Survey on real forms of the complex $A_2^{(2)}$-Toda equation and surface theory

The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for $k$-symmetric spaces over reductive Lie groups. In this survey we will show that to each of the five different types of real forms for a loop group of $A_2^{(2)}$ there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in $\mathbb {CP}^2$, minimal Lagrangian surfaces in $\mathbb {CH}^2$, timelike minimal Lagrangian surfaces in $\mathbb {CH}^2_1$, proper definite affine spheres in $\mathbb R^3$ and proper indefinite affine spheres in $\mathbb R^3$, respectively.


Introduction
Following the important work of Zakharov-Shabat [40] and Ablowitz-Kaup-Newell-Segur [1] in the 1970s, systematic constructions of hierarchies of integrable differential equations were developed. They were associated to a complex simple Lie algebra with various reality conditions given by finite order automorphisms. Mikhailov [27] first studied their reductions with various reality conditions given by finite order automorphisms. Drinfeld-Sokolov [18] constructed generalized KdV and mKdV hierarchies for any affine Kac-Moody Lie algebra using this ZS-AKNS scheme. In particular, the sine-Gordon equation and the sinh-Gordon equation are two real forms of the −1-flow or Toda-type equation in the mKdV-hierarchy for the simplest affine algebra A (1) 1 , which is a 2-dimensional extension of the loop algebra 1 of sl 2 C.
It is amazing that these two equations have already appeared in classical differential geometry for constant negative Gauss curvature surfaces (or pseudo-spherical surfaces) and constant mean curvature surfaces. For example, Bäcklund [2] constructed his famous transformation for pseudo-spheres around 1883, which produced many explicit solutions of the sine-Gordon equation ω xy = sin ω. This transformation and the higher flows in the hierarchy can be regarded as hidden symmetries of such submanifolds or differential equations. It has ever since become a central problem in geometry how to find special submanifolds in higher dimension and/or codimension which admit similar geometric transformations and have a lot of hidden symmetries, [35]. It is now natural to expect the answer to lie in integrable systems, as we will illustrate it further using next the rank 2 affine algebra A (2) 2 , which is a 2-dimensional extension of a loop subalgebra of sl 3 C, twisted by an outer automorphism σ, that is Λsl 3 C σ . Here the outer automorphism σ has order 6 and it is defined by σ(g)(λ) =σ(g(ǫ −1 λ)), for g(λ) ∈ Λsl 3 C, with ǫ = e πi/3 (the natural primitive sixth root of unity) andσ is the automorphism of sl 3 C given byσ Then a fundamental question for the affine algebra A 2 is, how many different real forms it has. In our case this means how many different real forms of Λsl 3 C σ there exist. The answer was given by [3,5,22]: there are 5 different real form involutions; T , (• A 3 ) τ (g)(λ) = Ad(I * P 0 ) g(1/λ), where I 2,1 = diag(1, 1, −1) and P 0 is as just above. Moreover, I * denotes I or I 2,1 .
It was Tzitzéica [39] who found a special class of surfaces in Euclidean geometry, which turns out to be equivalent to indefinite affine spheres in equi-affine geometry. They are related to the real form involution ( iA 3 ) given by τ (g)(λ) = g(λ) above. More precisely, the coordinate frame of an affine sphere with the additional loop parameter is fixed by the above real form involution. More recently, minimal Lagrangian surfaces in CP 2 or special Lagrangian cone in C 3 have been related to the involution (• CP 2 ) given by τ (g)(λ) = −g(1/λ) T , see [32] or [14].
In this survey, we relate all real forms of the affine algebra A where I * denotes I for the elliptic case I 2,1 for the hyperbolic case. Then each of the classes of surfaces can be characterized by some Tzitzéica equation 2 : (• CP 2 ) ω CP 2 zz + e ω CP 2 − |Q CP 2 | 2 e −2ω CP 2 = 0, Q CP 2 z = 0, (• CH 2 ) ω CH 2 zz − e ω CH 2 + |Q CH 2 | 2 e −2ω CH 2 = 0, Q CH 2 z = 0, take values in iR and Q iA 3 , R iA 3 take values in R, respectively.
It is known that the above equations are different real forms of the −1-flow in the corresponding A 2 -mKdV hierarchy, or the complex A 2 -Toda field equation; and the real groups are exactly the automorphism groups of the corresponding geometries.
The fifth equation ( iA 3 ) has been studied in the context of gas dynamics [21] and pseudohyper-complex structures on R 2 ×RP 2 [19], and it is also related to harmonic maps from R 1,1 to the symmetric space SL 3 R/SO 2,1 R. The fourth equation (• A 3 ) above can help construct semi-flat Calabi-Yau metrics and examples for the SYZ Mirror Symmetry Conjecture, see [29,20]. Specially the local radially symmetric solutions turn out to be Painlevé III transcendents. It is a striking universal feature of integrable systems that the same equation often arises from many unrelated sources. To further convince the reader of the great varieties here, we mention that minimal surfaces and Hamiltonian stationary Lagrangian surfaces in CP 2 and CH 2 [23] also correspond to solutions of integrable systems associated to sl 3 C, but with different automorphisms (of order 3 and order 4 respectively).
One should also observe that in [26] already all real forms of the affine algebra A (1) 1 have been related to constant mean curvature/constant Gaussian curvature surfaces in the Euclidean 3-space, the Minkowski 3-space or the hyperbolic 3-space.
The systematic construction from Lie theory above is just the starting point. It naturally gives rise to loop group factorizations, which in turn provide a method for constructing explicit solutions and symmetries of the equations. For example the classical Bäcklund and Darboux transformations have been generalized to dressing actions via loop group factorizations, see for examples Terng-Uhlenbeck [36] or Zakharov-Shabat [40]. The classical Weierstrass representation of minimal surfaces has also been generalized by Dorfmeister-Pedit-Wu, [15], using Iwasawa type loop group factorizations. Many interesting questions naturally arise by translating between holomorphic/meromorphic data and properties of special geometric objects or special solutions of integrable PDEs. Although the original DPW method only considered surfaces of conformal type (that is, associated with elliptic PDEs), it has also been generalized to surfaces of asymptotic line type (that is, associated with hyperbolic PDEs), such as constant negative Gaussian curvature surfaces given by sine-Gordon equation, [37]. Another way to get a very special class of solutions, called the finite type or finite gap solutions, has beautiful and deep links to geometries of algebraic curves or Riemann surfaces and stable bundles over them, the so-called Hitchin systems.
The paper is organized as follows: After discussing in the following sections one geometry for each real form of A (2) 2 we will compare their similarities and differences in Section 6 by the loop group method. To be self-contained and also to put this survey into a larger context, we discuss the classification of our real forms in the last Section 7 from a geometric point of view.

