octonionic planes and real forms of G2, F4 and E6 D. Corradetti 0 A. Marrani 0 D. Chester 0 R. Aschheim 0 MSC: 17C36 , 17C60, 17C90, 22E15, 32M15 Keywords: Exceptional Lie Groups, Jordan Algebra, Octonionic Projective Plane, Real Forms, Veronese embedding

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane OP 2 through the use of Veronese vectors. O Then we focus on their relation with the exceptional Jordan algebra J3 and show that the Veronese vectors are the rank-1 elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of G2, F4 and E6 groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.

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M X r a

 Introduction The Octonionic Pro jective Plane

Octonions O are, along with Real numbers R, Complex numbers C and Quaternions H, one of the four Hurwitz algebras, more specifically are the only unital non-associative normed division algebra. A pratical way to work with them is to consider their R8 decomposition, i.e., where {i0 = 1, i1, ..., i7} is a basis of R8 and the multiplication rules are mnemonically encoded in the Fano plane (Fig. 1) along with i2k = −1 for k = 1, ..., 7. We then define the octonionic conjugate of x as with the usual norm

kxk2 = xx = x20 + x21 + x22 + x23 + x24 + x25 + x26 + x27 and the inner product given by the polarisation of the norm, i.e., hx, yi = kx + yk2 − kxk2 − kyk2 = xy + yx.

In respect to this norm the Octonions are a composition algebra, i.e. kxyk = kxk kyk, which will be of paramount importance in the following sections. Finally, we denote with Os the split-octonionic algebra, whose definition can be found in [13, 14].

x =

7 X xkik k=0 x := x0i0 −

7 X xkik k=1 The Projective Plane It is a common practice defining a projective plane over an associative division algebra starting from a vector space over the given algebra, e.g. Rn+1, and then define the projective space as the quotient

RP n = Rn+1/ ∼ where x ∼ y if x and y are multiple through the scalar field, i.e. λx = y, λ ∈ R, λ 6= 0, x, y ∈ Rn+1. But, since the algebra of Octonions is not associative, we have that x (λμ) 6= (xλ) μ when λ, μ ∈ O. If we then try to define the equivalence relation as above, we then might have x ∼ y = xλ, and x ∼ z = x (λμ), but z not related to y since

z = x (λμ) 6= (xλ) μ = yμ.

Therefore the previous is not an equivalence relation and the quotient cannot be defined. A method for overcoming such an issue is based on determining an equivalent algebraic definition of the rank-one idempotent of the exceptional Jordan algebra in order define points in the projective plane, but here we want to use a direct and less known way to proceed making use of the Veronese vectors. Veronese coordinates Let V =∼ O3 ×R3 be a real vector space, with elements of the form

(xν ; λν )ν = (x1, x2, x3; λ1, λ2, λ3) where xν ∈ O, λν ∈ R and ν = 1, 2, 3. A vector w ∈ V is called Veronese if λ1x1 = x2x3, λ2x2 = x3x1, λ3x3 = x1x2 kx1k2 = λ2λ3, kx2k2 = λ3λ1, kx3k2 = λ1λ2. ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) Let H ⊂ V be the subset of Veronese vectors. If w = (xν ; λν )ν is a Veronese vector then also μw = μ (xν ; λν )ν is a Veronese vector, that is Rw ⊂ H. We define the Octonionic plane OP 2 as the geometry having this one-dimensional subspaces Rw as points, i.e.

OP 2 = {Rw : w ∈ H r {0}} .

Remark 2.1. The point in the projective plane is defined as the equivalence class Rw of the Veronese vector w, but, in order to determine an explicit relation between points in the projective plane and rank-one idempotent elements of the Jordan algebra J3O, we will choose when as representative of the class the vector v = (yν ; ξν )ν ∈ Rw such that ξ1 + ξ2 + ξ3 = 1. Then (yν ; ξν )ν are called Veronese coordinates of the projective point.

Projective lines We then define projective lines of OP 2 as the vectors orthogonal to the points Rw. Let β be the bilinear form over O3 × R3 defined as where w1 = x1ν ; λ1ν ν ,w2 = x2ν ; λ2ν ν ∈ O3 × R3. Then, for every Veronese vector w, corresponding to the point Rw in OP 2, we define a line ` in OP 2 as the orthogonal space

` := w⊥ = z ∈ O3 × R3 : β (z, w) = 0 .

