Linear phase space deformations with angular momentum symmetry

Motivated by the work of Leznov--Mostovoy, we classify the linear deformations of standard $2n$-dimensional phase space that preserve the obvious symplectic $\mathfrak{o}(n)$-symmetry. As a consequence, we describe standard phase space, as well as $T^{*}S^{n}$ and $T^{*}\mathbb{H}^{n}$ with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in $\mathbb{R}^{n+2}$.


Introduction and statement of the problem
The notions of momentum maps and symplectic reduction provide a very convenient formulation of integrability in classical mechanics [GS84,GS06]. A standard example is the integrability of central potential Hamiltonians such as the n-dimensional Kepler and harmonic oscillator problems, which can be understood in terms of dynamical (i.e. symplectic) symmetries for the groups SO(n + 1) and SU(n) [Len24,Fra65]. These principles can be exploited in more general contexts, such as the analogous integrability of the Kepler and harmonic oscillator problems on the round n-sphere studied by Higgs [Hig79]. The standard formulation of these problems can then be recovered from a limiting process, if we interpret the sectional curvature of the n-sphere as a deformation parameter yielding a commutative linear deformation of the Poisson structure in standard phase space.
We will address a natural generalization of the previous idea. A precise formulation relies on three facts (two of which are proved in appendix A): (i) (R n ⊕ R n , ω = n i=1 dx i ∧ dp i ) is symplectomorphic to a connected component O + 2n of a coadjoint orbit O 2n of G n = O(n) ⋉ H n , where H n is the 2n + 1-dimensional Heisenberg group. The action of O(n) on H n is the standard one on its R n ⊕ R n subgroup and trivial on the central extension element. (ii) g n = o(n)⋉h n is the orthogonal Lie algebra associated to a quadratic form Q 0 on R n+2 of isotropy index 2 and signature (n, 0), and O 2n is 2010 Mathematics Subject Classification. Primary 17B08, 17B56, 17B80; Secondary 53D20, 14H70. 1 identified with a Zariski open set in Gr 2 R n+2 , the Grassmannian of oriented 2-planes in R n+2 . (iii) (T * S n , ω) and (T * H n , ω) with their standard symplectic forms are symplectomorphic to coadjoint orbits (a connected component in the latter case) for deformations of g n that are respectively isomorphic to e(n + 1) = o(n + 1) ⋉ R n+1 and e(n, 1) = o(n, 1) ⋉ R n+1 . The deformation parameter is interpreted as the sectional curvature of an underlying configuration space (remark 8). The problem that we pose is the following: to classify all deformations of the Lie algebra g n and the subsequent coadjoint orbits that specialize to the previous examples. 1 At a technical level, the problem is equivalent to the understanding of the Lie algebra cohomology H 2 (g n , g n ). In virtue of (ii), the problem is related to the study of equivalence classes of deformations of a degenerate quadratic form in R n+2 of isotropy index 2 and signature (n, 0). Our treatment emphasizes the physical and geometric features of the problem while solving it, and is motivated by the work of Leznov-Mostovoy [LM03]. They studied the Kepler problem on a 3-dimensional family of deformations of standard phase space in the special case n = 3, interpreted as commutative deformations of the standard Poisson structure on C ∞ R 3 ⊕ R 3 . The main result of this article is a proof that the generators of the Leznov-Mostovoy deformations determine three cocycles spanning H 2 (g n , g n ) for arbitrary n ≥ 3.
Theorem 1. Let n ≥ 3. The space of infinitesimal deformations of g C n is three-dimensional. Every infinitesimal deformation is integrable, and the generic deformation is isomorphic to o(n + 2, C).
However, the induced linear deformations of g C n are not independent (corollary 1), marking a subtle difference between the infinitesimal and global pictures. The effective family of linear deformations of g C n turns out to be at most two-dimensional.
The phase space deformations that we will study are natural generalizations of the phase space of a manifold of constant sectional curvature, and contain the latter as particular cases (remark 8). Such deformations can be understood geometrically in terms of the Grassmannian of oriented planes in R n+2 , relative to a family of deformations of a degenerate quadratic form of signature (n, 0) and isotropy index 2. In particular, there is a generic regime of deformation parameters for which the induced symplectic manifold is compact. This leads to the possibility of extending the study of classical and quantum integrable systems on spaces of constant curvature to the most general phase space deformations that preserve a notion of angular momentum symmetry, by means of the study of the geometry of suitable momentum maps. We plan to address such a problem in the future.