Minimal Lagrangian surfaces in CP 2
In this section, we discuss a loop group formulation of minimal Lagrangian surfaces in the complex projective plane CP 2 . The detailed discussion can be found in [31] or [30]. In the following, the subscripts z andz denote the derivatives with respect to z = x + iy and z = x − iy, respectively, that is, 1.1. Basic definitions. We first consider the five-dimensional unit hypersphere S 5 as a quadric in C 3 ; where , is the standard Hermitian inner product in C 3 which is complex anti-linear in the second variable. Then let CP 2 be the two-dimensional complex projective plane and consider the Hopf fibration π : Moreover, the space H u = {v ∈ T u S 5 | v, u = 0} is a natural horizontal subspace. The form , is a positive definite Hermitian inner product on H u with real and imaginary components , = g( , ) + iΩ( , ).
Hence g is positive definite and Ω is a symplectic form. Put and SU 3 = {A ∈ U 3 | det A = 1}. We note U 3 = S 1 · SU 3 and that these are connected real reductive Lie groups with their centers consisting of multiples of the identity transformation. Then the groups U 3 and SU 3 act naturally on S 5 and CP 2 . The group U 3 acts transitively on both spaces. Moreover, this action is equivariant relative to π and holomorphic on CP 2 . Using the base point e 3 = (0, 0, 1) T it is easy to verify

