a “magic” approach to octonionic Rosenfeld spaces 12 December 2022 Alessio Marrania, Daniele Corradettib, David Chesterc, Raymond Aschheimc, Klee Irwinc a Instituto de Física Teorica, Departamento de Física, Universidad de Murcia, Campus de Espinardo, E-30100, Spain email: alessio.marrani@um.es b Universidade do Algarve, Departamento de Matemática, Campus de Gambelas, 8005-139 Faro, Portugal email: a55944@ualg.pt c Quantum Gravity Research, Topanga Canyon Rd 101 S., California CA 90290, USA email: DavidC@QuantumGravityResearch.org; Raymond@QuantumGravityResearch.org; Klee@QuantumGravityResearch.org In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfor- tunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation à la Rosenfeld. 1 arXiv:2212.06426v1 [math.RA] 13 Dec 2022 Contents 1 Introduction 3 2 Real forms of Magic Squares of order 3 and 2 4 2.1 Hurwitz algebras and their split versions . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Trialities and derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Euclidean and Lorentzian simple rank-3 Jordan algebras . . . . . . . . . . . . . . 6 2.4 Real forms of the Freudenthal-Tits Magic Square . . . . . . . . . . . . . . . . . . 7 2.5 Magic Square of order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Octonionic entries of Magic Squares of order 2 and 3 . . . . . . . . . . . . . . . . 8 3 “Magic” formulæ for Rosenfeld planes 9 3.1 Coset manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 “Magic” formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Octonionic Rosenfeld planes 12 4.1 O ' R⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 C⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 H⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 O⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 “Magic” formulæ for Rosenfeld lines 16 6 Octonionic Rosenfeld lines 19 6.1 O ' R⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.2 C⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3 H⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.4 O⊗ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 Conclusions 23 2 1 Introduction Around the half of the XX century, geometric investigations on the octonionic plane gave rise to a fruitful mathematical activity, culminating into the formulation of Tits-Freudenthal Magic Square [Fr65, Ti]. The Magic Square is an array of Lie algebras, whose entries are obtained from two Hurwitz algebras1 and enjoy multiple geometric and algebraic interpretations [BS00, BS03, SH, Sa, El02, El04], as well as physical applications [GST, DHW, CCM]. The entries m3(A1,A2) of the Tits-Freudenthal Magic Square can be defined equivalently [BS03] using the Tits [Ti], Barton-Sudbery [BS03] (also cf. [Ev]), and Vinberg [Vin66] construc- tions2, m3(A1,A2) =  der(A1)⊕ der(J3(A2)) +̇ ImA1 ⊗ J′3(A2); der(A1)⊕ der(A2) +̇ sa(3,A1 ⊗ A2); tri(A1)⊕ tri(A2) +̇ 3(A1 ⊗ A2), (1) where A1 and A2 are normed Hurwitz division algebras R,C,H,O or the split counterparts Cs,Hs,Os. Here, J3(A) denotes the Jordan algebra of 3 × 3 Hermitian matrices over A and J′3(A) its subspace of traceless elements. The space of anti-Hermitian traceless n × n matrices over A1 ⊗ A2 is denoted sa(n,A1 ⊗ A2). Details of the commutators and the isomorphisms between these Lie algebras can be found e.g. in [BS03]. The choice of division or split A1,A2 yields different real forms : for the Tits construction, further possibilities are given by allowing the Jordan algebra to be Lorentzian, denoted J2,1, as described in [CCM] (see Sec. 2.3 below); equivalently, one can introduce overall signs in the definition of the commutators between distinct components in the Vinberg or Barton-Sudbery constructions, as described in [ABDHN]. For a complete listing of all possibilities3, see [CCM]. Recently [WDM], Wilson, Dray and Manogue gave a new construction of the Lie algebra e8 in terms of 3 × 3 matrices such that the Lie bracket has a natural description as matrix commutator; this led to a new interpretation of the Freudenthal-Tits Magic Square of Lie algebras4, acting on themselves by commutation. While Tits was more interested in algebraic aspects, Freudenthal interpreted every row of the Magic Square in terms of a different kind of geometry, i.e. elliptic, projective, symplectic and meta-symplectic for the first, second, third and fourth row, respectively [LM01]. This means that, while the algebras taken in account were always the Hurwitz algebras R,C,H and O, all Lie groups were arising from considering different type of transformations such as isometries, collineations and homographies of the plane [Fr65]. On the other hand, Rosenfeld conceived every entry of the 4 × 4 array of the Magic Square as the Lie algebra of the (global) isometry group of a “generalized” projective plane, later called Rosenfeld plane [Ro98, Ro97]. Thus, while in Freudenthal’s framework the projective plane was always the same but, depending on the considered row of the Magic Square, the kind of transformations was changed; in Rosenfeld’s picture the group of transformations was always the isometry group, and different Lie groups appeared considering different planes over tensor product of Hurwitz algebras. To be more precise, since the exceptional Lie group F4 is the isometry group of the octonionic projective plane OP 2, then Rosenfeld regarded E6 as the isometry group of the bioctonionic “pro- 1Even though the original construction by Tits was not symmetric in the two entries (cf. the third of (1)), the square turned out to be symmetric and was, therefore, dubbed as “magic”. 2The ternary algebra approach of [Ka] was generalised by Bars and Günaydin in [BG] to include super Lie algebras. Generalizations to affine, hyperbolic and further extensions of Lie algebras have been considered in [Pa]. 3Concerning the extension of Freudenthal-Tits Magic Square to various types of “intermediate” algebras, the extension to the sextonions S (between H and O) is due to Westbury [Wes], and later developed by Landsberg and Manivel [LM01, LM06], whereas further extension to the tritonions T (between C and H) was discussed by Borsten and one of the present authors in [BM]. 4Magic Squares of order 2 of Lie groups have been investigated in [DHK]. Interestingly, the Freudenthal-Tits Magic Square of Lie groups was observed to be non-symmetric since [Yo85]. 