The work is organized as follows. Section 2 is dedicated to presenting a proof of theorem 1 following an application of the Hochschild-Serre spectral sequence. An argument for an alternative proof in terms of the geometry of quadratic forms is given in remark 1. The rest of the article is a series of applications of theorem 1. Section 3 describes the general family O 2n (ε ε ε) of coadjoint orbits, induced by deformations of g n , that correspond to the deformations of standard phase space (corollary 3). Section 4 describes some geometric structures in O 2n (ε ε ε) corresponding to the induced deformations of the position and momentum polarizations in phase space, and the Euclidean group momentum map corresponding to the free-motion Hamiltonian.
2. Deformations of the Lie algebra o(n) ⋉ h n It will be convenient to work with complex Lie algebras, since the real deformations of g n can be regarded as all possible real forms of a complex deformation of its complexification. The infinitesimal deformations of a complex Lie algebra g are described by the Lie algebra cohomology space H 2 (g, g) with respect to the adjoint representation [CE48,Ger64,LN67,Fia85]. We are interested in the case when g is a semidirect product of a semisimple Lie subalgebra k ⊂ g and an ideal h ⊂ g, i.e. g = k ⋉ h. Hence g is also a module for k and h. Let E p,q 2 = H p (k, H q (h, g)). It follows from Whitehead's lemma that E 1,1 2 = 0 and E 2,0 2 = 0. The Hochschild-Serre spectral sequence [HS53,Wei95] collapses from the E 2 -term for p + q = 2. Hence, restriction induces the isomorphism [HS53,theorem 13]). In order to describe the deformations of g C n , we will first consider the case k = o(n, C), h = C n so g = e C n = o(n, C) ⋉ C n , the complexification of the Euclidean Lie algebra in dimension n ≥ 3. In terms of a canonical basis {e i , l jk }, e C n is defined by the commutation relations [e i , e j ] = 0, [l ij , e k ] = δ ik e j −δ jk e i , [l ij , l kl ] = δ ik l jl +δ jl l ik −δ il l jk −δ jk l il .
We will now describe the space H 2 C n , e C  for some linear map l : C n → e C n . A 2-cocycle f is called invariant if ∀g ∈ o(n, C), , ·) − f (·, [g, ·]) = a coboundary.
Lemma 1. The space of infinitesimal deformations of the complexification of the Euclidean Lie algebra e C n , n ≥ 3, is one-dimensional and generated by the invariant 2-cocycle f (e i , e j ) = l ij .
Proof. Any 2-coboundary necessarily takes values in C n . Hence, any two invariant 2-cocycles taking values in o(n, C) are cohomologous if and only if they are equal.
That f is an invariant element in Z 2 C n , e C n under the o(n, C)-action is a routine computation. By the previous remark, the cohomology class of f is nontrivial.
It remains to show that up to a constant, this is the only possibility for f . First, let us assume that ∀i, j, f (e i , e j ) ∈ o(n, C), then the invariance condition (2.2) implies that for any g ∈ o(n, C) such that [g, e i ] = [g, e j ] = 0, [g, f (e i , e j )] is identically 0. From the structure of o(n, C), it follows that where the left hand side denotes the subalgebra of o(n, C) annihilated by ker ad e i | o(n,C) ∩ ker ad e j | o(n,C) .
Thus, f (e i , e j ) = cl ij for some c ∈ C is the only invariant cocycle with image lying in o(n, C). Now, let us assume that f is an invariant cocycle with f (e i , e j ) ∈ C n , for some i, j. We claim that the same holds for any other value of f . Indeed, for any k = i, j, the invariance of f under l ik implies that [l ki , f (e i , e j )] − f (e k , e j ) is equal to a 2-coboundary evaluated at e i ∧ e j , and therefore f (e k , e j ) ∈ C n . Moreover, for any l = i, j, k, a similar argument on the invariance of f under l jl shows that f (e k , e l ) ∈ C n . In conclusion, the image of an invariant 2-cocycle either lies fully in o(n, C) or in C n .
To conclude the classification, let us assume that f : 2 C n → C n is an arbitrary linear map. We claim that f is a 2-coboundary. To see this, observe that the linear space of such maps and the space of linear maps l : C n → o(n, C) are equidimensional, and the coboundary map l → [l(·), ·] + [·, l(·)] is a linear map between these spaces. A basis for the space of linear maps l : C n → o(n, C) is given by the set {l ijk (e m ) = δ im l jk } j<k . Since and the collection of the latter maps is obviously linearly independent, we conclude that the coboundary map is an isomorphism. Therefore, all linear maps f : 2 C n → C n are 2-coboundaries, and any invariant 2-cocycle taking values in C n is necessarily trivial in cohomology.