1.2.
Horizontal lift and fundamental theorem. We now consider a Lagrangian immersion f CP 2 from a Riemann surface M into CP 2 . Then it is known that on an open and contractible subset D of M, there exists a special lift into S 5 , that is, 11) and the integrability conditions are (1.12) ω CP 2 zz + e ω CP 2 − |Q CP 2 | 2 e −2ω CP 2 = 0, Q CP 2 z = 0. The first equation (1.12) is commonly called the Tzitzéica equation. From the definition of Q CP 2 in (1.8), it is clear that C CP 2 (z) = Q CP 2 (z) dz 3 is the holomorphic cubic differential of the minimal Lagrangian surface f CP 2 .
Remark 1.4. The fundamental theorem in Theorem 1.3 is still true for a minimal Lagrangian immersions into CP 2 .
1.4. Associated families of minimal surfaces and flat connections. From (1.12), it is easy to see that there exists a one-parameter family of solutions of (1.12) parametrized by λ ∈ S 1 ; The corresponding family {ω λ CP 2 , C λ CP 2 } λ∈S 1 then satisfies As a consequence, there exists a one-parameter family of minimal Lagrangian surfaces The family {f λ CP 2 } λ∈S 1 will be called the associated 6 family of f CP 2 . LetF λ CP 2 be the coordinate frame off λ CP 2 . Then the Maurer-Cartan form α λ CP 2 =Û λ CP 2 dz +V λ CP 2 dz ofF λ CP 2 for the associated family {f λ CP 2 } λ∈S 1 is given by U CP 2 and V CP 2 as in (1.11) where we have replaced Q CP 2 and Q CP 2 by λ −3 Q CP 2 and λ 3 Q CP 2 , respectively. Then consider the gauge transformation G λ given by Hence we will not distinguish between {f λ CP 2 } λ∈S 1 and {f λ CP 2 } λ∈S 1 , and both families will be called the associated family of f CP 2 , and F λ CP 2 will also be called the coordinate frame of f λ CP 2 . From the discussion just above we derive a family of Maurer-Cartan forms α λ CP 2 in (1.14) of minimal Lagrangian surfaces from D to CP 2 . They can be computed explicitly as It is clear that α λ CP 2 | λ=1 is the Maurer-Cartan form of the coordinate frame F CP 2 of f CP 2 . Then by the discussion in the previous section, we have the following theorem. Conversely, given a family of connections d + α λ CP 2 on D × SU 3 , where α λ CP 2 is as in (1.15), then d + α λ CP 2 belongs to an associated family of minimal Lagrangian immersions into CP 2 if and only if the connection is flat for all λ ∈ S 1 .

Minimal Lagrangian surfaces in CH 2
In this section, we discuss a loop group formulation of minimal Lagrangian surfaces in the complex hyperbolic plane CH 2 . Most of what we present can be found in [28]. We will use complex parameters and restrict generally to surfaces defined on some open and simply connected domain D of the complex plane C.
2.1. Basic definitions. The space CH 2 can be realized as the open unit disk in C 2 relative to the usual positive definite Hermitian inner product. But for our purposes it will be more convenient to realize CH 2 in the form It is natural then to consider on C 3 1 the indefinite Hermitian inner form of signature (1, 2) given by Vectors in C 3 1 satisfying u, u < 0 will be called "negative". Then the set (C 3 1 ) − of negative vectors and the "negative sphere" , and the natural (submersions) projections π : (C 3 1 ) − → CH 2 and π : H 5 1 → CH 2 will be the central objects of this section. (Note that we use the same letter for both projections.) This is called the Boothby-Wang type fibration, [7,11]. For later purposes we point out that the tangent space at u ∈ H 5 1 is is a natural horizontal subspace. The form , is a positive definite Hermitian inner product on H u with real and imaginary components , = g( , ) + iΩ( , ).
Hence g is positive definite and Ω is a symplectic form. Clearly, the isometry group of , will be of importance in our setting. Put We note U 2,1 = S 1 · SU 2,1 and that these are connected, real, reductive Lie groups with their centers consisting of multiples of the identity transformation.
The groups U 2,1 and SU 2,1 act naturally on H 5 1 and on CH 2 . The group U 2,1 acts transitively on both spaces. Moreover, this action is equivariant relative to π and holomorphic on CH 2 . Using the base point e 3 = (0, 0, 1) T it is easy to verify

