=(so(2,1)⊗C)⊕⨁i=1r-1Pi⊕⨁k≥2Multk+Dk+⊕⨁k≥2Multk-Dk-,

where there are r-1 unitary principal series representations Pi whose s parameter is suppressed in this formula, and Multk± discrete series representations Dk± of lowest (+) or highest (−) weight type and parameter k with 2≤k∈Z. By the Cartan–Chevalley involution there is a bijection between the set of lowest weight representations and the set of highest weight representations, so Multk+=Multk-. Because they appear with integral parameters, they lift to representations of SO(2, 1). Only single-valued representations appear in (2.27), but the decompositions take a more complicated form for the so(2,1)α subalgebras associated with timelike imaginary roots α.

Defining a Group Action on the Kac–Moody Algebra

A main goal of this paper is to see under what circumstances an exponential action of the imaginary root subalgebras so(2,1)α on the full KM algebra can be defined. For real root subalgebras this is always possible, yielding an action of the respective real root subgroups. This can be done with either real or complex coefficients (or even coefficients taking values in some ring or finite field) because the relevant representations are always finite-dimensional. This is the reason why the minimal KM groups are defined to be generated by the real root subgroups. However, defining larger KM groups including generators from the imaginary root subalgebras so(2,1)α is only possible subject to special restrictions. One option that has been extensively explored in the literature (see [8, 9] and references therein) is to consider only exponentials of positive imaginary root vectors. Here, we wish to explore an alternative option that allows us to exponentiate the action of any imaginary subalgebra so(2,1)α to get an imaginary root group action on completions of g and on completions of any integrable representation.

As we showed above, for any given subalgebra so(2,1)α the KM algebra decomposes into a direct sum of so(2,1)α-modules which have Hilbert space completions. More specifically, for each timelike imaginary root α (and each choice of a multicommutator E(α) of its root space gα) we have3.1 g(A)=(so(2,1)α⊗C)⊕Hα

with the associated ‘total’ Hilbert space Hα, which itself is the direct sum of infinitely many Hilbert spaces corresponding to the irreducible representations of this so(2,1)α. We have written g(A) on the left side of the equation above, but it should be understood as a certain vector space completion. The reason why we must now consider the completion of the KM algebra with respect to the norm (2.5), rather than algebraically in the sense of formal sums, is that the operators J± and J3 are no longer locally nilpotent. The key issue here is the fact that these operators and their exponentials are unbounded operators.5 This means that they can be defined only on a domain, that is, a dense subspace of any given Hilbert space. Dense subspaces are, for instance, obtained by considering finite linear combinations of the basis elements of the given representation space. But then one faces the problem that repeated action of different exponentials on any element of the dense subspace may throw one out of the domain, so a group action (corresponding to a repeated application of exponentials) is not possible in general.

The key idea which allows us to circumvent the difficulty with unbounded operators is to exponentiate the (adjoint) action of these operators on the KM algebra by exploiting our knowledge of SO(2, 1) unitary irreducible representations (UIRs), and the fact that this group action is defined on the full Hilbert space, and not just on dense subspaces. For the exponentiation of linear combinations of J+, J- and J3 in so(2,1)α this requires using the unitary operators3.2 U(w,r)=exp(iwJ++iw¯J-+irJ3)⇒U†=U-1

with w∈C and r∈R. This action corresponds to the compact real form of g(A), with anti-Hermitian generators3.3 i(J++J-),(J+-J-),iJ3

so this is not the exponential of the standard split real form. As the unitary representation spaces of SO(2, 1) are complex, the adjoint action here is on the complexified Lie algebra g(A)C.

On each of these representation spaces, hence on all of Hα and the KM algebra g(A), the action of the group SO(2,1)α or a cover is well defined. Concretely this is done as follows: for all UIRs we can evaluate the the action of SO(2, 1) on any basis vector vn by exploiting the action (C.1) to express the transformed function again in terms of the chosen basis:3.4 S∘vm=∑n∈ZUmn(S)vn

for any given Möbius transformation (or in a cover of the Möbius group). The infinite unitary matrix Umn(S) here is the same as in (C.9). To determine this matrix for any given unitary operator (3.2) we use formula (C.8) to convert the argument of the exponential into a real expression in terms of the real sl(2) generators {e,f,h} with real parameters {u,v,r}. With these data we can then compute the two-by-two matrix S and the transformation (C.1), which yields the coefficients Umn= Umn(S) upon expansion in the appropriate basis of the relevant function space, at least in principle. Conversely, given a matrix S we can re-express it in Iwasawa form and then convert each factor in the Iwasawa decomposition back to the complex form by reading (C.8) from right to left.

For the action on the KM algebra we simply replace the basis vectors by the corresponding elements of the KM algebra. Because the action is on complex functions the coefficients Umn(S) are in general complex, hence we have to implement the action of the complexified KM algebra g(A)C. In this way, we have therefore succeeded in exponentiating the associated imaginary root generator. The presence of non-trivial covers is reflected in the periodicity properties of (3.2) with respect to the rotation parameter r: for a k-fold covering of SO(2, 1) we have r∈R/2πkN.

Requiring the exponential of so(2,1)α to belong to the KM group therefore requires two generalizations of the common definition of KM group (either minimal or completed). The first one is that having both E(α) and F(α) to have well-defined exponentials requires a completion in both Borel directions. The second generalization is that we also have to consider covers. The ‘sheetedness’ of the cover is given by α2; as α2 varies we will exhaust all possible covers of SO(2, 1). We therefore find a Kac–Moody group structure that involves the universal cover of SO(2, 1).

In physics applications, the physical system is typically built on the symmetric space G/K of a (split real) Kac–Moody group divided by its maximal compact subgroup. Here, one has to make a choice of which Kac–Moody group to use. Using an Iwasawa parametrization of the symmetric space for the minimal group [45], a natural choice uses only a Borel subgroup with Lie algebra corresponding to the positive roots, both real and imaginary, of the KM algebra. The biggest group that can be associated with this Lie algebra is the maximal KM group using democratically one-parameter subgroups associated with all generators belonging to positive roots. One parametrization of the symmetric space G/K for the maximal KM group G can then be obtained using for example the standard form parametrization of [9, Thm. 8.51] although a different parametrization is common in physics. That this choice of Kac–Moody group is consistent with applications has been discussed for example in [46, 47]. Whether our more unitary choice is useful in physics remains to be seen.

Example: The Rank 2 Fibonacci Algebra Fib

In this section we discuss two kinds of decomposition for the specific example when the Cartan matrix is 2×2 with a12=-3=a21, in which case the KM algebra g(A) is called Fib because its real roots can be described by the Fibonacci numbers [28]. Figure 1, taken from [27], shows a diagram of some of the positive roots for Fib, including root multiplicities for the imaginary roots (black dots). The open circles show some of the positive real roots, with the simple roots, α1 and α2 at the left of the diagram. The central vertical line is the symmetry line of the outer automorphism which switches α1 and α2, but the two angled black lines are the lines fixed by the two simple Weyl group reflections, w1 and w2. The Weyl group W for Fib is the infinite dihedral group D∞=⟨w1,w2∣|w1w2|=∞⟩. Figure 2, taken from [48], shows some of the roots of Fib, both positive and negative, with the real roots on the red hyperbola labelled as Weyl conjugates of the simple roots, and the imaginary roots on blue hyperbolas in the light cone. The gray lines are the asymptotes of these hyperbolas, the null cone of zero norm points, but no roots of Fib have zero norm. The inner-most green lines are the fixed lines of the simple Weyl reflections, and other green lines are their images under the Weyl group action. In each half of the light cone, the wedge between the inner green lines is a fundamental domain for the action of W on that half of the lightcone, which is tessellated by W.Fig. 2 Partial Fib root system with real roots labelled and hyperbolas of constant square length shown. First published in [48], where the simple roots were labelled by βi instead of αi

Decomposition of Fib with respect to a real simple root

First we will discuss in some detail a (partial) decomposition of Fib into finite-dimensional modules with respect to the subalgebra sl(2)1 with basis {e1,f1,h1}. The following decomposition can be performed for either the complex Lie algebra or the split real form. Such a type of decomposition has been used frequently, especially in the physics literature [5, 16, 49, 50], as well as in the math literature [51], but we present it here for completeness and also for comparison with the case of the imaginary root that is discussed in Sect. 4.2 and the one of central interest in this work.