The bilinear form β also defines the elliptic polarity, i.e. the map π+ that corresponds points to lines and lines to points, i.e.

π+ (w) = w⊥, π+ w⊥ = w ( 12 ) where the orthogonal space to a vector is defined by the bilinear form β (·, ·), so that ( 11 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) ( 17 ) ( 18 ) (19) (20) π+ :w −→ {β (·, w) = 0}

` −→ w (x) 7→ R 0, 0, x; kxk2 , 1, 0 (∞) 7→ R (0, 0, 0; 1, 0, 0) . (x, y) 7→ R x, y, yx; kyk2 , kxk2 , 1 which is an homeomorphism. To complete the affine plane, we then have to extend the map to another set of coordinates, i.e. when ` is given by {β (·, w) = 0}. Explicitly, β (w1, w2) = 0 when 2x11x12 + 2x21x22 + 2x31x23 + λ11λ12 + λ21λ22 + λ31λ32 = 0.

In addition to the elliptic polarity defined above, we then define the hyperbolic polarity π−, which still has

π− (w) = w⊥, π− w⊥ = w but through the use of the bilinear form β− that has a change of sign in the last coordinate, i.e. β− (w1, w2) = 0 is given by

2x11x12 + 2x21x22 − 2x31x23 + λ11λ12 + λ21λ22 − λ31λ32 = 0.

A projective plane equipped with the hyperbolic polarity will be called hyperbolic plane and denoted as OH2.

The Affine Plane The octonionic projective plane is also the completion of the octonionic affine plane. The embedding of the affine plane can be explicited through the use of Veronese coordinates defining the map that sends a point (x, y) of the affine plane to the projective point R x, y, yx; kyk2 , kxk2 , 1 , i.e.

Remark 2.2. To show that the above is a Veronese vector and therefore that the map is well defined, we made essential use of alternativity of the Octonions and fact that Octonions are a composition algebra. In case of non-composition algebra, though the definition of the projective and hyperbolic planes would still be valid using Veronese coordinates, the geometry of these planes will not satisfy the basic axioms of projective and affine geometry and therefore they would have to be considered as "generalised" projective or hyperbolic planes.

Moreover, let [s, t] be a line in the affine plane OA2 of the form [s, t] = {(x, sx + t) : x ∈ O} where s is the slope of the line. Then [s, t] is mapped into the projective line orthogonal to the vector st, −t, −s; 1, ksk2 , ktk2 , i.e. (21) (22) (23) (24) Vertical lines [c] that are of the form {c} × (O) , are mapped into lines of OP 2 given by [c] 7→ R −c, 0, 0; 0, 1, kck2 ⊥ .

Finally the line at infinity [∞] is mapped to the orthogonal space of the vector [s, t] 7→ R st, −t, −s; 1, ksk2 , ktk2 ⊥

. [∞] 7→ R (0, 0, 0; 0, 0, 1)⊥ .

The Exceptional Jordan Algebra J3 (O) The exceptional Jordan algebra J3 (O) is the algebra of Hermitian three by three octonionic matrices with the Jordan product where I3 = diag(+, +, +), along with the symmetric trilinear form the quadratic form whose the previous bilinear form is a polarisation and the Freudenthal product, i.e.

We then have that

and the determinant Now, let Rw be a point in the projective plane OP 2, related to a vector in O3 × R3 with Veronese coordinates w = (xν ; λν )ν and consider the map from V =∼ O3 ×R3 into the space of three by three Hermitian matrices with octonionic coefficients, defined as (X, Y, Z) =

(X, Y ∗ Z) det (X) =

(X, X, X) .

 λ1 x3 x2  (xν ; λν )ν 7→  x3 λ2 x1  .

x2 x1 λ3 det (X) = λ1λ2λ3 − λ1 kx1k2 − λ2 kx2k2 − λ3 kx3k2 + 2Re ((x1x2) x3) (33) that, imposing the Veronese conditions translates to det (X) = 0.