2.2.
Horizontal lift and fundamental theorem. We now consider a Lagrangian immersion f CH 2 from a Riemann surface M into CH 2 . Then it is known that on an open and contractible subset D of M, there exists a special lift into H 5 1 , that is, and it is called a horizontal lift. Moreover, any two such horizontal lifts only differ by a constant multiplicative factor from S 1 . 8) and the integrability conditions are Note, the first of these two equations is one of the Tzitzéica equations (which differ from each other by some sign(s)). From the definition of Q CH 2 in (2.7), it is clear that is the holomorphic cubic differential of the minimal Lagrangian surface f CH 2 .
Remark 2.4. The fundamental theorem in Theorem 2.3 is still true for a minimal Lagrangian immersions into CH 2 .
2.4. Associated families and flat connections. From (2.9), it is easy to see that there exists a one-parameter family of solutions of (2.9) parametrized by λ ∈ S 1 . The corresponding family {ω λ CH 2 , C λ CH 2 } λ∈S 1 then satisfies ω λ CH 2 = ω CH 2 , C λ CH 2 = λ −3 Q CH 2 dz 3 . As a consequence, there exists a one-parameter family of minimal Lagrangian surfaces The family {f λ CH 2 } λ∈S 1 will be called the associated family of f CH 2 . LetF λ CH 2 be the coordinate frame off λ CH 2 . Then the Maurer-Cartan form α λ CH 2 =Û λ CH 2 dz +V λ CH 2 dz ofF λ CH 2 for the associated family {f λ CH 2 } λ∈S 1 is given by U CH 2 and V CH 2 as in (2.8) where we have replaced Q CH 2 and Q CH 2 by λ −3 Q CH 2 and λ 3 Q CH 2 , respectively. Then consider the gauge transformation G λ given by Hence we will not distinguish between {f λ CH 2 } λ∈S 1 and {f λ CH 2 } λ∈S 1 , and both families will be called the associated family of f CH 2 , and F λ CH 2 will also be called the coordinate frame of f λ CH 2 .
From the discussion just above we obtain that the family of Maurer-Cartan forms α λ CH 2 in (2.11) of a minimal Lagrangian surface f CH 2 : M → CP 2 can be computed explicitly as It is clear that α λ CH 2 | λ=1 is the Maurer-Cartan form of the coordinate frame F CH 2 of f CH 2 . Then by the discussion in the previous section, we have the following theorem.  Conversely, given a family of connections d + α λ CH 2 on D × SU 2,1 , where α λ CH 2 is as in (2.12), then d + α λ CH 2 belongs to an associated famiy of minimal Lagrangian immersions into CH 2 if and only if the connection is flat for all λ ∈ S 1 . 2 1 In this section, we discuss a loop group formulation of timelike minimal Lagrangian surfaces in the complex projective plane CH 2 1 . The detailed discussion can be found in [13]. Here we use that the subscripts u and v denote the derivatives with respect to u and v, respectively, that is,

Timelike minimal Lagrangian surfaces in CH
3.1. Basic definitions. Let and consider the three-dimensional complex Hermitian flat space C 3 2 with signature (2, 1).
Let H 5 3 be the indefinite sphere (note again that the signature of C 3 2 is (2, 1)) , w, w < 0} Then there exists the Boothby-Wang type fibration [7,11] π : H 5 3 → CH 2 1 given by w → C × w. The tangent space of H 5 3 at p ∈ H 5 3 is Moreover, the space H p = {w ∈ T p H 5 3 | w, p = 0} is a natural horizontal subspace. The form , is an indefinite Hermitian inner product on H u with real and imaginary components , = g( , ) + iΩ( , ).
Hence g is indefinite and Ω is a symplectic form. Put satisfying Aw, Aq = w, q }, and SU 2,1 = {A ∈ U 2,1 | det A = 1}. We note U 2,1 = S 1 · SU 2,1 and that these are connected real reductive Lie groups with their centers consisting of multiples of the identity transformation. Since, SU 2,1 and SU 2,1 are isomorphic groups, so they are both connected. The groups U 2,1 and SU 2,1 act naturally on H 5 3 and CH 2 1 . The group U 2,1 acts transitively on both spaces. Moreover, this action is equivariant relative to π and holomorphic on CH 2 1 . Using the base point e 3 = (0, 0, 1) T it is easy to verify

3.2.
Horizontal lift and fundamental theorem. We now consider a timelike Lagrangian immersion f CH 2 1 from a surface M into CH 2 1 . Then it is known that on an open and contractible subset D of M, there exists a special lift into H 5 3 , that is, holds, see [13]. The lift f CH 2 It is straightforward to see that F CH 2 1 takes values in U 2,1 , that is, For what follows it will be convenient to lift the mean curvature vector of f CH 2 It is easy to verify that the vectors f CH 2 v , H denotes the mean curvature vector, and Q CH 2 1 and R CH 2 1 are purely imaginary functions defined by u , H . Moreover, ℓ and m take values in iR.