In Fig. 1, the subalgebra sl(2)1 corresponds to the simple root α1, giving a direction for the decomposition of Fib into finite-dimensional irreducible sl(2)-modules. For 0≤m∈Z, we denote by V(m) the irreducible sl(2)-module with dim(V(m))=m+1. Since Fib has a symmetric Cartan matrix, the two choices for simple root, αi, yield symmetric decompositions, so it is enough to just look at one choice. Of the two open circles corresponding to simple roots, let the one towards the left be α1, and the one to the right be α2. In the decomposition of Fib with respect to sl(2)1, the first irreducible representation (irrep) to note is sl(2)1 itself, a copy of V(2). Since the dimension of the Cartan subalgebra (corresponding to the origin in Fig. 1) is 2, there should be another sl(2)1 irrep with a weight space in that Cartan. In fact, it is trivial to compute [e1,ah1+bh2]=(-2a+3b)e1 and [f1,ah1+bh2]=(2a-3b)f1, so that for 2a=3b we find the one-dimensional span of 3h1+2h2 is a trivial module V(0), and no other irrep has a non-trivial intersection with the Cartan subalgebra. The next irrep is the one generated by e2 with basis4.1 {e2,[e1,e2]=e12,[e1,[e1,e2]]=e112,[e1,[e1,[e1,e2]]]=e1112}

where we have used the notation for multibrackets from Sect. 2.1. This irrep V(3) corresponds to the root string α2, α2+α1, α2+2α1, α2+3α1 in the root diagram, whose end points are real roots and whose middle points are imaginary roots, each with multiplicity 1. There can be no other irreps with weights on that line of roots, so we go to the next parallel line starting with the imaginary root α1+2α2. That root space is one-dimensional with basis vector e212 since there is no way to get to that root space except by [e2,[e1,e2]]. The sl(2)1 irrep generated by that root space is 5-dimensional because the Weyl group reflection w1 sends α1+2α2 to4.2 w1(α1+2α2)=-α1+2(α2+3α1)=5α1+2α2

forcing the weights of the irrep V(4) to be the α1-string {mα1+2α2∣1≤m≤5} and a basis for that V(4) must be4.3 {e212,e1212,e11212,e111212,e1111212}.

But the dimension of the root space 3α1+2α2 is 2 so there must also be a trivial 1-dimensional module in that root space. To find an explicit basis vector for it, take an arbitrary linear combination of the two independent multibrackets in that root space, e11212 and e21112, and solve4.4 [f1,ae11212+be21112]=0.

The same linear condition will occur if f1 is replaced by e1, since this is a trivial module, and it will give a basis for a V(0) irrep in that root space. Those two irreps fill up that root string, so the decomposition process continues on the next parallel α1-string of roots starting with the real root α1+3α2, {mα1+3α2∣1≤m≤8}. The list of root multiplicities from Fig. 1 for that string is {1,2,3,4,4,3,2,1} so we see that there must be an irrep V(7) having all eight of those weights, as well as an irrep V(5) having the middle six weights, plus an irrep V(3) having the middle four weights, plus an irrep V(1) having the middle two weights. No other irreps occur in that string, and each one occurred only once. It would be straight-forward but tedious to find explicit basis vectors for those irreps, or just lowest weight vectors.

There would be no difference in the above decomposition if we were looking at the split real form FibR, but each irrep would be a real vector space. Some interesting patterns have been seen in such a Z-graded decomposition where the grading of the irreps is according to the coefficient of α2 in the α1-string. The obvious symmetry between positive and negative graded pieces means it suffices to understand the positively graded part. The Lie bracket respects the grading, of course, and the 0-graded piece is just sl(2)1 plus the trivial module in the Cartan. So the idea is to see how the irrep V(3) comprising the 1-graded piece, bracketed with itself is related to the 2-graded piece, which is the sum V(4)⊕V(0). Naturally, the bracket should correspond to anti-symmetric tensors in the tensor product V(3)⊗V(3), and we have complete information about such a tensor product decomposition of finite-dimensional irreps of sl(2) from the theory of Clebsch–Gordan. One finds that the wedge product V(3)∧V(3) exactly equals the sum V(4)⊕V(0). One expects to get the 3-graded piece by bracketing the 1-graded piece V(3) with the 2-graded piece, and so on recursively. But this expansion seems to just get more and more complicated as the grading increases, with no clear pattern emerging. A similar situation was encountered [19] in the decomposition of the rank-three hyperbolic F (mentioned in option (2) in the introduction) with respect to its A1(1) affine subalgebra, where the Z-grading was with respect to the “level” of the affine submodules. A clear answer for level 2 gave a closed generating function for infinitely many imaginary roots of F because level 1 was a single irrep whose multiplicities were exactly the values of the classical partition function. Higher levels were studied in [52], up to level 4, and in [53] up to level 3, but this method has never yielded a new insight into the full structure of a hyperbolic KM algebra.

In Fig. 3 we have a graphical display of some weights of the irreducible highest weight Fib-module with highest weight ρ=λ1+λ2 along with the weight multiplicities. The weights are determined by the action of the Weyl group W and the root-string properties of finite-dimensional sl(2)-modules. The multiplicities are determined recursively by the Racah–Speiser formula [54], which is valid in any irreducible highest weight module Vλ:4.5 Multλ(μ)=∑1≠w∈W(-1)ℓ(w)+1Multλ(μ+ρ-w(ρ)).

This is valid for any weight μ of Vλ not in the Weyl orbit of highest weight λ. All weights in the Weyl orbit W·λ have multiplicity 1. For Fib it is easy to compute ρ-w(ρ) and see that for w∈{w1,w2,w1w2,w2w1,w2w1w2,w1w2w1} it equals4.6 α1,α2,α1+4α2,4α1+α2,12α1+4α2,4α1+12α2.