To conclude the proof, it is a routine computation to verify that the bracket deformation of e C n determined by any choice of nontrivial invariant 2-cocycle, [e i , e j ] ε = εl ij , ε ∈ C, and with the rest of brackets kept the same, satisfies the Jacobi identity and defines a Lie algebra e C n (ε) that is isomorphic to o(n + 1, C) if ε = 0. In particular, the specialization ε ∈ R leads to the real forms o(n + 1) for ε > 0, and o(n, 1) for ε < 0.
Proof of theorem 1. The Heisenberg Lie algebra is defined as a central extension of the abelian Lie algebra C 2n , and o(n, C) acts trivially in the extension term. This implies that g C n = o(n, C) ⋉ h C n is a central extension of the Lie algebra o(n, C) ⋉ C 2n , and moreover, we have the following commutative diagram: In other words, there is an isomorphism It will be convenient to consider a canonical basis for h C n , {e 1 , . . . , e 2n , I}, with commutation relations [e i , e n+j ] = δ ij I, i, j = 1, . . . , n.
Any invariant cocycle f : 2 h C n → g C n is cohomologous to a cocycle f ′ satisfying f (e i , I) = 0 ∀i. To see this, notice that the cocycle condition satisfies (dl)(e i , I) = f (e i , I), and the claim follows. Moreover, a similar argument shows that any invariant cocycle is cohomologous to a cocycle with image in o(n, C) ⋉ C 2n . Indeed, assume that f (e i , e j ) = c ij I. Then the linear map l : h C n → g C n given by l(e k ) = δ ik c ij e j−n if j > n (or = −δ ik c ij e j+n if j ≤ n) satisfies (dl)(e i , e j ) = c ij I and zero otherwise. This concludes the proof of the isomorphism in cohomology. Thus, it is enough to understand the infinitesimal deformations of o(n, C) ⋉ C 2n .
Lemma 1 can be used in the classification of the independent invariant 2-cocycles f : C 2n → o(n, C) ⋉ C 2n . A similar argument as in the proof of lemma 1 shows that an invariant 2-cocycle which is not a 2-coboundary necessarily has image in o(n, C). When thought of as a o(n, C)-module, the subalgebra C 2n splits as a direct sum C n 1 ⊕C n 2 of invariant o(n, C)-subspaces. There is an induced splitting 2 C 2n = 2 C n 1 ⊕(C n 1 ⊗ C n 2 )⊕ 2 C n 2 , and any invariant 2-cocycle can be decomposed into three different components. The classification problem is then reduced to the classification of invariant 2-cocycles on each component. It follows from lemma 1 that the restriction to C n 1 and C n 2 determines two nontrivial one-dimensional spaces of invariant 2-cocycles, spanned by f 2 (e n+i , e n+j ) = l ij for i, j ≤ n, and zero otherwise.
These are the only possibilities that are supported in the invariant subspaces 2 C n 1 and 2 C n 2 . The remaining possibility would consist of an invariant 2-cocycle supported in C n 1 ⊗ C n 2 . There is an obvious choice, namely (2.5) f 3 (e i , e n+j ) = l ij for i, j ≤ n, and zero otherwise.
A similar argument as in the proof of lemma 1 shows that any other invariant 2-cocycle supported in C n 1 ⊗ C n 2 must be a multiple of f 3 . Therefore, any nontrivial invariant 2-cocycle is a linear combination of f 1 , f 2 and f 3 .
Let ε 1 , ε 2 , ε 3 ∈ C. The lift to g n of the Lie bracket deformations of o(n, C) ⋉ C 2n induced by the previous cocycles is determined by and additionally, while the remaining basis elements' Lie brackets are unchanged. It is a routine computation to verify that these Lie bracket deformations satisfy the Jacobi identity. Hence they integrate to a three-dimensional family of deformations g C n (ε 1 , ε 2 , ε 3 ). When ε 1 , ε 2 = 0, and ε 2 3 = ε 1 ε 2 , there is an isomorphism g C n (ε 1 , ε 2 , ε 3 ) ∼ = g C n (1, 1, 0) = o(n + 2, C). The details on this isomorphism and the full classification of deformations are described in proposition 1.