3 jective” plane (C⊗ O)P 2, E7 as the isometry group of the quaternoctonionic “projective” plane (H⊗ O)P 2, and E8 as the isometry group of the octooctonionic “projective” plane (O⊗ O)P 2. Despite Rosenfeld’s suggestive interpretation, it was soon realized that the planes identified by Rosenfeld did not satisfy projective axioms, and this explains the quotes in the term “projec- tive”, which would have to be intended only in a vague sense. It was the work of Atsuyama, followed by Landsberg and Manivel, to give a rigorous description of the geometry arising from Rosenfeld’s approach [Atsu2, LM01, AB]. Indeed (unlike O ' R⊗ O) C⊗ O, H⊗ O and O⊗ O are not division algebras, thus preventing a direct projective construction; moreover (unlike C⊗ O), Hermitian 3× 3 matrices over H⊗ O or O⊗ O do not form a simple Jordan algebra, so the usual identification of points (lines) with trace 1 (2) projection operators cannot be made [Ba]. Nonetheless, they are in fact geometric spaces, generalising projective spaces, known as “buildings”, on which (the various real forms of) exceptional Lie groups act as isometries. Build- ings where originally introduced by Tits in order to provide a geometric approach to simple Lie groups, in particular the exceptional cases, but have since had far reaching implications; see, for instance, [Ti2, Te] and Refs. therein. More specifically, Rosenfeld noticed that, over R, the isometry group of the octonionic pro- jective plane was the compact form of F4, i.e. F4(−52), while the other, non-compact real forms, i.e. the split form F4(4) and F4(−20), were obtained as isometry groups of the split-octonionic projective plane OsP 2 and of the octonionic hyperbolic plane OH2 [CMCAb, CMCAa], respec- tively. Thence, he proceeded in relating all real forms of exceptional Lie groups with projective and hyperbolic planes over tensorial products of Hurwitz algebras [Ro97, Ro98]. Despite being very insightful, to the best of our knowledge Rosenfeld’s approach was never formulated in a systematic way, and a large part of the very Rosenfeld planes were never really rigorously defined. In the present investigation, we give a systematic and explicit definition of all octonionic5 “Rosenfeld planes” as coset manifolds, by exploiting all real forms of Magic Squares of order 3 over Hurwitz algebras and their split versions, thus all possible real forms of Freudenthal-Tits Magic Square. In doing so, we will also find a total of 10 coset manifolds, namely 7 “planes” and 3 “lines”, whose definition is straightforward according to our procedure, but that apparently do not have any interpretation according to Rosenfeld’s approach. The plan of the paper is as follows. In Sec. 2 we will resume all the algebraic machinery used in this paper, i.e. Hurwitz algebras, their triality and derivation symmetries, Euclidean and Lorentzian cubic Jordan algebras, Magic Squares of order 3 and 2, etc. In Sec. 3 we will then present the “magic” formulæ which we will use in order to define the octonionic Rosenfeld planes in Sec. 4; then, we will do the same for octonionic Rosenfled lines in Secs. 5 and 6. Some final remarks and an outlook are given in Sec. 7. 2 Real forms of Magic Squares of order 3 and 2 2.1 Hurwitz algebras and their split versions Let the octonions O be the only unital, non-associative, normed division algebra with R8 de- composition of x ∈ O given by x = 7∑ k=0 xkik, (2) where {i0 = 1, i1, ..., i7} is a basis of R8 and the multiplication rules are mnemonically encoded in the Fano plane (see left side of Fig. 1), along with i2k = −1 for k = 1, ..., 7. Let the norm 5The quaternionic, complex and real “Rosenfeld planes” and “Rosenfeld lines” can be obtained as proper sub-manifolds of their octonionic counterparts, so we will here focus only on the latter ones. 4 Figure 1: Multiplication rule of octonions O (left) and of split-octonions Os (right) as real vector space R8 in the basis {i0 = 1, i1, ..., i7}. In the case of the octonions i20 = 1 and i2k = −1 for k = 1...7, while in the case of split-octonions i2k = +1, for k = 1, 2, 3 and i 2 k = −1 for k 6= 1, 2, 3. N (x) be defined as N (x) := 7∑ k=0 (xkik) 2 , (3) and its polarisation as 〈x, y〉 := N (x+ y)−N (x)−N (y) . Then the real part of x is R (x) := 〈x, 1〉 and x := 2 〈x, 1〉 − x. We then have that N (x) = xx, 〈x, y〉 = xy + yx,and that N (xy) = N (x)N (y) , (4) or, in other words, that octonions are a composition algebra with respect to the Norm N defined by (3). We now define as the algebra of quaternions H as the subalgebra of O generated by the elements {i0 = 1, i1, i2, i3} and the algebra of complex numbers C as the subalgebra of O generated by the elements {i0 = 1, i1}. Thus, the split-octonions Os can be defined as the only unital, non-associative, normed algebra with R8 decomposition of x ∈ Os given by (2) where, again, {i0 = 1, i1, ..., i7} is a basis of R8, but the multiplication rules are encoded in a different variant of the Fano plane, given by the right side of Fig. 1, along with i2k = 1 for k = 1, ..., 3 and i 2 k = −1 for k = 4, 5, 6, 7 . Norm and conjugation are defined as in the octonionic case and, as for the division octonions, we obtain that split octonions are a composition algebra with respect to the Norm N defined by (3). We then define the algebra of split-quaternions Hs as the subalgebra of Os generated by {i0 = 1, i1, i4, i5} and the split-complex (or hypercomplex ) numbers Cs as the subalgebra of Os generated by {i0 = 1, i4}. 2.2 Trialities and derivations Let A be an algebra and End (A) the associative algebra of its linear endomorphisms. Then, the triality algebra tri (A) is the Lie subalgebra of so (A)⊕ so (A)⊕ so (A) defined as tri (A) := { (A,B,C) ∈ End (A)⊕3 : A (xy) = B (x) y + xC (y) ,∀x, y ∈ A } , (5) where the Lie bracket are those inherited as a subalgebra. Derivations are a special case of trialities of the form (A,A,A) and therefore they again form a Lie algebra, defined as der (A) := {A ∈ End (A) : A (xy) = A (x) y + xA (y) ,∀x, y ∈ A} , (6) 5 R C H O tri (A) Ø u (1)⊕ u (1) so (3)⊕ so (3)⊕ so (3) so (8) tri (As) Ø so (1, 1)⊕ so (1, 1) sl2 (R)⊕ sl2 (R)⊕ sl2 (R) so (4, 4) der (A) Ø Ø so (3) g2(−14) der (As) Ø Ø sl2 (R) g2(2) A (A) Ø u1 su2 Ø A (As) Ø so1,1 sl2 (R) Ø Table 1: Triality and derivation algebras of Hurwitz algebras. Moreover we also added the algebra A (A) := tri (A) so (A) , that will be used in Sec. 3.2 in the definition of the Rosenfeld planes. where the bracket is given by the commutator. In the case of A = C,H,O we intended so (A) as so (2), so (4) and so (8) respectively; while, for their split companions, we intended so (As) as so (1, 1), so (2, 2) and so (4, 4). Triality and derivation Lie algebras over Hurwitz algebras are summarized in Table 1. 