For the weights of Vρ shown in Fig. 3, this recursion only needed Weyl group elements of length ℓ(w)≤2, so only the first four shifts in the last list. The same algorithm was applied to the fundamental representation Vλ1 and the results are shown in Fig. 4. The point of displaying those weight diagrams is to help understand how those modules decompose under the action of subalgebras of Fib like the root subalgebra sl(2)1 or an so(2,1)α subalgebra for an imaginary root α that we discuss in the next section. Fig. 3 Some weights of irreducible Fib module Vρ with highest weight ρ=λ1+λ2 showing multiplicities

Decomposition of Fib with respect to an imaginary subalgebra

Let ei, fi, hi for i=1,2, be the generators of Fib with the KM relations coming from the Cartan matrix A. The positive imaginary root of Fib with lowest height is α=α1+α2, corresponding the the multi-bracket e12=-e21, which we take as E(α) in the sense of Sect. 2.2. Then F(α)=-ω(E(α))=-f12=f21, and H(α)=h1+h2. We find that α2=-2, and using the Jacobi identity, we compute the bracket4.7 [E(α),F(α)]=[e12,f21]=[[e12,f2],f1]+[f2,[e12,f1]]=[[e1,h2],f1]+[f2,[h1,e2]]=[3e1,f1]+[f2,-3e2]=3(h1+h2)

so N=3 in formula (2.11). This gives us the basis of so(2,1)α as in (2.16),4.8 J3=-h1+h22,J+=e126,J-=f216.

We begin to find the decomposition of Fib into a direct sum of irreducible so(2,1)α-modules, where the action of so(2,1)α is the adjoint action in Fib. It is easy to check that4.9 0=[J3,h1-h2]=[J±,h1-h2]

so the one-dimensional subspace spanned by v0=h1-h2 is a trivial so(2,1)α-module. Note that for any root β=n1α1+n2α2 we have β(h1-h2)=5(n1-n2), so for any root vector xβ∈gβ, 15[h1-h2,xβ]=(n1-n2)xβ so the operator 15(h1-h2) provides a Z-grading on Fib corresponding to the horizontal position of the root β in Fig. 1.

Since J3 and v0 are independent, they form a basis for the Cartan subalgebra of Fib, so there cannot be any other irreducible so(2,1)α-modules in the decomposition having a non-trivial intersection with the Cartan. In particular, this means that only discrete series modules (highest or lowest weight modules) can occur on the central symmetry line of roots nα=n(α1+α2). Looking at parallel lines just to the side shifted by either adding α1 or α2, let us see what so(2,1)α-modules are generated by the simple root vectors e1 and e2. We find that for i=1,2,4.10 [J3,ei]=-12[h1+h2,ei]=-12αi(h1+h2)ei=12ei,[J+,e1]=-16e112,[J-,e1]=36f2,[J+,e2]=-16e212,[J-,e2]=-36f1

and then, using the Jacobi identity, we get4.11 [J-,[J+,e1]]=-16[f21,e112]=2e1and[J+,[J-,e1]]=32e1

so the Casimir operator on e1 gives4.12 Ωe1=(J3J3-J-J+-J+J-)e1=-134e1.

A similar calculation for e2 (or using the symmetry exchanging subscripts 1 and 2), gives4.13 [J-,[J+,e2]]=-16[f21,e212]=2e2and[J+,[J-,e2]]=32e2

so4.14 Ωe2=-134e2.

This means there are two principal series so(2,1)α-modules generated by these two simple root vectors, one with weights in the line of roots α1+nα, and the other in the line of roots α2+nα. In both cases we find the parameter s such that s(s-1)=-134 to be4.15 s=1±iq2withq=12.

Going back to the center line of symmetry, Fig. 1 shows that for β=2(α1+α2), the β root space has dimension 1, and it is easy to check that e1212=e2112 is a basis for it. To verify that it is a lowest weight vector killed by J- we use that4.16 [f1,e1212]=4e212and[f2,e1212]=4e112

as well as4.17 [f1,e112]=4e12and[f2,e212]=4e12

to compute4.18 [f12,e1212]=[[f1,e1212],f2]+[f1,[f2,e1212]]=4[e212,f2]+4[f1,e112]=0.

Since β(h1+h2)=-4 we have4.19 [J3,e1212]=-12[h1+h2,e1212]=-12β(h1+h2)e1212=2e1212

so the parameter s=2 and4.20 Ωe1212=s(s-1)e1212=2e1212.

Staying on the center line the next space reached by J+ when acting on e1212 is the root space of β=3(α1+α2) which has dimension 3 and so there should be two lowest weight vectors in that space. A basis of the 3(α1+α2) root space is given by4.21 e112212,e121212ande211212.

Acting with J+ on e1212 leads to4.22 J+e1212=16e121212-e211212.

One can check that the following are independent lowest weight vectors for the action of J-4.23 ℓ1=e121212+e211212andℓ2=e112212+3e211212.

The corresponding Casimir eigenvalues are given from their J3 eigenvalues:4.24 J3ℓ1ℓ2=3003ℓ1ℓ2,

so that in both cases Ω=6.

Let us also determine a lowest weight representation off the center line. There must be one in the root space of 3α1+2α2 that has the basis4.25 e11212ande21112.

The lowest weight combination annihilated by J- is4.26 3e11212+4e21112

with J3 eigenvalue s=52 and so Ω=154.

The fractional s-value for a discrete series also appears in4.27 exp(irJ3)e11212=exp(5ir/2)e11212,

so that r∈R/(4πZ), showing that we are dealing with a cover of SO(2, 1).

Decomposition of highest weight Fib-representations

Up until now, we have discussed the decomposition of the KM algebra, gC(A), with respect to an imaginary subalgebra, so(2,1)α. But one also has the decomposition of any highest or lowest weight representation of the KM algebra with respect to the action of that imaginary subalgebra. In that situation only discrete series of so(2,1)α representations can occur in the decomposition, so some of the complications coming from the continuous series do not arise.

As an illustration, we present here two examples of highest weight representations of the rank 2 hyperbolic KM algebra, Fib, Vρ and Vλ1, whose partial weight diagrams are shown in Figs. 3 and 4. In each case, let vλ be a highest weight vector of weight λ in Vλ, so that J+(vλ)=0 and from (2.12) we get4.28 J3·vλ=λ(H(α))α·αvλ=λ·αα·αvλ.

From (B.1) we have4.29 Ω(vλ)=(J3(J3+1)-2J-J+)(vλ)=J3(J3+1)(vλ)=λ·αα·αλ·αα·α+1vλ.

As we did in the previous section, use the positive imaginary root α=α1+α2 of Fib corresponding the the multi-bracket E(α)=e12, giving the formulas in (4.8). Then we have4.30 J3(vλ)=-12(h1+h2)·vλ=-12λ(h1+h2)vλ

and in particular,4.31 J3(vλ1)=-12λ1(h1+h2)vλ1=-12vλ1sos=12

and4.32 J3(vρ)=-12ρ(h1+h2)vρ=-vρsos=1.

Therefore,4.33 Ω(vλ1)=-14vλ1andΩ(vρ)=0.