satisfies the Maurer-Cartan equations if and only if
1 is a conformal Lagrangian immersion and let f CH 2 1 denote one of its horizontal lifts and F CH 2 1 the corresponding coordinate frame (3.7). Then α CH 2 have the form (3.9) and their coefficients satisfy the equations stated in Corollary 3.2.
Conversely, given a functions ω CH 2 1 , H on D together with a cubic differential Q CH 2

3.3.
Timelike minimal Lagrangian surfaces CH 2 1 . If we restrict to minimal timelike Lagrangian surfaces, then the equations (3.9) together with ℓ = m = 0 show that the determinant of the coordinate frame is a constant (in S 1 ). So we can, and will, assume from here on that the horizontal lift of the given minimal immersion into CH 2 1 is scaled (by a constant in S 1 ) such that the corresponding coordinate frame F CH 2 1 is in SU 2,1 . It is clear that 13 the Maurer-Cartan form α CH 2 dv of the minimal surface is given by The integrability conditions stated in the corollary above then are The first equation (3.12) is again one of the Tzitzéica equations. From the definition of Q CH 2 1 in (3.10), it is clear that is the purely imaginary cubic differential of the timelike minimal Lagrangian surface f CH 2 1 . Conversely, let C CH 2 1 be a cubic differential and let ω CH 2 1 be a solution of (3.12). Then there exists a frame F CH 2 1 taking values in U 2,1 and a timelike minimal Lagrangian surface given by 3.4. Associated families of minimal surfaces and flat connections. From (3.12), it is easy to see that there exists a one-parameter family of solutions of (3.12) parametrized by As a consequence, there exists a one-parameter family of timelike minimal Lagrangian surfaces {f λ and λ 3 R CH 2 1 , respectively. Then consider the gauge transformation G λ given by 1 } λ∈R + , and both families will be called the associated family of f CH 2 1 , and F CH 2 1 will also be called the coordinate frame of f λ : M → CH 2 1 can be computed explicitly as (3.15) α λ Then by the discussion in the previous section, we can characterize a minimal Lagrangian immersion in CH 2 1 in terms of a family of flat connections. Conversely, given a family of connections d + α λ belongs to an associated famiy of minimal Lagrangian immersions into CH 2 1 if and only if the connection is flat for all λ ∈ R + .

Definite Proper Affine Spheres
In this section, we discuss a loop group formulation of definite proper affine spheres. The detailed discussion can be found in [16,17]. The general theory of affine submanifolds can be found in [33]. We will use again complex coordinates and again restrict to surfaces defined on some simply-connected open subset D of C.

4.1.
Basic definitions and results. Classical affine differential geometry studies the properties of an immersed surface f A 3 : D → R 3 which are invariant under the equi-affine transfor- The following form in local coordinates (u 1 , u 2 ) is naturally an equi-affine invariant: which induces an equi-affinely invariant quadratic form conformal to the Euclidean second fundamental form, called the affine metric g, by Λ = g ⊗ vol(g). Although the Euclidean angle is not invariant under affine transformations, there exists an invariant transversal vector field ξ along f (D) defined by ξ = 1 2 ∆f , called the affine normal. Here ∆ is the Laplacian with respect to g.
Another way to find the affine normal up to sign is by modifying the scale and direction of any transversal vector field (such as the Euclidean normal) to meet two natural characterizing conditions: is tangent to the surface for any X ∈ T p D, (ii) ξ A 3 and g induce the same volume measure on D: The formula of Gauss, * Y into tangential and transverse component, induces a torsionfree affine connection ∇ on D. Its difference with the Levi-Civita connection ∇ g of g is measured by the affine cubic form defined as: . It is actually symmetric in all 3 arguments. The affine shape operator S defined by the formula of Weingarten: , is self-adjoint with respect to g. The affine mean curvature H and the affine Gauss curvature K are defined as In the following we assume that the affine metric g is definite. This means that f A 3 (D) is locally strongly convex and oriented (since its Euclidean second fundamental form is positive definite). Then there exist conformal coordinates (x, y) ∈ D, that is, Then it is known that the affine normal ξ A 3 of a Blaschke immersion can be represented in the form The affine normal ξ A 3 points to the concave side of the surface, and the orientation given by i dz ∧ dz or du ∧ dv is consistent with the orientation induced by ξ A 3 . This z coordinate essentially defines D as a uniquely determined Riemann surface.
Alternatively we are studying affine-conformal immersions f of any Riemann surface D into R 3 : The first condition here reflects that f A 3 is affine-conformal. Moreover, we introduce a function Q A 3 by Then direct computations derive the fundamental affine invariants: g = 2e ω A 3 |dz| 2 by (4.1) and C A 3 = Q A 3 dz 3 + Q A 3 dz 3 by (4.2) and (4.3). We also have The shape operator S has the special form Maurer-Cartan form and Tzitzéica equation. The relations discussed above can also be illustrated by computing the evolution equations for the positively oriented frame Theorem 4.1. The Maurer-Cartan form The compatibility condition ( is equivalent to the two structure equations:  ). Assume f A 3 : D → R 3 is an affine-conformal immersion. Define ω A 3 , Q A 3 , H and the frame F A 3 as above. Then its affine metric is g = 2e ω A 3 |dz| 2 , its affine cubic form is C A 3 = Q A 3 dz 3 + Q A 3 dz 3 , and they satisfy the compatibility conditions (4.10) and (4.11), which are also equivalent to the flatness of having the form (4.9). Conversely, given a positive symmetric 2-form g = 2e ω A 3 |dz| 2 and a symmetric 3-form C A 3 = Q A 3 dz 3 + Q A 3 dz 3 on D ⊂ C such that H defined by (4.10) satisfies (4.11), then there exists a surface (unique up to affine motion) such that g, C A 3 are the induced affine metric and affine cubic form respectively.