Remark 1. There is yet another way to describe the linear deformations of g C n , and in particular the invariant 2-cocycles generating H 2 g C n , g C n , in terms of the geometry of the of the quadratic space C n+2 , Q C 0 . Any deformation Q C ε ε ε of the quadratic form Q C 0 induces a deformation of the orthogonal Lie algebra o C n+2 , Q C 0 ∼ = g C n , in such a way that if a new quadratic formQ C ε ε ε is induced by an orthogonal transformation of Q C ε ε ε , the corresponding deformations g C n (ε ε ε) andg C n (ε ε ε) are isomorphic.
Consider the canonical basis {v 1 , . . . , v n+2 } of C n+2 , together with the correspondence It is straightforward to verify that the deformations of the corresponding bilinear form of Q 0 along the totally isotropic plane W = Span{v n+1 , v n+2 }, parametrized as induce the deformation g C n (ε 1 , ε 2 , ε 3 ). Therefore, we conclude a posteriori that the map Remark 2. The special cases n = 1, 2 were excluded from the proof since g 1 = h 1 and o(2) is abelian. However, it follows from remark 1 that the spaces H 2 (g n , g n ) are still three-dimensional when n = 1, 2.

Deformation of special coadjoint orbits
Let us assume that n ≥ 3. The three-parameter family of deformations of g n can be conveniently prescribed as a deformation of the Lie-Poisson structure on a basis for g ∨ n . Let us consider the dual coordinates {l ij } 1≤i<j≤n in o(n) ∨ with canonical commutation relations (3.1) {l ij , l kl } = δ ik l jl − δ il l jk + δ jl l ik − δ jk l il together with the Darboux coordinates {x i , p i } n i=1 , and the central extension coordinate I in h ∨ n , on which the coordinates l ij act as The commutation relations (3.1)-(3.2) do not admit nontrivial deformations as a consequence of the simplicity of o(n), and in particular, they are not affected by the integration of the cocycles (2.3)-(2.5). On the other hand, the linear deformations of the induced Lie-Poisson bracket of the chosen basis for g ∨ n manifest in the remaining commutation relations. Let ε 1 , ε 2 and ε 3 be complex parameters corresponding to the cocycles f 1 , f 2 and f 3 in H 2 g C n , g C n . The Lie-Poisson bracket deformations in g C n ∨ take the explicit form Remark 3. Let ε ε ε = (ε 1 , ε 2 , ε 3 ). For every λ ∈ C * , the nonzero triples ε ε ε = (ε 1 , ε 2 , ε 3 ) and ε ε ε ′ = λ·(ε 1 , ε 2 , ε 3 ) define isomorphic Lie algebras under scaling of generators. Therefore, in order to describe the different isomorphism classes of nontrivial deformations of g C n , it is sufficient to consider them in terms of a stratification of the projective plane P(ε ε ε).
Corollary 2. In terms of the previous stratification of R 3 \ {(0, 0, 0)}, the isomorphism type of the real forms g n (ε ε ε) for ε ε ε = (0, 0, 0) is 3.1. Special coadjoint orbits. The rank of a semi-simple Lie algebra g is equal to the dimension of the center of its universal enveloping algebra U (g)-a space generated by the so-called Casimir invariants. Z(U (g)) can be equivalently described in terms of the Lie-Poisson structure in C ∞ (g ∨ ).
For o(n+2, C) and the dual basis {l ij } 1≤i<j≤n+2 with Poisson brackets (3.1), the Casimir invariants can be determined explicitly as the homogeneous polynomials There is an analogous description of the coadjoint orbits in o(n + 2, C) ∨ isomorphic to the homogeneous space O(n + 2, C)/SO(2, C) × O(n, C), and which are the minimal nontrivial orbits when n = 2, 4 [Wol78]. The next result is described in [BS97].
Lemma 2 ( [BS97]). The 2n-dimensional coadjoint orbits in o(n + 2) ∨ are isomorphic to the homogeneous space SO(n + 2)/SO(2) × SO(n) and form a 1-dimensional algebraic family determined by the collection of quadratic equations Remark 6. The set of quadratic equations (3.4)-(3.5) identify the given coadjoint orbits with the Grassmannian Gr 2 R n+2 of oriented 2-planes in R n+2 , as they can be understood as a 2 : 1 lift of the classical Plücker embedding.
The Plücker relations indicate that equations (3.5) are overdetermined, and can be generated by any subcollection of n 2 equations containing a given fixed element l ij , i.e. l n+1n+2 .