The fractional value of s for the lowest weight vector vλ1 means that the relevant group acting on this irrep will be a cover of SO(2, 1).Fig. 4 Some weights of irreducible Fib fundamental module Vλ1 with highest weight λ1 showing multiplicities

Examining the (partial) weight diagrams of these two modules in Figs. 3 and 4, we see that the vertical line of weights going down from the highest weight contains the discrete series (B.8) module Ds- for s=12 in Vλ1 and for s=1 in Vρ. In that vertical line of weights {λ1-nα∣0≤n∈Z} for Vλ1 the weight multiplicities shown in Fig. 4 are {1,1,2,6,17,50,151,461} corresponding to 0≤n≤7. Since the weight spaces in Ds- are each 1-dimensional, the “top” summand in the decomposition for that line accounts for the first two 1’s on that list, and decreases each of the following numbers by 1. So the next summand is determined by a highest weight vector (killed by J+) of weight λ=λ1-2α. We do not explicitly compute that highest weight vector here, but it is straightforward to find it as a linear combination of basis vectors in that 2-dimensional weight space of Vλ1. Since α(h1+h2)=-2, we see that the next summand is a discrete series module with s=52, which accounts for one of the dimensions in each of the list of multiplicities, reducing the list to {0,0,0,4,15,48,149,459}. In general, if there are any highest weight vectors in that column with weight λ1-nα, the eigenvalue of J3 on such vectors will be4.34 -12(λ1-nα)(h1+h2)=-2n+12sos=2n+12

is the corresponding value of parameter s for each copy of the discrete series module Ds- at that weight in the decomposition. Clearly this decomposition process continues, giving a 4-dimensional space of highest weight vectors with weight λ1-3α, and thus, four copies of Ds- with s=72, reducing each of the remaining numbers by 4, leaving the list {0,0,0,0,11,44,145,455}. There will be 11 copies of Ds- with s=92, and 33 copies with s=112, and 101 copies with s=132, and 310 copies with s=152, etc. Each column of weights in the diagram has a top weight, and each weight below has a multiplicity, so the process above produces a list of summands consisting of copies of Ds- for values of s determined by the weight. For example, the column to the right of λ1 starts with λ1-α1 and consists of weights of the form {λ1-α1-nα∣0≤n∈Z}. Since α1(h1+h2)=-1, the eigenvalue of J3 on such weight vectors is -12(λ1-α1-nα)(h1+h2)=-(n+1) corresponding to s=n+1. In Fig. 4 we see the list of multiplicities is {1,1,3,9,26,80,246}, giving a list of multiplicities of discrete series modules for that column as differences. The complete decomposition involves doing that process for every column in the weight diagram. The reader is invited to carry out part of this process for Vρ using the multiplicities shown in Fig. 3.

Now we can apply formulas for the action of the group SO(2,1)α and its covers on each of the discrete series summands, Ds-, in the decomposition of any highest weight representation Vλ of gC(A). These can be understood as an exponentiation of the imaginary Lie subalgebra, so(2,1)α, as operators on Vλ. The infinite sums involved can be understood as converging with respect to a Hermitian form on Vλ which has been defined in [1], uniquely determined by (vλ,vλ)=1 and (X(v),w)=(v,X†(w)) for every v,w∈Vλ and every X∈gC(A). Since a highest weight vector, vλ∈Vλ is only determined up to a scalar, the same is true of the form.

Decomposition of Fib with respect to another imaginary subalgebra

Another positive imaginary root of Fib, not in the Weyl group orbit of α1+α2, is α=2α1+3α2, whose root space is 2-dimensional with basis {e21212,e12221}, so we could take either one of these as E(α) in the sense of Sect. 2.2. For this section we choose E(α)=e21212 so that F(α)=-ω(E(α))=f21212, and H(α)=2h1+3h2. We find that α2=-10, and using the Jacobi identity, we compute the bracket4.35 [E(α),F(α)]=[e21212,f21212]=288(2h1+3h2)

so N=288 in formula (2.11). This gives us the basis of so(2,1)α4.36 J3=-2h1+3h210,J+=e21212245,J-=f21212245.

The bracket calculation above used brackets from Sect. 4.2 as well as the following bracket:4.37 [e1212,f1212]=-96(h1+h2).

We begin to find the decomposition of Fib into a direct sum of irreducible so(2,1)α-modules whose weights will be on lines parallel to the line through α. Since (2α1+3α2)(h2)=0, It is easy to see that4.38 0=[J3,h2]=[J±,h2]

so the one-dimensional subspace spanned by v0=h2 is a trivial so(2,1)α-module. Since J3 and v0 are independent, they form a basis for the Cartan subalgebra of Fib, so there cannot be any other irreducible so(2,1)α-modules in the decomposition having a non-trivial intersection with the Cartan. In particular, this means that only discrete series modules (highest or lowest weight modules) can occur on the line of roots nα=n(2α1+3α2). Looking at Fig. 5 we see six parallel lines of roots, three on each side of that line, where unbroken α root strings could contain principal series representations. Those lines are each of the form {μ+nα∣n∈Z} for μ in the set4.39 {-α1,α2,α1+2α2,α1+α2,-α2,α1}

Let us see what principal series so(2,1)α-modules are generated by root vectors in gμ for the roots μ in that set.

For μ=-α1 the basis root vector is f1 and for μ=-α2 the basis root vector is f2 and we find4.40 [J3,f1]=-110[2h1+3h2,f1]=110α1(2h1+3h2)f1=-12f1,[J3,f2]=-110[2h1+3h2,f2]=110α2(2h1+3h2)f1=0,[J+,f1]=-165e2212,[J-,f1]=-1245f121212,[J+,f2]=-145e1212,[J-,f2]=-1245f221212

and then, using the Jacobi identity, we get4.41 [J-,[J+,f1]]=-1720[f21212,e2212]=15f1,

4.42 [J+,[J-,f1]]=-12880[e21212,f121212]=710f1,

4.43 [J-,[J+,f2]]=-1480[f21212,e1212]=35f2,

4.44 [J+,[J-,f2]]=-12880[e21212,f221212]=35f2.

so the Casimir operator on f1 and f2 gives4.45 Ωf1=(J3J3-J-J+-J+J-)f1=14-15-710f1=-1320f1,

4.46 Ωf2=(J3J3-J-J+-J+J-)f2=0-35-35f2=-65f2.

Further calculations give the following values of the Casimir operator on basis vectors in μ root spaces for the other values of μ. For μ=α1 the basis root vector is e1, for μ=α2 the basis root vector is e2, for μ=α1+α2 the basis root vector is e12 and for μ=α1+2α2 the basis root vector is e212. The results are:4.47 Ωe1=-1320e1,Ωe2=-65e2,Ωe12=-1320e12,Ωe212=-1320e212.

Since the Casimir operator values above are all less than -1/4, these principal series representations are not complementary, consistent with our conjecture. Also, note that the Lie algebra automorphism, ω, commutes with the Casimir operator, Ω, so if Ωx=λx then Ω(ω(x))=λω(x). Since ω(ei)=-fi, for i=1,2, this explains why the Ω eigenvalues of ei and fi are equal. Furthermore, the root -α1-α2 is in the same 2α1+3α2 root string as α1+2α2, and ω(e12)=f12 is in the -α1-α2 root space, so that explains why the Ω eigenvalues of e12 and e212 are equal.Fig. 5 Some positive roots of Fib with root multiplicities and six lines parallel to the line through α=2α1+3α2 where principal series might occur for so(2,1)α

In the Fib example above and in Sect. 4.2 no complementary series representations arose among the finitely many continuous series so(2,1)α-representations appearing in the decomposition of Fib. We have performed similar checks for the rank-two hyperbolic algebra with Cartan matrix 2-4-42 as well as the rank-three hyperbolic algebra F studied in [19] and found no complementary series representations. Based on this data we make the following conjecture whose further analysis we leave to future work.

Conjecture: In the decomposition of any hyperbolic KM Lie algebra g with respect to so(2,1)α for positive imaginary root, α, the principal series representations that occur are not complementary series.