2.3 Euclidean and Lorentzian simple rank-3 Jordan algebras Let A be a composition algebra and Hn (A) be the set of Hermitian n×n matrices with elements in A such that X† := X t = X, (7) where the conjugation is the one pertaining to A itself. When6 n = 2 or 3, we define the simple, rank-n Euclidean Jordan algebra Jn (A) as the commutative algebra over Hn (A) with the Jordan product ◦ defined by the anticommutator X ◦ Y := 1 2 (XY + Y X) =: {X,Y } , ∀X,Y ∈ Hn (A) , (8) where juxtaposition denotes the standard ’rows-by-columns’ matrix product. Moreover, we define J′n (A) as the subspace of the Jordan algebra orthogonal to the identity I (namely, the subspace of traceless matrices in Jn (A)), endowed with the product inherited7 from Jn (A) (see e.g. [BS03, p.8]). In particular, for n = 3 the simple, rank-3 Euclidean Jordan algebra J3 (A) has elements X :=  a x1 x2 x1 b x3 x2 x3 c  ∈ J3 (A) , (9) where x1, x2, x3 ∈ A and a, b, c ∈ R. The Hermiticity condition (7) can be generalized by inserting a pseudo-Euclidean metric : ηX†η = X, (10) η := diag −1, ...,−1 ︸ ︷︷ ︸ p times , 1, ..., 1 ︸ ︷︷ ︸ n−p times  . (11) Correspondingly, one defines the simple, rank-n pseudo-Euclidean Jordan algebra Jn−p,p (A) ' Jp,n−p (A). In particular, by setting n = 3, the simple, rank-3 Lorentzian Jordan algebra 6Excluding A = O and Os, one can actually consider any n ∈ N. 7Note that J′n (A) is not closed under the restriction of the Jordan product ◦ of Jn (A) to J′n (A) itself. However, one can deform ◦ into the Michel-Radicati product ∗, defined as X ∗ Y := X ◦ Y − Tr(X◦Y ) n I ∀X,Y ∈ J′n (A), under which J′n (A) is closed [MR70, MR73]. 6 J2,1 (A) ' J1,2 (A), whose elements read as X :=  a x1 x2 −x1 b x3 −x2 x3 c  ∈ J1,2 (A) , (12) where x1, x2, x3 ∈ A and a, b, c ∈ R. 2.4 Real forms of the Freudenthal-Tits Magic Square As anticipated in (1), given A1 and A2 two composition algebras, the corresponding entry of Tits-Freudenthal Magic Square m3 (A1,A2) is a Lie algebra that can be realized in at least three different, but equivalent ways, on which we will now briefly comment (without the explicit definition of the Lie brackets, that can be found e.g. in [BS03]). 1. Tits construction [Ti] is given by the first line of the r.h.s. of (1) : m3 (A1,A2) := der(A1)⊕ der(J3(A2)) +̇ ImA1 ⊗ J′3(A2), (13) thus involving the derivation Lie algebra of A1 and the derivation Lie algebra of the simple, rank-3 Euclidean Jordan algebra J3 (A2) over A2. Tits construction, despite being not manifestly symmetric under the exchange A1 ↔ A2, is the most general of the three constructions presented here, since it holds for any alternative algebras A1 and A2, as long as it is possible to define a Jordan algebra over A2 itself. As mentioned above, Tits construction can be generalized to J2,1(A2), thus yielding8 m1,2 (A1,A2). 2. A more symmetric approach was pursued by Vinberg [Vin66], who obtained the formula given by the second line of the r.h.s. of (1), m3 (A1,A2) = der(A1)⊕ der(A2) +̇ sa(3,A1 ⊗ A2). (14) This formula is manifestly symmetric under the exchange A1 ↔ A2, but it only holds if A1 and A2 are both composition algebras; it involves only the derivation Lie algebras of A and B, as well as the 3 × 3 antisymmetric matrices over A1 ⊗ A2. As mentioned above, a suitable deformation of (14) is possible, in order to give rise to m1,2 (A1,A2). 3. Another symmetric formula under the exchange A1 ↔ A2 was obtained by Barton and Sudbery [BS03] (also cf. [Ev]), and it is given by the third line of the r.h.s. of (1), m3 (A1,A2) = tri(A1)⊕ tri(A2) +̇ 3(A1 ⊗ A2), (15) involving the triality algebras tri (A1) and tri (A2), together with three copies of the tensor product A1⊗A2. Again, as mentioned above, a suitable deformation of (15) is possible, in order to give rise to m1,2 (A1,A2). Other, different versions of the construction of Freudenthal-Tits Magic Square have been devel- oped by Santander and Herranz [SH, Sa], Atsuyama [Ats, Atsu2] and Elduque [El04, El02, El18], all involving composition algebras (even though Elduque’s construction involves flexible compo- sition algebras instead of alternative composition algebras [El18]). 8Since J2,1(A2) ' J1,2(A2), it holds that m2,1(A1,A2) ' m1,2(A1,A2). 7 2.5 Magic Square of order 2 Inspired by the works of Freudenthal and Tits, Barton and Sudbery [BS03] also considered a different Magic Square, based on 2× 2 matrices, and defined by the following formula : m2 (A1,A2) := so (A ′ 1)⊕ der (J2 (A2))⊕ (A′1 ⊗ J′2 (A2)) , (16) which also enjoys a “Vinberg-like” equivalent version as m2 (A1,A2) = so (A ′ 1)⊕ so (A′2)⊕ sa2 (A1 ⊗ A2) , (17) that was more recentely used at Lie group level in [DHK]. Formula (16) can be generalized to involve simple, rank-2 Lorentzian Jordan algebras J1,1 (A2) (see Sec. 2.3), thus obtaining the Lorentzian version of the Magic Square of order 2, m1,1 (A1,A2) := so (A ′ 1)⊕ der (J1,1 (A2))⊕ ( A′1 ⊗ J′1,1 (A2) ) , (18) which we will explicitly evaluate in the treatment below (for the first time in literature, to the best of our knowledge). 2.6 Octonionic entries of Magic Squares of order 2 and 3 In order to rigorously define all possible octonionic “Rosenfeld planes” and “Rosenfeld lines” over R (i.e. all possible real forms thereof), we need to isolate all octonionic and split-octonionic entries of all real forms of the Freudenthal-Tits Magic Square (order 3) [CCM, BM, ABDHN, BS03] and of the Magic Square of order 2 [BS03], i.e. we need to consider the entries mα (A1,A2) in which at least one of A1 and A2 is O or Os, for all possible real forms, namely for α = 3 (Euclidean order 3), 1, 2 (Lorentzian order 3), 2 (Euclidean order 2) and 1, 1 (Lorentzian order 2). The octonionic entries of the Euclidean Magic Squares of order 3 m3 (i.e., of all Euclidean real forms of Freudenthal-Tits Magic Square) are A m3 (A,O) m3 (As,O) m3 (Os,A) m3 (As,Os) R f4(−52) f4(−52) f4(4) f4(4) C e6(−78) e6(−26) e6(2) e6(6) H e7(−133) e7(−25) e7(−5) e7(7) O e8(−248) e8(−24) e8(−24) e8(8) The entries of the table above comprise all real forms of e8 and e7, and most of the real forms of e6 and f4. The missing real forms f4(−20) and e6(−14) can be recovered as the octonionic entries of the Lorentzian Magic Squares of order 3 m2,1 (i.e., of all Lorentzian real forms of Freudenthal-Tits Magic Square), given by A m1,2 (A,O) m1,2 (As,O) m1,2 (Os,A) m1,2 (As,Os) R f4(−20) f4(−20) f4(4) f4(4) C e6(−14) e6(−26) e6(2) e6(6) H e7(−5) e7(−25) e7(−5) e7(7) O e8(8) e8(−24) e8(−24) e8(8) On the other hand, the octonionic entries of the Euclidean Magic Squares of order 2 m2 are9 9The explicit form of m2 (As,Bs) is, as far as we know, not present in the current literature. We will present a detailed treatment elsewhere, and here we confine ourselves to report its octonionic column only. 8 A m2 (A,O) m2 (As,O) m2 (Os,A) m2 (As,Os) R so (9) so (9) so (5, 4) so (5, 4) C so (10) so (9, 1) so (6, 4) so (5, 5) H so (12) so (10, 2) so (8, 4) so (6, 6) O so (16) so (12, 4) so (12, 4) so (8, 8) Finally, the octonionic entries of the Lorentzian Magic Squares of order 2 m1,1 are given by10 A m1,1 (A,O) m1,1 (As,O) m1,1 (Os,A) m1,1 (As,Os) R