Moreover, let X] be the image of a non-zero element X under the adjoint

O (]-)map of J3 , which is given by (cf. Example 5 of [12]) From this explicit expression, it is immediate to realize that the Veronese conditions are equivalent to the vanishing of X]. Then, by the Str(J3 (O))-invariant definition of the rank of an element of J3 (O) [9], one obtains that the Veronese conditions are equivalent to the rank-1 condition for an element of J3 (O).

Thus, from the knowledge of the orbit stratification of J3O under the nontransitive action of its reduced structure group Str0 (J3 (O)) ' E6(−26), it follows that the Veronese conditions for a non-zero element of J3 (O) are equivalent to imposing that such an element belongs to the (unique) rank-1 orbit of E6(−26) in J3 (O) (cf. [?], and Refs. therein).

Now we want to show that in the rank-1 (unique) orbit of J3 (O), idempotency is equivalent to the condition of unitary trace. In order to do that, let us consider the element X2 and let us impose the condition of X being of rank= 1. Since this condition is equivalent to the Veronese conditions, one obtains from which it follows that X2 = X if and only if i.e. if and only if tr (X) = 1. Thus, the idempotency condition for rank-1 elements of J3O is equivalent to the condition of unitary trace. 4

Lie Groups of Type G2, F4 and

of Collineations

 E6 as Groups

We are now interested in the motions and symmetries of the octonionic projective plane. More specifically we are interested in collineations that are transformations of the projective plane that send lines into lines. If the collineation preserves the elliptic polarity or the hyperbolic polarity is then called elliptic or hyperbolic motion. Elliptic and hyperbolic motions are an equivalent characterization of the isometries of the projective or hyperbolic plane respectively, thus the elliptic motion group of the projective plane will be indicated as Iso OP 2 and the hyperbolic motion group as Iso OH2 .

Collineations of the Octonionic Projective Plane A collineation is a bijection ϕ of the set of points of the plane onto itself, mapping lines onto lines. (35) (36) It is straightforward to see that the identity map is a collineation, as the inverse ϕ−1 and the composition ϕ ◦ ϕ are if ϕ, ϕ0 are both collineation. Therefore the 0 set Coll OP 2 of collineations is a group under composition of maps. It also has a proper subgroup of order three generated by the triality collineation that permutes three special points of the affine/projective octonionic plane, i.e. the origin of coordinate (0, 0), the point at the origin of the line at infinity which has coordinate (0) and the point at infinity of the line at the infinity which has affine coordinate (∞). In Veronese coordinates these three points are images of the following vectors (0, 0) −→ R (0, 0, 0; 0, 0, 1) (0) −→ R (0, 0, 0; 0, 1, 0) (∞) −→ R (0, 0, 0; 1, 0, 0) and the triality collineation τ is given by

(x1, x2, x3; λ1, λ2, λ3) 7→ (x2, x3, x1; λ2, λ3, λ1) that is a cyclic permutation of order three that leaves invariant the Veronese vectors. This means that it induces a bijection τ on OP 2 that is unseen by the bilinear form β and therefore maps lines into lines, since lines are constructed as the ortogonal space of a vector through the bilinear form β.

Let us now consider the transformations Ta,b of O3 × R3 into itself defined on the Veronese coordinates as x1 −→x1 + λ3a x2 −→x2 + λ3b x3 −→x3 + bx1 + x2a + λ3ba λ1 −→λ1 + hx2, ai + λ3 kbk2 λ2 −→λ2 + hx1, ai + λ3 kak2 Those are in fact translations on the affine plane corresponding to the transformation (x1, x2) −→ (x1 + a, x2 + b) and they all induce collineations Ta,b on OP 2.

It can be shown that all collineations are generated by the interplay between a translation and the conjugation of a power of the triality collineation, i.e. are of the form

Ta,b, τ Ta,bτ −1, τ 2Ta,bτ −2.