A Real Forms of sl(2,C)

We here summarize some very basic facts about two kinds of real forms of the complex Lie algebra sl(2,C) of 2×2 complex matrices with trace 0. This simple 3-dimensional Lie algebra has basisA 1 e=0100,f=0010,h=100-1

with the Lie brackets [e,f]=h, [h,e]=2e and [h,f]=-2f. Its finite-dimensional representations play a crucial role in the representation theory of semi-simple Lie algebras over C, as well as in the definition of Kac–Moody Lie algebras. A real form of a complex Lie algebra gC is a Lie algebra g over R such that g⊗C is isomorphic to gC. For example, the split real form sl(2,R) of sl(2,C) is just the real span of the basis {e,f,h}, and it can be understood as the fixed points in sl(2,C) of the involution τ defined by τ(ae)=a¯e, τ(af)=a¯f and τ(ah)=a¯h for any a∈C.

An important point is that for any A∈GL(2,C), and any real subalgebra a of sl(2,C), the conjugate AaA-1 is clearly a real subalgebra of sl(2,C) isomorphic to a. For example, if a=sl(2,R), then each AaA-1 is a split real form of sl(2,C), but the entries of its matrices can be complex. It is therefore somewhat misleading to speak of “the split real form” of sl(2,C) unless one understands this equivalence of conjugates. The same consideration applies to “the compact real form” of sl(2,C), and means that there can be infinitely many conjugate versions of it. Below we will give some explicit realizations of these two kinds of real forms. Of course, if A is a real matrix, the conjugation AaA-1 amounts to a real change of basis within a which means AaA-1=a.

Two real forms of sl(2,C) can be found as real Lie algebras of 3×3 matrices. By definition,A 2 so(3)={A∈R3×3∣At=-A}

is the real Lie algebra of anti-symmetric matrices which has a basisA 3 M1=0000010-10,M2=001000-100,M3=010-100000

with the Lie brackets [M1,M2]=M3, [M2,M3]=M1 and [M3,M1]=M2. This is clearly isomorphic to the real Lie algebra R3 where the Lie bracket is just the cross product. It is also the real Lie algebra of matrices determined by the standard dot product in R3, X·Y=XtY. The adjoint of A∈R3×3 with respect to this dot product is the unique A∗ such that (AX)·Y=X·(A∗Y) so (AX)tY=XtAtY=X·(AtY) gives A∗=At=-A for A∈so(3). For any real symmetric 3×3 matrix, η, we have a symmetric bilinear form Bη(X,Y)=XtηY and the associated Lie algebra isA 4 so(Bη)={A∈R3×3∣Atη=-ηA}.

With respect to Bη(X,Y) the adjoint of A is determined by Bη(AX,Y)=Bη(X,A∗Y), that is, (AX)tηY=Xtη(A∗Y) so XtAtηY=Xtη(A∗Y) so A∗ is determined by the condition Atη=ηA∗. For A∈so(Bη) this says A∗=-A. For η=I3 this is just so(3), but if we use η=diag(-1,1,1) we getA 5 so(2,1)={A∈R3×3∣Atη=-ηA}

which has a basisA 6 J0=0000010-10,J1=010100000,J2=001000100

with the Lie brackets [J0,J1]=-J2, [J1,J2]=J0 and [J2,J0]=-J1. We wish to show that the real Lie algebras so(2,1) and sl(2,R) are isomorphic. First note that the following real linear combinationsA 7 J3=-J2andJ±=12(J0±J1)

satisfy the bracket relationsA 8 [J3,J±]=±J±and[J+,J-]=-J3.

Then the following elements satisfy the bracket relations for the standard basis of sl(2,R):A 9 H=2J3=-2J2,E=2J+=J0+J1andF=-2J-=-J0+J1.

Going back to the basis (A 3) of so(3), the following complex linear combinationsA 10 M±=i(M1±iM2)andMz=iM3

satisfy the bracket relations:A 11 [Mz,M±]=±M±and[M+,M-]=2Mz.

The slight rescalingA 12 E=M+,F=M-andH=2Mz

gives the bracket relations for the standard basis of sl(2,R). The compact real form so(3) and the split real form sl(2,R) are not isomorphic as real Lie algebras, but they are related by the complex linear map above. A simple real transformation of the basis {M+,M-,Mz} gives the bracket relations of so(2,1). We have the adjoints Mi∗=-Mi for 1≤i≤3, and since the standard complex dot product is sesquilinear, complex coefficients get conjugated in an adjoint, so we getA 13 Mz∗=Mzand(M±)∗=M∓.

B Abstract Algebra: Unitary Representations

This appendix contains a brief review of unitary irreducible representations of the groups SO(2, 1) and SL(2,R) (and further covers) for the readers convenience. Some general references are [44, 55–58], see also the recent thesis [59] that discusses automorphic aspects. In Appendix C, we present functional realizations of these abstract representations.

The starting point is the Lie algebra so(2,1) with commutation relations given by (2.17). We are looking for unitary irreducible representations, that is complex vector spaces admitting an inner product such that the Lie algebra generators satisfy the hermiticity properties in (2.15). The so(2,1) Casimir operator, which commutes with so(2,1), is given byB.1 Ω=J3J3-J+J--J-J+=J3(J3-1)-2J+J-=J3(J3+1)-2J-J+.

Since the representation is irreducible, by Schur’s lemma, Ω is a constant scalar operator on the entire irreducible representation. From these relations we immediately obtainB.2 2J+J-=J3(J3-1)-Ω,2J-J+=J3(J3+1)-Ω

for any irreducible representation.

B.1 Discrete series

Let us first consider the discrete series representations that admit a highest or lowest weight. Since the discussions are fully analogous in the two cases, we restrict mainly to highest weight representations.

Given a highest weight state w0 we have J+w0=0 and we denote its J3-eigenvalue by J3w0=-sw0. We assume that there is an Hermitian inner product (·,·) on the representation space and we normalize the highest weight state to have norm ||w0||=1. From (B.2) we get immediately the Casimir eigenvalue:B.3 0=2(w0,J-J+w0)=s(s-1)-Ω⇒Ω=s(s-1).

The first excited state has norm (J-w0,J-w0)=(w0,[J+,J-]w0)=+s whence we conclude that unitarity requires real s andB.4 s>0.

Continuing in this way, we see that the J3 eigenstates w-n=(J-)nw0 in this representation can be labeled by their J3 eigenvalues {|-s-n⟩} with 0≤n∈Z. Since J3 is self-adjoint, eigenvectors with distinct eigenvalues are orthogonal. Using (s+n)(s+n-1)-s(s-1)=n(2s-1+n) we find thatB.5 ((J-)nw0,(J-)nw0)=12[n(2s-1+n)]((J-)n-1w0,(J-)n-1w0)

(for n≥1), so all eigenstates have positive norm squaredB.6 ||w-n||2=∏k=1n12k(2s-1+k)=n!2n∏k=1n(2s-1+k).

Hence the representation is unitary. Similarly, for lowest weight representations we have J-w0=0 and J3w0=+sw0 and the J3 eigenstates wn=(J+)nw0 are {|s+n⟩} with 0≤n∈Z. We thus see that for both highest and lowest weight representations unitarity is implied by (B.4).

There are at this point no further restrictions besides s>0; in particular, there is no a priori reason to exclude non-integer values of s>0. It is only for single-valued representations of SO(2, 1) that one must have exp(2πiJ3)=1, in which case s must be a positive integer. Other rational values can occur for covers of SO(2, 1) and, we will see that, in fact, all values s∈Q+ can appear in the KM algebra.