so (8, 1) so (8, 1) so (5, 4) so (5, 4) C so (8, 2) so (9, 1) so (6, 4) so (5, 5) H so (8, 4) so (10, 2) so (8, 4) so (6, 6) O so (8, 8) so (12, 4) so (12, 4) so (8, 8) 3 “Magic” formulæ for Rosenfeld planes In his study of the geometry of Lie groups [Ro97], Rosenfeld defined its “projective” planes (A⊗ B)P 2 as the completion of some affine planes obtained as non-associative modules over the tensor algebra A ⊗ B. He then argued the form of the matrices composing the linear transfor- mations associated with the Lie algebra of the collineations that preserved the polarity, i.e. the isometries of the plane. This approach allowed him to relate all real forms of exceptional Lie groups [Ro93, Ro98] with isometries of suitable “projective” hyperbolic spaces. As mentioned above, unfortunately the procedure carried out by Rosenfeld was not rigorous, since many of theorems used in [Ro97] do not extend to the case of algebras that are not alternative nor of composition, as it is in the case of many algebras listed in Table 2. We present here a general and rigorous way to define the spaces considered by Rosenfeld, in terms of their isometry and isotropy groups, namely using the theory of coset manifolds. 3.1 Coset manifolds Since our “magic” formulæ define all Rosenfeld planes as coset manifolds, it is worth briefly reviewing them, and their relation to homogenous spaces. An homogeneous space is a manifold on which a Lie group acts transitively, i.e. a manifold on which is defined an action ρg from G ×M in M such that ρe (m) = m for e the identity in G and m ∈ M and for which, given any m,n ∈ M it exists a a not necessarely unique g ∈ G such that ρg (m) = n. Within this framework, the isotropy group Isotm (G) is the set formed by the elements of G that fix the point m ∈M under the action of G, i.e. Isotm (G) = {g ∈ G : ρg (m) = m} . (19) Since, by definition of group action, we have that ρe (m) = m and ρgh (m) = ρg (ρh (m)), then K = Isotm (G) is a closed subgroup of G and, moreover the natural map from the quotient space G/K in M given by gK 7→ gm is a diffeomorphism (see [Ar] and Refs. therein). Given a Lie groupG and a closed subgroupK < G, then the coset spaceG/K = {gK : g ∈ G} is endowed with a natural manifold structure inherited by G and is, therefore, called a coset manifold. Notice that the action of G on the coset manifold G/K, given by the translation τg defined as τg (m) = gm, (20) 10The Lorentzian magic square of order two yield to three algebras, i.e. so (8, 1), so (8, 2) and so (8, 4) which are not covered in other magic squares. The explicit forms of m1,1 (A,B), m1,1 (As,B) and m1,1 (As,Bs) are, as far as we know, not present in the current literature. We will present a detailed treatment elsewhere, and here we confine ourselves to report their octonionic rows and columns only. 9 Algebra Comm. Ass. Alter. Flex. Pow. Ass. C⊗ C Yes Yes Yes Yes Yes C⊗ H No Yes Yes Yes Yes H⊗ H No Yes Yes Yes Yes C⊗ O No No Yes Yes Yes H⊗ O No No No No No O⊗ O No No No No No Table 2: Properties of the algebra A ⊗ B where A,B are Hurwitz algebras. As for the property an algebra A is said to be commutative if xy = yx for every x, y ∈ X; it is defined as associative if satisfies x (yz) = (xy) z; alternative if x (yx) = (xy)x; flexible if x (yy) = (xy) y and, finally, power-associative if x (xx) = (xx)x. for every g ∈ G andm ∈ G/K, is transitive, i.e. for everym,n ∈ G/K it exists a (not necessarily unique) g ∈ G such that n = gm, and thus the coset manifold G/K is an homogenous space. On the other hand, since multiple groups can act transitively on the same manifold with different isotropy groups, then an homogeneous space can be realised in multiple way as a coset manifold. Moreover, a close look to the definitions shows that the isotropy group Isotm (G/K) is exactly the closed subgroup K.11 In general, the holonomy subgroup and the isotropy subgroup have the same identity-connected component; so, if one assumes that G/K is simply-connected, they are equal (see e.g. [Be, He] and Refs. therein). Moreover, let G be a Lie group and K a closed and connected subgroup of G, denoting with g and k their respective Lie algebras, then the coset manifold G/K is reductive if there exists a subspace m such that g = k⊕m and { [k, k] ⊂ k, [k,m] ⊂ m, (21) while if in addition to (21) we also have [m,m] ⊂ k, (22) then the space is symmetric. All Rosenfeld planes are symmetric coset manifolds. It is also worth noting that for any coset manifold G/K the structure constants of the Lie algebra g of the Lie group G define completely the structure constants of the manifold, thus the invariant metrics, and all the metric-dependent tensors, such as the curvature tensor, the Ricci tensor, etc. Indeed, let {E1, ..., En} be a basis for g in such a way that {E1, ..., Em} are a basis for k, which we will also call {K1, ...,Km} for readibility reasons, and {Em+1, ..., En} a basis for m that we will also denote as {Mm+1, ...,Mn}. Then consider the structure constants of the algebra g, i.e. [Ej , Ek] = n∑ i=1 CijkEi, (23) for every j, k ∈ {1, ..., n}. Conditions for reductivity in (21) are then translated in  [Kj ,Kk] = m∑ i=1 CijkKi for j, k ∈ {1, ...,m} , [Kj ,Mk] = n∑ i=m+1 CijkMi for j ∈ {1, ...,m} , k ∈ {m+ 1, ..., n} , (24) 11In particular, the origin of G/K is, by definition, the point at which theK-invariance is immediately manifest. 10 while, on the other hand, from [Mj ,Mk] = m∑ i=1 CijkKi+ n∑ i=m+1 CijkMi, (25) we deduce that if Cijk = 0 for all i, j, k ∈ {m+ 1, ..., n}, we have also a symmetric space. From the structure constants Cijk, all geometrical invariants of the coset manifold can be obtained (see [FF17] for all technical details) such as the Riemann tensor over the coset manifold G/K which is given by Rabcd = n∑ e=m+1 1 λ2 ( 1 8 CaedC e bc − 1 8λ CabeC e cd − CaecCebd ) − 1 2λ2 m∑ i=1 CabiC i cd, (26) and that, in case of symmetric spaces, i.e. when Cijk = 0 for i, j, k ∈ {m+ 1, ..., n}, reduces drastically to Rabcd = − 1 2λ2 m∑ i=1 CabiC i cd, (27) for every a, b, c, d ∈ {m+ 1, ..., n}. In fact, what is relevant for our purposes is that the Lie group G, which acts as isometry group, and its closed subgroup K, which acts as isotropy group, define completely all the metric properties and geometric invariants of the Rosenfeld planes. On the other hand, instead of working on Lie groups G andK, i.e. with the isometry group and the isotropy group respectively, in the following sections we will work with their respective Lie algebras g and k, in order to recover the appropriate G and K and thus define the Rosenfeld plane. 