From another perspective, collineations transform lines of OP 2 in lines of OP 2. This is equivalent to find all the linear transformations A of V in itself such that the image of Veronese vectors is still a Veronese vector A (H) ⊂ H. If this condition is fulfilled, the linear transformation A in End (V ) will induce

O3 × R3

A −→

ϕ −→

O3 × R3

↑

OP 2 iso OP 2 = g2 ⊕ sa ( 3 ) Since all linear multiple of the transformation A will produce the same collineation ϕ, to have a bijection between linear transformations and collineations we have to impose also det (A) = 1. That is that the group of collineation Coll OP 2 is SL (V, H) := {A ∈ End (V ) ; A (H) ⊆ H; det (A) = 1} . (44) If we also impose the preservation of the elliptic polarity, i.e. of the bilinear form β, we will then have the group of elliptic motion Iso OP 2 that is

SU (V, H) = {A ∈ End (V ) ; A (H) ⊆ H; det (A) = 1; tr (A) = 1} . Those two groups are in fact two exceptional Lie Groups, i.e.

Coll OP 2 ∼= SL (V, H) =∼ E6

Iso OP 2 ∼= SU (V, H) =∼ F4.

The identification of this two group is done through a direct determination of the generators as in [13]; instead, we will here follow Rosenfeld in [4] focusing on the Lie algebra of the group of collineations on OP 2, i.e. coll OP 2 , which is given by coll OP 2

= g2 ⊕ a3 (O) where g2 =∼ der (O) and a3 are the three by three matrices on O with null trace, i.e. tr (A) = 0. The dimension count on the possible generators of this algebra, since the only condition you have is to have null trace, i.e. tr (A) = 0, gives as only condition on the condition on the trace, i.e. a33 = − a11 + a22 , and therefore we have 8 entries of dimension 8 and dimRa3 = 64. We therefore have

dimRcoll OP 2 ∼= 78 = 64 + 14.

Since coll OP 2 is a Lie group, simple and of dimension 78, then it must be of E6 type.

Isometries of the Plane Again, following Rosenfeld [4] we look at the elliptic motions of OP 2, which are the collineations that preserve also the polarity π+ or equivalently the form β; they are given by (43) (45) (46) (47) (48) (49) (50) (51) where we notated sa ( 3 ) the skew-Hermitian matrices with null trace. Here the elements of sa ( 3 ) are of the form with aij = aji, a33 = − a11 + a22 and Re a11 = Re a22 = 0. We therefore have 3 coefficient of dimension 8, 2 entries of dimension 7 and therefore dimRsa ( 3 ) = 38 so that

dimRiso OP 2 ∼= 52 = 38 + 14 and the group of elliptic motion Iso OP 2 , being simple and of dimension 52, is of the F4 type.

Moreover we can proceed as in [4] to find the collineations that preserve the hyperbolic polarity π− or equivalently the form β− we previously defined. Here the element of the Lie algebra are of the form (52) (53) (54) (55) (56) (57) (58) (59) where A, B and C are automorphisms with respect to the sum over O and that satisfy

B (sx) = C (s) A (x) .

Proof. A collineation ϕ that fixes (0, 0), (0) and (∞), also fixes the x-axis and yaxis and all lines that are parallel to them. This means that the first coordinate is the image of a function that does not depend on y and the second coordinate is image of a fuction that does not depend of x, i.e. (x, y) 7→ (A (x) , B (y)) and with a11 = a22 + a33 and Re a11 count of the dimension of = Re a22 = 0, therefore leading to the same deducing that Iso OH2 is again an F4 type group.

Collineations with a Fixed Triangle or Quadrangle We are now interested in studying the collineations ϕ on the affine plane that fix every point of 4, i.e. ϕ ((0, 0)) = (0, 0), ϕ ((0)) = (0) and ϕ ((∞)) = (∞).

Proposition 1. The group Γ (4, O) of collineations that fix every point of 4 are transformations of this form then C ( 1 ) = 1 and, therefore A = B = C and therefore A is an automorphism of O. We then have the following Proposition 2. The collineations that fix every point of of the type are transformations ( 1, 1 ) 7→ (A ( 1 ) , B ( 1 )) = ( 1, 1 ) (x, y) 7→ (A (x) , A (y))

(s) 7→ (A (s))

If we want this to be a collineation, the points of the line [s, t] must all belong to the same line which can be easily identified setting x = 0, i.e. the image of [s, t] is the line that joins the points (0, B (t)) and (C (s)). We now have that the condition for (A (x) , B (sx + t)) to be in the image of [s, t] is

B (sx + t) = C (s) A (x) + B (t) .