Note that the vectors w±n are not normalized, so we adopt the notationB.7 vn:=wn||wn||

for the rescaled and normalized vectors which give an orthonormal basis of the relevant representation space. We also define the following notation for these discrete representation spaces:B.8 Ds±=⨁0≤n∈ZCv±n.

Since all scalars are real in the formulas for the actions of J3 and J± on these representations, we could have used Rv±n in the above summation and gotten a real Hilbert space of so(2,1) in this basis. Changing to the standard basis using the formulas of Appendix A shows, however, that we should really consider the representation space as a complex Hilbert space. The Casimir Ω=s(s-1) can be negative (but ≥-1/4) for general discrete series representations of the cover.

B.2 Principal series representations

Next we consider a continuous irreducible representation, Ps, labeled by a complex parameter s such that the Casimir Ω acts as the scalar s(s-1), and for which the spectrum of J3 is unbounded from both above and below. Hermiticity of J3 implies that its spectrum is real and irreducibility that all weight multiplicities are equal to one. Assume there exists a ‘minimal’ eigenvector wp in Ps satisfying J3wp=pwp with 0≤p<1. Since we do not require the representations to be single-valued there does not need to be a spherical vector6 and normalized to unity, viz.B.9 ||wp||2=1.

Since J3 is Hermitian (2.15), p is real, and so will be all other eigenvalues. It follows from the bracket relations (2.17) that wp±1=J±wp are also eigenvectors for J3, but they are no longer normalized to unity. Applying the raising and lowering operators repeatedly, we get a J3 eigenbasis starting from wpB.10 wp+n:=J+nwp,wp-n:=J-nwpforn∈N

with spectrum of eigenvalues {p+n∣n∈Z}. The norms of these eigenvectors can be calculated inductively using J3wp=pwp. Using (B.2) one easily proves the following recursion relationB.11 ||wp+n||2=J+nwp,J+nwp=J+n-1wp,(J-J+)J+n-1wp=12[(p+n)(p+n-1)-s(s-1)]||wp+n-1||2

Similarly, we findB.12 ||wp-n||2=12[(p-n)(p-n-1)-s(s-1)]||wp-n+1||2.

In fact, it is sufficient to just prove that for any J3 eigenvector, wx with eigenvalue, x,B.13 ||wx±1||2=12(x(x±1)-s(s-1))||wx||2

(this relation follows directly from (B.2)). Therefore, for unitarity we must have (p+n)(p+n-1)>s(s-1) for all n∈Z, so if x=p+n, we may choose n such that 1>x≥0 so 0>x-1≥-1 so 0≥x(x-1)>s(s-1). The minimum value of the parabola y=x(x-1) is -1/4, so we will be sure of this condition for all n and p when Ω=s(s-1)<-1/4. Then all norms are positive and we obtain a unitary representation. Below we write Ω for its value on Ps.

Among the continuous representations one further distinguishes between principal series for which Ω≤-14, and complementary series for which -14<Ω<0. The distinction arises since -1/4 is the minimum of the parabola s(s-1) for real s; the unitary principal series requires complex s. The relation above gives the following closed formula for the norms of the eigenvectors:B.14 ||wp±n||2=12n∏k=0n-1[(p±k)(p±(k+1)-Ω]

Writing Ω=s(s-1) also for the principal series representations we have s=12(1+iq) with q∈R, thus Ω=-14(1+q2)=-ss¯. The parameter q is determined up to sign and its value depends on the example.

Defining the orthonormal basis as in (B.7) we denote denote the corresponding representation space byB.15 Ps=⨁n∈ZCvp+n.

B.3 Finite-dimensional representations

For completeness we also mention finite-dimensional irreducible modules of sl(2,R), see for instance [60]. These will be denoted by V(m) for the (m+1)-dimensional module and the only unitary case is the trivial representation V(0). The non-unitary ones play a role in the decomposition with respect to real roots α, discussed for example in Sect. 4.1.

The module V(m) can be characterised by having a highest weight vector vm with eigenvalue m/2 under J3 and satisfying J+vm=0. The Casimir eigenvalue on such an irreducible representation is Ω=14m(m+2) in the normalization (B.1). In particular, for the adjoint representation with m=2, the Casimir is Ω=2. The Casimir spectrum on the non-unitary representations V(m) overlaps with that of the unitary representations.

C Hilbert Space Realizations

For convenience we here summarize the known Hilbert (function) space realizations of the unitary representations of SL(2,R) and of its covers.

C.1 Functional representations of SL(2,R)

For all the discrete series representations of the group SL(2,R) on Hilbert spaces of complex-valued square integrable functions, G(z), the left action is defined byC.1 G(z)→(S·G)(z):=Gaz+bcz+d(cz+d)-2s,forS-1=abcd∈SL(2,R)

with integer s. For the continuous series representations the factor (cz+d)-2s is replaced by |cz+d|-2s. With St taken from each of the one-parameter subgroups, {exp(te)∣t∈R}, {exp(tf)∣t∈R} and {exp(th)∣t∈R}, for fixed z, the linear term in the Taylor expansion of L(t)=(St·G)(z) obtained from limt→0L′(t), gives the differential operators representing the sl(2,R) basis vectors (A 1) to beC.2 E=-ddz,F=z2ddz+2sz,H=-2zddz-2s

where we use capital letters to distinguish these operators from the abstract Lie algebra elements they represent. The basis vectors7C.3 j3=i2(e-f),j+=122(-i(e+f)+h),j-=122(-i(e+f)-h)

in sl(2,C) satisfy the (2.17) Lie brackets [j3,j±]=±j± and [j+,j-]=-j3 but also satisfy X†η+ηX=0 for η=0110, which means this real Lie algebra is su(1,1). The differential operators corresponding to these complex linear combinations of the operators in (C.2) give us an explicit realization of the algebra (2.17) by the differential operatorsC.4 J3=-i2(1+z2)ddz-isz,J+=122-i(z-i)2ddz-2s(1+iz),J-=122-i(z+i)2ddz+2s(1-iz).

One still has to specify the Hilbert space on which these operators act, and the scalar product (·,·) that defines the norm. The variable z is real for the continuous representations and complex for the discrete representations (in which case we write z=x+iy).

For all realizations we insist on the hermiticity propertiesC.5 (J3)†=J3,(J±)†=J∓

where hermiticity is defined with respect to the given scalar product. From these expressions it is straighforward to compute the CasimirC.6 Ω=-J+J--J-J++J3J3=s(s-1)

In terms of the operators (2.17) this action corresponds to the exponentialC.7 exp(iwJ++iw¯J-+irJ3)

with appropriate parameters w=u+iv∈C and r∈R determined from the matrix S. Note that if we use the formulas (C.3), the expressionC.8 iwj++iw¯j-+irj3=12(u(e+f)-vh)+12r(e-f)

is a real linear combination of {e,f,h}, so it is in sl(2,R) and its exponential is in SL(2,R). We note that with respect to the Kac bilinear form we have that H†=-H, E†=-E and F†=-F.

The map S∈SL(2,R) is represented with respect to an orthonormal basis of functions, vn, by a unitary matrixC.9 S·vn=∑m∈ZUmn(S)vm.

Because the Hilbert space consists of complex valued functions the infinite matrix Umn is necessarily complex. Unitarity means thatC.10 ∑k∈ZUkm∗Ukn=δmn

so in particular all sums that arise are manifestly convergent.