3.2 “Magic” formulæ As seen in the previous section, the geometry of a coset manifold is fully determined by the (global) isotropy Lie group G and by its (local) subgroup given by the isotropy Lie group K. We now characterize the Rosenfeld “projective” planes by specifying the real form of the Lie groups that will act as isometry and isotropy groups, while the Lie brackets - and therefore the metrical properties of the space - are those arising from Tits construction. It is here worth noting that the classification of the real forms of simple Lie groups makes use of the character χ, defined as the difference of the cardinality of non-compact and compact generators, i.e. χ := #nc−#c; consequently, Rosenfeld planes will have a corresponding χ. Let A and B be two Hurwitz algebras, in their division or split versions. Moreover, let the Lie group A (A) be such that its Lie algebra is Lie(A (A)) ≡ A(A) := tri (A) so (A), cfr. Table 1. Moreover, letMα (A,B) be the Lie group with Lie algebra given by the (A,B)-entry of the Magic Square mα (A,B), namely12 Lie(Mα (A,B)) = mα (A,B). In order to characterize “projective” Rosenfeld planes over tensor products of Hurwitz algebras in terms of coset manifolds with isometry (resp. isotropy) Lie groups whose Lie algebras are entries of real forms of the Magic Square of order 3 (resp. 2), we now introduce the following three different classes of locally symmetric, (pseudo-)Riemannian coset manifolds, that we name as Rosenfeld planes : 1. The projective Rosenfeld plane (A⊗ B)P 2 ' M3 (A,B) M2 (A,B)⊗A (A)⊗A (B) . (28) 12For the pseudo-orthogonal Lie algebras, we will generally consider the spin covering of the corresponding Lie group. 11 2. The hyperbolic Rosenfeld plane (A⊗ B)H2 ' M1,2 (A,B) M2 (A,B)⊗A (A)⊗A (B) . (29) 3. The pseudo-Rosenfeld plane (A⊗ B) H̃2 ' M1,2 (A,B) M1,1 (A,B)⊗A (A)⊗A (B) . (30) Eqs. (28), (29) and (30) are named “magic” formulæ, since they characterize the Rosenfeld planes as homogeneous (symmetric) manifolds, with isometry (resp. isotropy) groups whose Lie algebras are given by the entries of some real forms of the Magic Square of order 3 (resp. 2), with further isotropy factors given by the Lie groups A associated to the algebras A and B defining the tensor product associated to the class of Rosenfeld plane under consideration. As previously noticed, the term “projective” and “hyperbolic” are here to be intended in a vague sense since none of the octonionic Rosenfeld planes with dimension greater than 16 satisfy axioms of projective or hyperbolic geometry. Nevertheless, those adjectives are not arbitrary since the notion of such projective planes can be made precise as in [Atsu2, AB] and it then agrees with the above definitions. 4 Octonionic Rosenfeld planes We now consider the “magic” formulæ (28)-(30) in the cases13 in which A and/or B is O or Os : this will allow us to rigorously introduce the octonionic Rosenfeld planes, which all share the fact that their isometry group is a real form of an exceptional Lie group of F- or E- type; however, it is here worth anticipating that a few real forms of Rosenfeld planes with exceptional isometry groups cannot be characterized in this way. 4.1 O ' R⊗ O The simplest case concerns the tensor product R ⊗ O, which is nothing but O : in fact, this was the original observation by Rosenfeld that started it all, yielding to the usual octonionic projective, split-octonionic and hyperbolic plane, i.e. OP 2, OH2 and OsP 2, respectively. Within the framework introduced above, the starting point is given by the Cayley-Moufang plane over C, namely by the octonionic projective plane over C, i.e. by the locally symmetric coset manifold having as isometry group the complex form of F4, and as isotropy group Spin (9,C), i.e. OP 2C ' FC4 Spin (9,C) . (31) In this coset space formulation, the tangent space of OP 2C can be identified with the 16C- dimensional spinor representation space of the isotropy group Spin (9,C). Then, by specifying the formulæ (28), (29) and (30) for A = R and B = O or Os, we obtain all real forms14 of (31) as Rosenfeld planes over R ⊗ O ' O or over R ⊗ Os ' Os; they are summarized by the following15 Table [CMCAb] : 13As mentioned above, this restriction does not imply any loss of generality, as far as the other, non-octonionic Rosenfeld planes can be obtained as suitable sub-manifolds of the octonionic Rosenfeld planes. 14By real forms of OP 2C, we here mean the cosets with isometry groups given by all real (compact and non- compact) forms of F4, and with isotropy group given by all (compact and non-compact) real forms of Spin(9) which are subgroups of the corresponding real form of F4. 15Recall that A(R) = A(O) = A(Os) = ∅. 12 Plane Isometry Isotropy #nc #c χ OP 2 F4(−52) Spin (9) 0 16 −16 OH2 F4(−20) Spin (9) 16 0 16 OH̃2 F4(−20) Spin (8, 1) 8 8 0 OsP 2 F4(4) Spin (5, 4) 8 8 0 along with OsP 2 ' OsH2 ' OsH̃2. (32) 4.2 C⊗ O In the bioctonionic case, i.e. for C⊗O, the starting point is given by the bioctonionic projective plane over C, i.e. by the locally symmetric coset manifold having as isometry group the complex form of E6, and as isotropy group Spin(10,C)⊗U1, i.e. (C⊗ O)P 2C ' EC6 Spin (10,C)⊗ (U1)C , (33) which is a Kähler manifold, and has been recently treated in [CMCAa]. In this case, the tangent space of the coset manifold (33) is the ( 16C,+ ⊕ 16C,− ) representation of the isotropy group Spin (10,C)⊗U1. Then, by specifying the formulæ (28), (29) and (30) for A = C or Cs and B = O or Os, we ob- tain all real forms16 of (33) which can be expressed as Rosenfeld planes over C(or Cs)⊗O(or Os); they are summarized by the following17 Table : Plane Isometry Isotropy #nc #c χ (C⊗ O)P 2 E6(−78) Spin (10)⊗U1 0 32 −32 (C⊗ O)H2 E6(−14) Spin (10)⊗U1 32 0 32 (C⊗ O)H̃2 E6(−14) Spin (8, 2)⊗U1 16 16 0 (C⊗ Os)P 2 E6(2) Spin (6, 4)⊗U1 16 16 0 (Cs⊗O)P 2 E6(−26) Spin (9, 1)⊗ SO (1, 1) 16 16 0 (Cs⊗Os)P 2 E6(6) Spin (5, 5)⊗ SO (1, 1) 16 16 0 along with (C⊗ Os)P 2 ' (C⊗ Os)H2 ' (C⊗ Os) H̃2, (34) (Cs⊗O)P 2 ' (Cs⊗O)H2 ' (Cs⊗O) H̃2, (35) (Cs⊗Os)P 2 ' (Cs⊗Os)H2 ' (Cs⊗Os) H̃2, (36) expressing the fact that projective, hyperbolic and pseudo Rosenfeld planes involving Cs and/or Os are all isomorphic. The first four manifolds of the above Table, having a U1 factor in the isotropy group, are Kähler manifolds, whereas the last two, having a SO(1, 1) factor in the isotropy group, are pseudo-Kähler manifolds. 16By real forms of (C⊗ O)P 2C, we here mean the cosets with isometry groups given by all real (compact and non-compact) forms of E6, and with isotropy group given by all (compact and non-compact) real forms of Spin(10,C)⊗ (U1)C which are subgroups of the corresponding real form of E6. 17Recall that A(C) = u1, and A(Cs) = so1,1. 