Now, if B is an automorphism with respect to the sum over O, we then have the condition B (sx) = C (s) A (x). Conversely if B (sx) = C (s) A (x) is true that B (sx + t) = B (sx) + B (t), and B is an automorphism with respect to the sum.

Let us consider the quadrangle given by the points (0, 0), ( 1, 1 ), (0) and (∞), that is = 4 ∪ {( 1, 1 )}, and consider the collineations that fix the . Since in addition to the previous case we also have to impose (s) 7→ (C (s)). Now consider the image of a point on the line [s, t]. The point is of the form (x, sx + t) and its image goes to where A is an automorphism of O.

Moreover, since Aut (O) = G2(−14) and Aut (Os) = G2( 2 ) [18], we have the following Corollary 1. The group of collineations Γ ( , O) that fix (0, 0), ( 1, 1 ), (0) and (∞) is isomorphic to Aut (O). Therefore Γ ( , O) is isomorphic to G2(−14), while in the case of split octonions Os is isomorphic to G2( 2 ).

It can be shown that the group of collineations Γ (4, O) is in fact the double cover of SO8 (R), i.e. Spin8 (R), that we define here as

Spin (O) = n(A, B, C) ∈ O+ (O)3 : A (xy) = B (x) C (y) ∀x, y ∈ Oo (66) where O+ is the connected component of the orthogonal group with the identity. Proposition 3. The Lie algebra Lie (Γ (4, O)) of the group of collineation that fixes (0, 0) , (0) and (∞) tri (O) = n(T1, T2, T3) ∈ so (O)3 : T1 (xy) = T2 (x) y + xT3 (y) o (67) while the Lie algebra Lie (Γ ( , O)) of the group of collineation that fixes (0, 0), ( 1, 1 ), (0) and (∞) is der (O) = {T ∈ so (O) : T (xy) = T (x) y + xT (y)} . (68) Proof. Γ (4, O) is a Lie group since it is a closed subgroup of the Lie group of collineations. We will find directly its Lie algebra considering the elements A, B, C ∈ Γ (4, O) in a neighbourhood of the identity and writing them as (A, B, C) −→ (Id + T1, Id + T2, Id + T3) where T1, T2, T3 ∈ so (O). Imposing the condition A (xy) = B (x) C (y) and then we obtain

(Id + T1) (xy) = (Id + T2) (x) (Id + T3) (x) which, considering 2 = 0, yields to

 We then have the following

The second part of the proposition is obtained imposing T1 = T2 = T3 = T .

T1 (xy) = T2 (x) y + xT3 (y) . Γ (4, O) ∼=Spin (O) ∼= Spin8 (R) Γ ( , O) ∼=Aut (O) ∼= G2(−14) Lie (Γ (4, O)) =∼tri (O) ∼= so (O)

Lie (Γ ( , O)) =∼der (O) ∼= g2(−14). Γ (4, Os) =∼Spin (Os) =∼ Spin( 4,4 ) (R) Γ ( , O) ∼=Aut (O) ∼= G2( 2 ) (69) (70) (71) (72) (73) (74) (75) (76) (77) (78) and, passing to Lie algebras, we obtain

By considering the split Octonions Os, previous formulas yield to and, passing to Lie algebras, we obtain

Lie (Γ (4, Os)) =∼tri (Os) =∼ so (Os) =∼ so4,4

Lie (Γ ( , O)) =∼der (O) ∼= g2( 2 ).

Resuming all the findings, following Yokota [18, p.105] in the definition of real forms of E6, we then have the following motion groups arising from the octonionic and split-octonionic projective and hyperbolic plane, i.e. Proj. Space

OPC2 OP 2 OsP 2 OsH2 OH2

Collineation group

E6C E6(−26)

E6( 6 )

E6( 2 ) E6(−14)

Isometry group

F4C F4(−52)

F4( 4 )

F4( 4 ) F4(−20) Γ ( )

C

G2 G2(−14)

G2( 2 )

G2( 2 ) G2(−14) 5

Classification of the Octonionic Pro jective Planes Thus, the space of rank-1 idempotent elements of J3 (O) enjoys the following expression as an homogeneous space (79) (81) (82) (83) (84) F4(−52) ,

Spin9 which is a compact Riemannian symmetric space, of (geodesic) rank = 1 and of real dimension dimR

F4(−52) Spin9 = dimR F4(−52) − dimR (Spin9) = 52 − 36 = 16, (80) as expected from the number of degrees of freedom characterizing rank-1 idempotents of J3 (O) itself. Since the unitary trace condition is imposed on top of Veronese conditions, the coset (79) is a (proper) submanifold of Orank 1 (J3 (O)), i.e.