C.1.1 Discrete series

As explained above, the function space realizations of discrete series representations correspond to lowest or highest weight representations, depending on whether there is a ground state killed by J- or by J+, respectively. In the first case, z is a complex variable z=x+iy and the functions f(z), g(z) are holomorphic in the upper half-plane H. The scalar product isC.11 (f,g):=∫Hdxdyy2s-2f(z)¯g(z)

Importantly, the operators (C.4) are Hermitian with respect to (C.11) only for s>1 because only then the boundary term arising from integration by parts vanishes.

The normalized ground state φ0 of a lowest weight representation satisfying J-φ0=0 and J3φ0=sφ0 is given byC.12 φ0(z)=2s-1π22s-11(z+i)2s

becauseC.13 ∫x∈R∫y>0dxdyy2s-2(x2+(y+1)2)-2s=41-2sπ2s-1

for s>12, and the excited state φn with J3-eigenvalue n+s isC.14 φn(z)=(J+)nφ0(z)=An(z-i)n(z+i)2s+n

withC.15 A0=2s-1π22s-1andAn=A02n∏k=0n-1(2s+k)=A02nΓ(2s+n)Γ(2s)

because we have the recursionC.16 ||φn||2=12n(2s-1+n)||φn-1||2=2-nΓ(n+1)Γ(2s+n)Γ(2s)||φ0||2.

The functions φn are not normalized.

Similar definitions apply for highest weight representations with a ground state such that J+φ0=0 and J3φ0=-sφ0, where the integral (C.11) is now to be performed over the lower half-plane, and we write φ-n=(J-)nφ0(z) for n∈N.

We use the following notation for these two kinds of function spaces which are each irreducible so(2,1) representation spacesC.17 Ds±=⨁n≥0Cφ±n(z)

but C could be replaced by R since all the coefficients An are real. We wish to define a Lie algebra module isomorphism which is also a (real or complex) Hilbert space isometryC.18 Φs±:Ds±→Ds±.

Using the orthonormal basis of Ds± in (B.8), define Φs±(v0)=φ0(z) and require thatC.19 Φs±((J±)nv0)=(J±)nΦs±(v0)=(J±)nφ0(z).

For completeness let us mention that the functional realization of the discrete series representation can be equivalently done on the Poincaré disk |w|<1 by means of the standard Cayley transformationC.20 w=z-iz+i⟺z=i1+w1-w

mapping the upper half plane (with coordinate z) to the interior of the unit disk (with coordinate w). The differential operators (C.4) can be easily converted using the Cayley transformation, leading for example toC.21 J3=wddw+s1+w1-w.

The integral (C.11) becomes for w=u+iv and s a positive integerC.22 (f,g)=∫|w|<1dudv(1-|w|)2s-2(1-w)2s(1-w¯)2sf(w)¯g(w).

The J3-eigenfunctions φn in (C.14) are then replaced by functions proportional to (1-w)2swn. In the inner product (C.22) such functions have the normC.23 ∫|w|<1dudv(1-|w|2)2s-2|w|2n=π2s-1Γ(2s)Γ(n+1)Γ(2s+n)

from a standard representation of the Euler beta function. When discussing covers of SO(2, 1) this formula must be analytically continued to non-integer values of s>0 and corresponding Hilbert spaces can be defined [56], see also Sect. C.2. We also note that by omitting the factor (2s-1)-1 one can redefine the norm to be finite and positive for all s>0 [56].

C.1.2 Principal series representations

A concrete realization of the principal series representations of SL(2,R) is provided by the space L2(R) of complex valued square integrable functions over the real line R (so x is now real) with the scalar productC.24 (f,g):=∫-∞∞f(x)¯g(x)dx

with the above operator realizations. Keeping in mind that ((x+a)2ddx)†=-(x+a¯)2ddx-2(x+a¯), with an extra linear term, one sees that the hermiticity properties (C.5) are only satisfied ifC.25 s=12(1+iq)⇒Ω=s(s-1)=-14(1+q2)≤-14

with q∈R. The J3-eigenfunctions are given byC.26 J3φm(x)=mφm(x)⇒φm(x)=1π1(1+x2)si-xi+xm.

Note that we are here considering representations of SL(2,R) so that m∈Z. Furthermore, ||φm(x)||=1 becauseC.27 ∫-∞∞φm(x)¯φm(x)dx=1π∫-∞∞dx1+x2=1.

To solve the differential equation J3φm(x)=mφm(x) we have also used the identityC.28 exp(2iarctan(x))=i-xi+x.

When we use the differential operators representing J± from (C.4), we getC.29 J±φm(x)=-(m±s)2φm±1(x)

It follows that for n≥1 we haveC.30 (J+)nφm(x)=An,sφm+n(x)and(J-)nφm(x)=A-n,sφm-n(x)

whereC.31 A±n,s=(-1)n(m±s)(m±1±s)⋯(m±(n-1)±s)(2)n

For 0≤p<1 denote by Πs(p)=⨁n∈ZCφp+n the irreducible representation space of functions (with m replaced by p+n in (C.26)) we have just found for the Lie algebra so(2,1) with operator basis given in (C.4). Using the basis {wp+n∣n∈Z} of Ps in (B.14), we can define a module isomorphism Φs:Ps→Πs(p) which is also a Hilbert space isometry, as follows. Begin by setting Φs(wp)=φp. Then letC.32 Φs(wp±1)=Φs(J±wp)=J±Φs(wp)=J±φp=-(p±s)2φp±1,Φs(wp±2)=Φs(J±wp±1)=J±Φs(wp±1)=(p±s)2J±φp±1=(p±s)2(p±1±s)2φp±2

so in general, for any n∈N, letC.33 Φs(wp±n)=Φs((J±)nwp)=(J±)nΦs(wp)=(J±)nφp=A±n,sφp±n.

To check that this is a Hilbert space isometry we need to see thatC.34 ||wp±n||2=||A±n,sφp±n||2=A±n,sA±n,s¯=12n∏k=0n-1(p±k±s)(p±k±s¯)

matches the answer in (B.14). For each k we haveC.35 (p±k±s)(p±k±s¯)=(p±k)2±(p±k)(s+s¯)+ss¯=(p±k)2±(p±k)+ss¯

does matchC.36 ((p±k)(p±(k+1)-Ω)=((p±k)(p±k±1)+ss¯).

C.1.3 Complementary series

Finally we turn to the complementary series with real 0<s<1. In this case we have Casimir eigenvalues -14<Ω<0. These are discussed (in the compact SU(1, 1) picture) in [61], see also [57, 58, 62] for further discussions. Even though we have not found any Kac–Moody algebra where they arise, we give some details here for completeness.

A model is provided by complex-valued functions on the real line. The inner product is given byC.37 (f,g)=∫R2f(x)¯g(y)|x-y|2s-2dxdy

and the Hilbert space is the completion with respect to this norm. We will show this in the appendix assuming f,g∈ L1(R)∩ L2(R). We will now show that we haveconvergence

positive definiteness

operators have right hermiticity properties

providedC.38 0<s<1⇒-14<Ω<0

Because f and g are in L1(R)∩ L2(R) we can write them as Fourier integralsC.39 f(x)=∫Ra(k)e2πikxdk,g(x)=∫Rb(k)e2πikxdk.