13 Finally, and more importantly, there are two real forms of (33), namely the locally symmetric, pseudo-Riemannian coset Kähler manifolds X32,I := E6(2) SO∗(10)⊗U1 , #nc = 20,#c = 12⇒ χ = 8; (37) X32,II := E6(−14) SO∗(10)⊗U1 , #nc = 12,#c = 20⇒ χ = −8, (38) whose isotropy Lie group has the corresponding Lie algebra which is not an entry of any real form of the Magic Square of order 2. In other words, since the Lie algebra so∗(10) does not occur in any real form of the Magic Square of order 2 (see Secs. 2.5 and 2.6), the symmetric Kähler manifolds (37) and (38) cannot be characterized as Rosenfeld planes over C(or Cs)⊗O(or Os). 4.3 H⊗ O In the quaternoctonionic case, i.e. for H⊗O, the starting point is given by the quaternoctonionic “projective” plane over C, i.e. by the locally symmetric coset manifold having as isometry group the complex form of E7, and as isotropy group Spin(12,C)⊗ SL (2,C), i.e. (H⊗ O)P 2C ' EC7 Spin (12,C)⊗ SL (2,C) , (39) which is a quaternionic Kähler manifold, and whose tangent space is given by the ( 32(′),2 ) C representation18 of the isotropy group Spin (12,C)⊗ SL (2,C). Then, by specifying the formulæ (28), (29) and (30) for A = H or Hs and B = O or Os, we ob- tain all real forms19 of (39) which can be expressed as Rosenfeld planes over H(or Hs)⊗O(or Os); they are summarized by the following20 Table : Plane Isometry Isotropy #nc #c χ (H⊗ O)P 2 E7(−133) Spin (12)⊗ SU (2) 0 64 −64 (H⊗ O)H2 E7(−5) Spin (12)⊗ SU (2) 64 0 64 (H⊗ O)H̃2 E7(−5) Spin (8, 4)⊗ SU (2) 32 32 0 (Hs⊗O)P 2 E7(−25) Spin (10, 2)⊗ SL (2,R) 32 32 0 (Hs⊗Os)P 2 E7(7) Spin (6, 6)⊗ SL (2,R) 32 32 0 along with (H⊗ O)H̃2 ' (H⊗ Os)P 2 ' (H⊗ Os)H2 ' (H⊗Os) H̃2, (40) (Hs⊗O)P 2 ∼= (Hs⊗O)H2 ∼= (Hs⊗O) H̃2, (41) (Hs⊗Os)P 2 ∼= (Hs⊗Os)H2 ∼= (Hs⊗Os) H̃2. (42) The planes that do not involve split algebras, namely the ones having a SU(2) factor in the isotropy group, are quaternionic Kähler manifolds, whereas all the ones involving split algebras, namely the ones having a SL(2,R) factor in the istropy group, are para-quaternionic Kähler manifolds. 18For the possible priming of the semispinor 32 of Spin(12), see e.g. [Min]. 19By real forms of (H⊗ O)P 2C, we here mean the cosets with isometry groups given by all real (compact and non-compact) forms of E7, and with isotropy group given by all (compact and non-compact) real forms of Spin(12,C)⊗ SL2 (C) which are subgroups of the corresponding real form of E7. 20Recall that A(H) = su2, and A(Hs) = sl2(R). 14 Again, there are three real forms of (39), namely the locally symmetric, pseudo-Riemannian (para-)quaternionic coset manifolds X64,I := E7(7) SO∗ (12)⊗ SU (2) , #nc = 40,#c = 24⇒ χ = 16; (43) X64,II := E7(−5) SO∗ (12)⊗ SL (2,R) , #nc = 32,#c = 32⇒ χ = 0; (44) X64,III := E7(−25) SO∗ (12)⊗ SU (2) , #nc = 24,#c = 40⇒ χ = −16, (45) whose isotropy Lie group -up to the factor SU(2) or SL(2,R)- has the corresponding Lie algebra which is not an entry of any real form of the Magic Square of order 2. In other words, since the Lie algebra so∗(12) does not occur in any real form of the Magic Square of order 2 (see Secs. 2.5 and 2.6), the symmetric (para-)quaternionic Kähler manifolds (43)-(45) cannot be characterized as Rosenfeld planes over H(or Hs)⊗O(or Os). It should however be noticed that the isotropy group of (43)-(45) admits an interpretation in terms of real forms of the Magic Square of order 3, namely X64,I ' M3 (Hs,Os) M3 (Hs,H)⊗A(H) , (46) X64,II ' M3 (H,Os) M3 (H,Hs)⊗A(Hs) , (47) X64,III ' M3 (Hs,O) M3 (Hs,H)⊗A(H) , (48) but still the rationale (if any) of such formulæ is missing, and it is surely not the one underlying the “magic” formulæ (28)-(30). 4.4 O⊗ O In the octooctonionic case, i.e. for O ⊗ O, the starting point is given by the octoooctonionic “projective” plane over C, i.e. by the locally symmetric coset manifold having as isometry group the complex form of E8, and as isotropy group Spin(16,C), i.e. (O⊗ O)P 2C ' EC8 Spin (16,C) , (49) whose tangent space is given by the ( 128(′) ) C representation21 of Spin (16,C). Then, by specifying the formulæ (28), (29) and (30) for A = O or Os and B = O or Os, we ob- tain all real forms22 of (49) which can be expressed as Rosenfeld planes over O(or Os)⊗O(or Os); they are summarized by the following Table : Plane Isometry Isotropy #nc #c χ (O⊗ O)P 2 E8(−248) Spin (16) 0 128 −128 (O⊗ O)H2 E8(8) Spin (16) 128 0 128 (O⊗ O)H̃2 E8(8) Spin (8, 8) 64 64 0 (Os⊗O)P 2 E8(−24) Spin (12, 4) 64 64 0 21Again, for the possible priming of the semispinor 128 of Spin(16), see e.g. [Min]. 22By real forms of (O⊗ O)P 2C, we here mean the cosets with isometry groups given by all real (compact and non-compact) forms of E8, and with isotropy group given by all (compact and non-compact) real forms of Spin(16,C) which are subgroups of the corresponding real form of E8. 15 along with (O⊗ O)H̃2 ' (Os⊗Os)P 2 ' (Os⊗Os)H2 ' (Os⊗Os) H̃2, (50) (O⊗ Os)P 2 ' (O⊗ Os)H2 ' (O⊗Os) H̃2. (51) Again, there are two real forms of (49), namely the locally symmetric, pseudo-Riemannian coset manifolds X128,I := E8(8) SO∗ (16) , #nc = 72,#c = 56⇒ χ = 16; (52) X128,II := E8(−24) SO∗ (16) , #nc = 56,#c = 72⇒ χ = −16, (53) whose isotropy Lie group has the corresponding Lie algebra which is not an entry of any real form of the Magic Square of order 2. In other words, since the Lie algebra so∗(16) does not occur in any real form of the Magic Square of order 2 (see Secs. 2.5 and 2.6), the symmetric manifolds (52)-(53) cannot seemingly be characterized as Rosenfeld planes over O(or Os)⊗O(or Os). It should however be noticed that the isotropy group SO∗(16) of (52)-(53) admits an interpretation in terms of the Magic Square of order 4, namely X128,I ' M3 (Os,Os) M4 (H,Hs) ; (54) X128,II ' M3 (O,Os) M4 (H,Hs) , (55) but still the rationale (if any) of such formulæ is missing, and it is surely not the one underlying the “magic” formulæ (28)-(30). 5 “Magic” formulæ for Rosenfeld lines Let Spin(A) the spin covering Lie group whose Lie algebra is Lie(Spin (A)) = so (A), namely the Lie algebra which preserves the norm of the Hurwitz algebra A. In order to characterize “projective” Rosenfeld lines over tensor products of Hurwitz algebras in terms of coset manifolds with isometry and isotropy Lie groups whose Lie algebras respectively are entries of real forms of the Magic Square of order 2 and norm-preserving Lie algebras of the involved Hurwitz algebras, we now introduce a variation of (28) and (29), giving rise to the following two different classes of locally symmetric, (pseudo-)Riemannian coset manifolds, that we name as Rosenfeld lines : 1. The “projective” Rosenfeld line (A⊗ B)P 1 ' M2 (A,B) Spin (A)⊗ Spin (B) . (56) 2. The hyperbolic Rosenfeld line (A⊗ B)H2 ' M1,1 (A,B) Spin (A)⊗ Spin (B) . (57) Eqs. (56) and (57) are named “magic” formulæ, since they characterize the Rosenfeld lines as homogeneous (symmetric) manifolds, with isometry and isotropy Lie groups respectively given by the entries of some real forms of the Magic Square of order 2 and by norm-preserving Lie algebras of the involved Hurwitz algebras. 