F4(−52) ⊂ Spin9

E6(−26)

Spin9,1 n R16 .

The space (79) of rank-1 idempotent (or, equivalently, trace-1) elements of J3 (O) can be identified with the (compact real form of the) octonionic projective plane OP 2, which is the largest octonionic (projective) geometry; this can also be hinted from the fact that the tangent space to the coset F4(−52)/Spin9 transforms under the isotropy group Spin9 as its spinor irreducible representation 16, which can indeed be realized as a pair of octonions [16] :

F4(−52)

Spin9 f4(−52) = so9 ⊕ 16 ⇒ T ' 16 (of Spin9) ' O ⊕ O.

 Thus, one obtains that which gives an alternative definition of the octonionic projective plane. From the table in the previous section, we can then classify all possible octonionic planes. We start from the complexification of the Cayley plane

Orank 1 (J3 (O))

∪ OP 2 ∼ = ∼ =

E6(−26) Spin9,1nR16

∪ F4(−52)

Spin9 OP 2 (C) '

F4 (C)

Spin9 (C) and define three different real forms of the plane: a totally compact real coset, that identifies with OP 2; a totally non compact which is OH2; a pseudoRiemannian real coset that we will define as OHe 2. Those octonionic planes will be defined taking as isometry group F4(−52) and F4(−20), while the last real form F4( 4 ) will yield to projective planes on the split-octonionic algebra Os, i.e.

OP 2 ' F4(−52)

Spin9 OH2 ' F4(−20)

Spin9 Moreover, if we consider the type of the plane, i.e. the cardinality of noncompact and compact generators (#nc, #c), and the character χ, i.e. the difference between the two, χ = #nc − #c, we then note that: the totally compact plane, i.e. the classical Cayley plane or the octonionic projective plane OP 2, is of type ( 0, 16 ) and character χ = 16; the totally non-compact one, i.e. hyperbolic octonionic plane OH2, is of type ( 16, 0 ) and character χ = −16; while the other two planes named OHe 2 and OsHe 2 are of type ( 8, 8 ) and character χ = 0. 6

Conclusions

We have presented an explicit construction of the octonionic projective and hyperbolic planes and showed how Lie groups of type G2, F4 and E6 arise naturally as groups of motion of such planes. The fact that all different real forms of E6, F4 and G2 can be recovered from similar constructions is of the uttermost physical importance since different physical theories require different real forms of Lie groups. Compact and non-compact forms of G2 are notoriously known to be isomorphic to automorphisms of Octonions and split Octonions. In this paper we show how they can be thought as the subgroup of collineations that fix a quadrangle of the Projective plane over Octonions and Split-Octonions. Different compact and non compact real forms of E6 and F4 are related to different but analogue geometric frameworks such as projective planes or hyperbolic planes over the algebra of Octonions and split Octonions. Indeed while we recover E6(−26) and F4(−52) as collineation and isometry group of the octonionic projective plane OP 2, we have E6( 6 ) and F4( 4 ) for the split case OsP 2. The hyperbolic plane over Octonions and split-Octonions lead to E6(−14) and F4(−20) in the octonionic case OH2 or to E6( 2 ) and F4( 4 ) in the split case OsH2. The only real form left out is the compact E6(−78) which is obtained as isometry group of the complex Cayley plane or projective Rosenfeld plane over Bioctonions (C ⊗ O) P 2 [6]. Moreover, we have classified all octonionic and split-octonionic projective planes as symmetric spaces. 7

Acknowledgments

The work of D.Corradetti is supported by a grant of the Quantum Gravity Research Institute. The work of AM is supported by a “Maria Zambrano" distinguished researcher fellowship, financed by the European Union within the NextGenerationEU program.

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