Then, after changing variablesC.40 v=12(x-y),u=12(x+y),

one can rewrite the integralC.41 (f,g)α=∫R2f(x)¯g(y)|x-y|αdxdy=∫R2∫R2a(k)¯b(ℓ)e2πi(kx-ℓy)|x-y|αdxdydkdℓ=2α+1∫R2∫R2a(k)¯b(ℓ)e2πi(k(u+v)-ℓ(u-v))|v|αdudvdkdℓ=∫R∫Ra(k)¯b(k)e2πikv|v|αdvdk=2-απ-1-αΓ(1+α)sinπ|α|2∫|k|-1-αa(k)¯b(k)dk,

where we usedC.42 ∫Re2πinv|v|αdv=-2-α|n|-1-απ-1-αΓ(1+α)sinπα2.

For convergence at small k and large v we must haveC.43 -1<Re(α)<0

where the lower bound comes from the pole of the Γ-function at 0, while the upper limit comes from requiring convergence of the integral over Fourier coefficients in (C.41). In particular, it follows from these formulas that (f,f)α converges for this choice and is positive definite.

The admissible values for α in terms of s can be fixed by demanding that the operators be Hermitian. To this aim consider the adjunction of the operators (C.4) with respect to this inner product, assuming that s∈R. For J3 one gets:C.44 (f,J3g)α=∫R2f(x)¯-i2(1+y2)ddyg(y)-isyg(y)|x-y|αdxdy=∫R2f(x)¯g(y)i2(1+y2)ddy|x-y|α+iy(1-s)|x-y|αdxdy.

Compute similarly, using ddx|x-y|α=-ddy|x-y|α, thatC.45 (J3f,g)α=∫R2i2(1+x2)ddxf(x)¯+isxf(x)¯g(y)|x-y|αdxdy=∫R2f(x)¯g(y)i2(1+x2)ddy|x-y|α-ix(1-s)|x-y|αdxdy.

Therefore the difference isC.46 (f,J3g)α-(J3f,g)α=∫R2f(x)¯g(y)i2(y2-x2)ddy|x-y|α+i(1-s)(x+y)|x-y|αdxdy.

Let us focus on the term in parentheses:C.47 i2(y2-x2)ddy|x-y|α+i(1-s)(x+y)|x-y|α=i2α(x-y)(x+y)|x-y|α-1sgn(x-y)+i(1-s)(x+y)|x-y|α=i2α+i(1-s)(x+y)|x-y|α,

such that this vanishes whenC.48 α=2s-2,

making J3 Hermitian with respect to this inner product.

The condition -1<Re(α)<0 translates for real s intoC.49 12<s<1,

which is half of the interval 0<s<1 and the other half is covered by analytic continuation using the functional relation between s and 1-s.

We also demonstrate the other Hermiticity relation. For J+ one finds:C.50 (f,J+g)α=122∫R2f(x)¯-i(y-i)2ddyg(y)-2s(1+iy)g(y)|x-y|αdxdy=122∫R2f(x)¯g(y)i(y-i)2ddy|x-y|α-2i(s-1)(y-i)|x-y|αdxdy.

andC.51 (J-f,g)α=122∫R2i(x-i)2ddxf(x)¯+2s(1+ix)f(x)¯g(y)|x-y|αdxdy=122∫R2f(x)¯g(y)i(x-i)2ddy|x-y|α+2i(s-1)(x-i)|x-y|αdxdy.

Therefore the difference isC.52 (f,J+g)α-(J-f,g)α=122∫R2f(x)¯g(y)(-i(x-y)(x+y-2i)ddy|x-y|α-2i(s-1)(x+y-2i)|x-y|α)dxdy.

The term in parentheses becomesC.53 -i(x-y)(x+y-2i)ddy|x-y|α-2i(s-1)(x+y-2i)|x-y|α=iα(x-y)(x+y-2i)|x-y|α-1sgn(x-y)-2i(s-1)(x+y-2i)|x-y|α=i(x+y-2i)|x-y|α(α-2(s-1)),

so that for α=2s-2 one has (J+)†=J- as required and consistent with (C.48).

C.2 Representations of covers

The functional realization on covers proceeds by a very similar method. We explain this in the case of the discrete series on an N-fold cover G~ of SL(2,R), meaning a 2N-fold cover of SO(2, 1). Having an N-fold cover, means that the group G~ formally consists of pairs (S,ζ) with S∈SL(2,R) and ζ is an element of the cyclic group of order N that can be represented by Nth roots of unity. The product on such pairs involves a cocycle that defines the extension. We will not require the precise form of this cocycle (see e.g. [59]) since its definition will be implicit in our construction.

LetC.54 S-1=abcd∈SL(2,R)

and choose an Nth root μ of the linear function cz+d where z is in the upper half plane. We demand that μ is holomorphic on the upper half plane, and by construction μ(z)N=cz+d. Such a function is given up the choice of a root of unity and there are N such roots. Therefore the pairs (S,μ) are what is needed to describe the N-fold cover of SL(2,R).

The action of a such a pair (S,μ) on holomorphic functions G(z) on the upper half plane isC.55 G(z)↦((S,μ)·G)(z)=1μ(z)2sGaz+bcz+d

and generalizes (C.1) to the case when s∈12NN. The infinitesimal action of the Lie algebra sl(2,R) in terms of differential operators is unchanged from this definition, in agreement with the fact that the differential operators (C.2) satisfy the sl(2,R) Lie algebra for any complex s.

From this action one deduces the product on the covering groupC.56 (S1,μ1)·(S2,μ2)=(S1S2,(μ1∘S2)μ2),

where the second entry denotes the holomorphic functionC.57 (μ1∘S2)μ2(z)=μ1a2z+b2c2z+d2μ2(z)

that intertwines the product of the roots with the action of SL(2,R) on the upper half plane.

The Hilbert space of the discrete series of G~ consists of all holomorphic functions with finite norm with respect to the (analytically continued) norm (C.11). For fractional 0<s<1 this represents a more stringent requirement than holomorphicity on the upper half but has been discussed in detail in the literature, see for example [56]. In particular, the hermiticity properties (2.15) and unitarity of the representation can be maintained.

For the principal series the holomorphic functions on the upper half plane are taken to the boundary real line.

Acknowledgements

We are grateful to Lisa Carbone, Thibault Damour, Walter Freyn, Benedikt König, Robin Lautenbacher and Timothée Marquis for discussions and the referees for useful comments. AF gratefully acknowledges support from his department and from the Max Planck Institute for Gravitational Physics during several visits related to this work. This work was supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740209).

Funding

Open Access funding enabled and organized by Projekt DEAL. Open Access funding enabled and organized by Projekt DEAL.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

1 See however [11] for a discussion of some aspects of timelike imaginary roots, and [12] for partial evidence associating imaginary roots to higher order corrections in M theory.

2 We note that, according to the results of [30], generators associated with root spaces of different positive imaginary roots generate a free Lie algebra under some mild assumptions.

3 We could also take linear combinations of different elements of gα, but that would not affect the main argument.

4 The fundamental group of SL(2,R) is also Z. Bargmann’s classification [44] only addresses representations of SL(2,R), not of its higher covers.

5 As is well known the usual operator norm does not exist for unbounded operators. However, this is not in contradiction with (2.18) because the norm of the Lie algebra elements induced by the bilinear form (2.5) is different from the standard operator norm.

6 For p=0 the vector wp is called a ‘spherical vector’.

7 This change of basis is similar to the change to the so-called ‘compact basis’ that appears for example in [58]. However, there are a few sign differences and a rescaling by a factor 2 to obtain (2.17).

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