16 Figure 2: Cartan classification, Satake diagram, character χ of exceptional Lie groups F4 and E6 and related octonionic projective or hyperbolic Rosenfeld plane of which they are isometry group. 17 Figure 3: Cartan classification, Satake diagram, character χ of exceptional Lie groups E7 and E8 and related octonionic projective or hyperbolic Rosenfeld plane of which they are isometry group. 18 6 Octonionic Rosenfeld lines We now consider the “magic” formulæ (56)-(57) in the cases23 in which A and/or B is O or Os : this will allow us to rigorously introduce the octonionic Rosenfeld lines; however, again, it is here worth anticipating that a few real forms of octonionic Rosenfeld lines cannot be characterized in this way. 6.1 O ' R⊗ O Within the framework introduced above, the starting point is given by the octonionic projective line over C, which is nothing but the 8-sphere S8C over C, i.e. the locally symmetric coset manifold having as isometry group Spin(9,C) and as isotropy group Spin (9,C) : OP 1C ' Spin (9,C) Spin (8,C) . (58) In this coset space formulation, the tangent space of OP 1C can be identified with the 8(v,s,c),C representation of Spin (8,C) where d4-triality [Por] allows to equivalently choose v, s, or c. Then, by specifying the formulæ (56) and (57) for A = R and B = O or Os, we obtain all real forms24 of (58) which can be expressed as Rosenfeld lines over R⊗ O ' O or over R⊗ Os ' Os; they are summarized by the following Table : Plane Isometry Isotropy #nc #c χ OP 1 Spin (9) Spin (8) 0 8 −8 OH1 Spin (8, 1) Spin (8) 8 0 8 OsP 1 Spin (5, 4) Spin (4, 4) 4 4 0 along with OsH1 ' OsP 1. It should then be remarked that the octonionic line OP 1 can be identified with the 8-sphere S8 ≡ S8R over R, the hyperbolic line OH1 with the 8-hyperboloid H8 and the split-octonionic line OsP 1 with the Kleinian 8-hyperboloidH4,4 (for some applications to physics and to non-compact versions of Hopf maps, see e.g. [Ha]). The real forms of OP 1C (58) have the general structures Spin (p, 9− p) Spin (p, 8− p) structure I or Spin (p, 9− p) Spin (p− 1, 9− p) structure II , (59) with p = 0, 1, ..., 9, which gets exchanged under p↔ 9− p. OP 1, OH1 and OsP 1 ' OsH1 respectively correspond to structure I with p = 0, 8, 4 (or structure II with p = 9, 1, 5). All other real forms of OP 1C (58) cannot be characterized as Rosenfeld lines over O (or Os). 6.2 C⊗ O In the bioctonionic case, i.e. for C⊗O, the starting point is given by the bioctonionic projective line over C, i.e. by the locally symmetric coset manifold having as isometry group Spin (10,C) with Spin (8,C)⊗U(1) as isotropy group : (C⊗ O)P 1C ' Spin (10,C) Spin (8,C)⊗ (U (1))C . (60) 23Also in this case, this restriction does not imply any loss of generality, as far as the other, non-octonionic Rosenfeld lines can be obtained as suitable sub-manifolds of the octonionic Rosenfeld lines. 24By real forms of OP 1C, we here mean the cosets with isometry groups given by all real (compact and non- compact) forms of Spin(9,C), and with isotropy group given by all (compact and non-compact) real forms of Spin(8,C) which are subgroups of the corresponding real form of Spin(9,C). 19 In this coset space formulation, the tangent space of (C⊗ O)P 1C can be identified with the 8(v,s,c),C+⊕ 8(v,s,c),C− representation of Spin (8,C)⊗ (U (1))C, where, as above, d4-triality [Por] allows to equivalently choose v, s, or c. Then, by specifying the formulæ (56) and (57) for A = C or Cs and B = O or Os, we obtain all real forms25 of (60) which can be expressed as Rosenfeld lines over C(or Cs)⊗O(or Os); they are summarized by the following Table : Plane Isometry Isotropy #nc #c χ (C⊗ O)P 1 Spin (10) Spin (8)⊗U (1) 0 16 −16 (C⊗ O)H1 Spin (8, 2) Spin (8)⊗U (1) 16 0 16 (C⊗ Os)P 1 Spin (6, 4) Spin (4, 4)⊗U (1) 8 8 0 (Cs ⊗ O)P 1 Spin (9, 1) Spin (8)⊗ SO (1, 1) 8 8 0 (Cs ⊗ Os)P 1 Spin (5, 5) Spin (4, 4)⊗ SO (1, 1) 8 8 0 along with (C⊗ Os)P 1 ' (C⊗ Os)H1; (61) (Cs⊗O)P 1 ' (Cs⊗O)H1; (62) (Cs⊗Os)P 1 ' (Cs⊗Os)H1. (63) The real forms of (C⊗ O)P 1C (60) have the general structures Spin (p, 10− p;C) Spin (p, 8− p;C)⊗ U1 structure I or Spin (p, 10− p;C) Spin (p− 2, 10− p;C)⊗ U1 structure II , (64) Spin (p, 10− p;C) Spin (p− 1, 9− p;C)⊗ SO (1, 1) structure III , (65) Y16 := SO∗ (10) SO∗ (8)⊗U (1) ' SO∗ (10) Spin (6, 2)⊗U (1) , (66) with p = 0, 1, ..., 10, such that structures I and II get exchanged (while structure III is invari- ant) under p↔ 10− p. The planes (C⊗ O)P 1, (C⊗ O)H1, (C⊗ Os)P 1 respectively correspond to structure I with p = 0, 8, 6 (or structure II with p = 10, 2, 4), whereas (Cs⊗O)P 1 and (Cs⊗Os)P 1 respectively correspond to structure III with p = 9 (or p = 1) and p = 5. All other real forms of (C⊗ O)P 1C (60), and in particular (66), cannot be characterized as Rosenfeld lines over C(or Cs)⊗O(or Os). 6.3 H⊗ O In the quateroctonionic case, i.e. for H⊗ O, the starting point is given by the quateroctonionic projective line over C, i.e. by the locally symmetric coset manifold having as isometry group Spin (12,C) with Spin (8,C)⊗Spin(4,C) as isotropy group : (H⊗ O)P 1C ' Spin (12,C) Spin (8,C)⊗ Spin (4,C) . (67) 25By real forms of (C⊗ O)P 1C, we here mean the cosets with isometry groups given by all real (compact and non-compact) forms of Spin(10,C), and with isotropy group given by all (compact and non-compact) real forms of Spin(8,C)⊗ (U (1))C which are subgroups of the corresponding real form of Spin(10,C). 20 In this coset space formulation, the tangent space of (H⊗ O)P 1C can be identified with the ( 8(v,s,c),2,2 ) representation of Spin(8,C)⊗Spin(4,C), where we used Spin(4,C) 'SL(2,C)⊗SL(2,C) and, as above, d4-triality [Por] allows to equivalently choose v, s, or c. Then, by specifying the formulæ (56) and (57) for A = H or Hs and B = O or Os, we obtain all real forms26 of (67) which can be expressed as Rosenfeld lines over H(or Hs)⊗O(or Os); they are summarized by the following Table : Plane Isometry Isotropy #nc #c χ (H⊗ O)P 1 Spin (12) Spin (8)⊗ SU (2)⊗ SU (2) 0 32 −32 (H⊗ O)H1 Spin (8, 4) Spin (8)⊗ SU (2)⊗ SU (2) 32 0 32 (H⊗ Os)P 1 Spin (8, 4) Spin (4, 4)⊗ SU (2)⊗ SU (2) 16 16 0 (Hs ⊗ O)P 1 Spin (10, 2) Spin (8)⊗ SL (2,R)⊗ SL (2,R) 16 16 0 (Hs ⊗ Os)P 1 Spin (6, 6) Spin (4, 4)⊗ SL (2,R)⊗ SL (2,R) 16 16 0 along with (H⊗ Os)P 1 ' (H⊗ Os)H1; (68) (Hs⊗O)P 1 ' (Hs⊗O)H1; (69) (Hs⊗Os)P 1 ' (Hs⊗Os)H1. (70) The real forms of (H⊗ O)P 1C (67) have the general structures Spin (p, 12− p;C) Spin (p, 8− p;C)⊗ SU(2)⊗2 structure I or Spin (p, 12− p;C) Spin (p− 4, 12− p;C)⊗ SU(2)⊗2 1 structure II , (71) Spin (p, 12− p;C) Spin (p− 1, 9− p;C)⊗ SL(2,C)R structure III or Spin (p, 12− p;C) Spin (p− 3, 11− p;C)⊗ SL(2,C)R structure IV , (72) Spin (p, 12− p;C) Spin (p− 2, 10− p;C)⊗ SL(2,R)⊗2 structure V , (73) Y32 := SO∗ (12) SO∗ (8)⊗ SO∗ (4) ' SO∗ (12) Spin (6, 2)⊗ SU (2)⊗ SL (2,R) , (74) with p = 0, 1, ..., 12, such that structures I and II (and structures III and IV ) get exchanged (while structure V is invariant) under p↔ 12− p. (H⊗ O)P 1, (H⊗ O)H1 and (H⊗ Os)P 1 respectively correspond to structure I with p = 0, 8, 4 (or structure II with p = 12, 4, 8), whereas (Hs⊗O)P 1 and (Hs⊗Os)P 1 respectively correspond to structure V with p = 2 (or p = 10) and p = 6. All other real forms of (H⊗ O)P 1C (67), and in particular (74), cannot be characterized as Rosenfeld lines over H(or Hs)⊗O(or Os). 