Applying corollary 2 and lemma 2 to the generic deformations g n (ε ε ε), ε ε ε ∈ R 3 , and letting x i = l in+1 , p i = l in+2 , and I = l n+1n+2 , equation (3.4) becomes The remaining equations do not depend on the deformation parameters. We emphasize the ones containing {I, x i , p j }, Notice that the subcollection (3.7) generalizes the usual definition of angular momentum and generate (3.5), while equations (3.8) generalize the vector analysis relations l · x = l · p = 0.
Remark 7. It follows from remarks 1 and 6 that over the open set R ++ , (3.6)-(3.7) correspond to the equations that determine a 2 : 1 lift of the Plücker embedding, identifying O 2n (ε ε ε) with the Grassmannian of oriented planes in R n+2 . The different solutions of the quadratic equation (3.6) correspond to the different choices of orientation of a given 2-plane in R n+2 . If we consider the degeneration ε ε ε → 0, the limiting equations (3.9) I = ±1 and ± l ij = x i p j − x j p i , define two disjoint orbits O + 2n and O − 2n in g ∨ n , each isomorphic to R n ⊕ R n . In turn, the degeneration of the canonical symplectic structure determined by the Kirillov-Konstant-Souriau symplectic form [RSTS94,Kir04] on O 2n (ε ε ε) corresponds to the symplectic structure on standard phase space dx i ∧ dp i for each of the two orbits in g ∨ n . The existence of two limiting connected components corresponds to the limiting degenerations in the work of Higgs [Hig79] for the cotangent bundles T * S n as the sectional curvature is allowed to vanish (cf. remark 8).
Remark 8. Over the lift of the flat conic pr −1 (C R ) = C + ∪ C − (and in particular, in the lines ε 2 = ε 3 = 0 and ε 1 = ε 3 = 0), the coadjoint orbits O 2n (ε ε ε) degenerate to a manifold isomorphic to T * S n if ε ε ε ∈ C + and two copies of T * H n if ε ε ε ∈ C − . A proof of this fact is given in proposition 2. Although the symplectic structure inherited in T * S n (resp. T * H n ) is the standard one (see remark 10), the variables x i and p j do not define Darboux coordinates. Instead, their commutation relations resemble physically the result of adding an external magnetic field in standard phase space [Nov82,Per90]. From a physical point of view, the study of dynamical problems over the complete family of coadjoint orbits O 2n (ε ε ε) can also be interpreted as the study of deformations of dynamical systems on n-manifolds of constant sectional curvature (which is equal to ε 2 when ε 1 = ε 3 = 0).
Observe that on C − , only the connected component O + 2n (ε ε ε) is of physical significance, as it degenerates to the component O + 2n corresponding to the value I = 1 when ε ε ε → 0.
Appendix A. Standard phase space as a coadjoint orbit Proposition 3. There is a symplectomorphism Proof. It is convenient to identify a suitable set of generators and relations on the dual space g ∨ n . Let {x 1 , p 1 , . . . , x n , p n , I} be a set of standard dual variables for the Heisenberg Lie algebra h n , and let {l ij } 1≤i<j≤n be dual variables for the orthogonal Lie algebra o(n). By definition, the symplectic structure of a coadjoint orbit is determined by the Lie-Poisson bracket on C ∞ (g ∨ n ). Consider the 2n-dimensional coadjoint orbit O 2n ⊂ g ∨ n determined by fixing the values I = ±1, together with the angular momentum relations The canonical commutation relations {x i , p j } = δ ij will follow if we restrict to the connected component O + 2n given by I = 1. The correspondence of symplectic O(n)-actions readily follows.
Proof. Recall that any bilinear form (·, ·) on a vector space V induces a Lie algebra structure on 2 V in terms of the orthogonal endomorphisms v ∧ w → L v∧w (u) := (v, u)w − (w, u)v, [GS84,GS06]. A direct computation shows that the Lie algebra structure on 2 R n+2 induced by the quadratic form Q 0 (a 1 , . . . , a n+2 ) = a 2 1 + · · · + a 2 n is isomorphic to g n under the dual correspondence a i ∧ a j → l ij 1 ≤ i < j ≤ n, a i ∧ a n+1 → x i , a i ∧ a n+2 → p i , 1 ≤ i ≤ n, a n+1 ∧ a n+2 → I.
Remark 11. Under the correspondences described in propositions 3 and 4, the Howe pair (O(n), SL(2, R)) of (R n ⊕ R n , ω) [KKS78,How89] is induced by the maximal compact subgroup G ⊂ O R n+2 , Q 0 ∼ = G n and the group G ′ of endomorphisms of the unique totally isotropic plane in R n+2 , Q 0 preserving the area element a n+1 ∧ a n+2 = 1.