6.4 O⊗ O In the octooctonionic case, i.e. for O ⊗ O, the starting point is given by the octooctonionic projective line over C, i.e. by the locally symmetric coset manifold having as isometry group Spin (16,C) with Spin (8,C)⊗Spin(8,C) as isotropy group : (O⊗ O)P 1C ' Spin (12,C) Spin (8,C)⊗ Spin (8,C) . (75) 26By real forms of (H⊗ O)P 1C, we here mean the cosets with isometry groups given by all real (compact and non-compact) forms of Spin(12,C), and with isotropy group given by all (compact and non-compact) real forms of Spin(8,C)⊗Spin(4,C) which are subgroups of the corresponding real form of Spin(12,C). 21 In this coset space formulation, the tangent space of (O⊗ O)P 1C can be identified with the ( 8(v,s,c),8(v,s,c) ) representation of Spin(8,C)⊗Spin(8,C), where, as above, d4-triality [Por] al- lows to equivalently choose v, s, or c (in all possible pairs for the two Spin(8,C) factors of the isotropy group). Then, by specifying the formulæ (56) and (57) for A = O or Os and B = O or Os, we obtain all real forms27 of (75) which can be expressed as Rosenfeld lines over O(or Os)⊗O(or Os); they are summarized by the following Table : Plane Isometry Isotropy #nc #c χ (O⊗ O)P 1 Spin (16) Spin (8)⊗ Spin (8) 0 64 −64 (O⊗ O)H1 Spin (8, 8) Spin (8)⊗ Spin (8) 64 0 64 (O⊗ Os)P 1 Spin (12, 4) Spin (8)⊗ Spin (4, 4) 32 32 0 (Os ⊗ Os)P 1 Spin (8, 8) Spin (4, 4)⊗ Spin (4, 4) 32 32 0 along with (O⊗ Os)P 1 ' (O⊗ Os)H1, (76) (Os⊗Os)P 1 ' (Os⊗Os)H1. (77) The real forms of (O⊗ O)P 1C (75) have the general structures Spin (p, 16− p;C) Spin (p, 8− p;C)⊗ Spin (8,C) structure I or Spin (p, 16− p;C) Spin (p− 8, 16− p;C)⊗ Spin (8,C) structure II , (78) Spin (p, 16− p;C) Spin (p− 1, 9− p;C)⊗ Spin (1, 7;C) structure III or Spin (p, 16− p;C) Spin (p− 7, 15− p;C)⊗ Spin (1, 7;C) structure IV , (79) Spin (p, 16− p;C) Spin (p− 2, 10− p;C)⊗ Spin (2, 6;C) structure V or Spin (p, 16− p;C) Spin (p− 6, 14− p;C)⊗ Spin (2, 6;C) structure V I , (80) Spin (p, 16− p;C) Spin (p− 3, 11− p;C)⊗ Spin (3, 5;C) structure V II or Spin (p, 16− p;C) Spin (p− 5, 13− p;C)⊗ Spin (3, 5;C) structure V III , (81) Spin (p, 16− p;C) Spin (p− 4, 12− p;C)⊗ Spin (4, 4;C) structure IX , (82) Y64 := SO∗ (16) SO∗ (8)⊗ SO∗ (8) ' SO∗ (16) Spin (6, 2)⊗ Spin (6, 2) , (83) with p = 0, 1, ..., 16, such that structures I and II, III and IV , V and V I, V II and V III get exchanged (while structure IX is invariant) under p↔ 16− p. The planes (O⊗ O)P 1, (O⊗ O)H1 and (O⊗ Os)P 1 respectively correspond to structure I with p = 0, 8, 4 (or structure II with p = 16, 8, 12), whereas (Os⊗Os)P 1 corresponds to structure IX with p = 8. All other real forms of (O⊗ O)P 1C (75), and in particular (83), cannot be characterized as Rosenfeld lines over O(or Os)⊗O(or Os). 27By real forms of (O⊗ O)P 1C, we here mean the cosets with isometry groups given by all real (compact and non-compact) forms of Spin(16,C), and with isotropy group given by all (compact and non-compact) real forms of Spin(8,C)⊗Spin(8,C) which are subgroups of the corresponding real form of Spin(16,C). 22 7 Conclusions In this work, in order to provide a rigorous characterization of Rosenfeld “projective” spaces over (rank-2) tensor products of (division or split) Hurwitz algebras, we have introduced some “magic” formulæ (28)-(30) (for planes) and (56)-(57) (for lines), which allowed us to characterize many Rosenfeld spaces as symmetric (pseudo-)Riemannian cosets, whose isometry and isotropy groups have Lie algebras which are entries of real forms of the order-3 (Freudenthal-Tits [Ti, Fr65]) Magic Square, or of the order-2 (Barton-Sudbery [BS03]) Magic Square. For “projective” planes, the application of the “magic” formulæ (28)-(30) to the case in which at least one of the two Hurwitz algebras in the associated tensor product is given by the octonions O or by the split octonions Os, allows us to retrieve all (compact and non-compact) real forms of the corresponding octonionic Rosenfeld planes, except for a limited numbers of pseudo-Riemannian symmetric cosets, named X32,I , X32,II , X64,I , X64,II ,X64,III , X128,I and X128,II , and respectively given by (37), (38), (43), (44), (45), (52) and (53). All such cosets share a common property : up to some possible rank-1 Lie group factor, their isotropy group is given by the non-compact Lie group SO∗ (N) for N = 10, 12, 16. The fact that such spaces are not encompassed by our “magic” formulæ (28)-(30) is ultimately due to the fact that the corresponding Lie algebra so∗(N) does not occur in any real form of the Magic Square of order 2 (except for the case N = 8, for which it holds the special isomorphism so∗(8) ' so(6, 2)). For “projective” lines, the application of the “magic” formulæ (56)-(57) to the case in which at least one of the two Hurwitz algebras in the associated tensor product is given by the octonions O or by the split octonions Os, allows us to retrieve some (compact and non-compact) real forms of the corresponding octonionic Rosenfeld lines, but still a number of other real forms of Rosenfeld lines is left out. Again, there are spaces, such as Y16, Y32 and Y64 (respectively given by (66), (74) and (83)) that have their isometry and isotropy groups containing factors SO∗(M) for M = 4, 8, 10, 12, 16; but there are also other pseudo-Riemannian spaces left out, given by (59), (64)-(65), (71)-(73) and (78)-(82). The fact that some pseudo-Riemannian symmetric spaces are not covered by the classification yielded by the “magic” formulæ (28)-(30) and (56)-(57), which are related to the entries of the Magic Squares of order 2 and 3, means that not all real forms of Rosenfeld spaces can be realized as “projective” lines or planes over (rank 2) tensor products of (division or split) Hurwitz algebras. If one still believe that Rosenfeld’s approach was right after all, one might want to extend our “magic” formulæ to involve not only unital, alternative composition algebras, i.e. (division or split) Hurwitz algebras, but also other algebras which are not unital or alternative. 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