Definite affine spheres.
A definite affine sphere is defined to be any affine surface with definite Blaschke metric having all affine normals meet at a common point which will be called its center, or where all affine normals are parallel. Equivalently an affine sphere is defined to be any "umbilical" affine surface (that is, S is a scalar function multiple of the identity everywhere).
By the matrix form (4.7) of the shape operator S, a definite affine sphere necessarily satisfies Q A 3 z = 0, that is, Q A 3 is holomorphic. Then the above Codazzi equation (4.11) implies Hz = 0, whence H = const., since H is real.
4.3.1. Types of affine spheres. So far we know that definite affine spheres have constant affine mean curvature H. Then a definite affine sphere is called elliptic, parabolic or hyperbolic, when its affine mean curvature H is positive, zero or negative respectively.
When H = 0, it is also called "improper"; and ξ A 3 is a constant vector which will usually be set to (0, 0, 1) t by some equi-affine transformation. Its center is at infinity. The only complete ones are paraboloids.
When the shape operator S in (4.7) satisfies S = HI = 0, the corresponding affine sphere will be called "proper". In this case we obtain ) with some f A 3 0 being the center of the affine sphere. For simplicity, we will always make f A 3 0 = 0 by translating the surface. (1) Elliptic definite affine spheres have centers 'inside' the surfaces and the only complete ones are ellipsoids. But the center of a hyperbolic definite affine sphere is 'outside'. They were considered in Calabi's conjecture for hyperbolic affine hyperspheres of any dimension (proved by Cheng-Yau [10], et al): Inside any regular convex cone C, there is a unique properly embedded or complete (with respect to the affine metric) hyperbolic affine sphere which has affine mean curvature −1, has the vertex of C as its center, and is asymptotic to the boundary ∂C. Conversely any properly embedded or complete hyperbolic affine sphere is asymptotic to the boundary of the cone C given by the convex hull of itself and its center. (2) It is clear that Qdz 3 is a globally defined holomorphic cubic differential (that is, in and only if f (D) is part of a quadric surface. So Q is nonzero except for the quadrics. Near any point z 0 which is not any of the isolated zeroes of Q one could make a holomorphic coordinate change to normalize Q to a nonzero constant, but we will not do that now, since then we have no control over the behaviour of Q "far away" from z 0 . The zeroes of Q will be called "planar" points of the affine sphere.
(3) We remark that the immersion is analytic for any definite affine sphere, since the defining equation is a fully nonlinear Monge-Ampere type elliptic PDE, see for example [6, §76].
It is easy to see that the Maurer-Cartan form dz of a definite affine sphere can be computed as In summary we obtain the governing equations for definite affine spheres in R 3 :

4.4.
A family of flat connections. From now on we will consider exclusively the case of proper definite affines spheres. Then we can and will scale the surface by a positive factor to normalize H = ±1. The following observation is crucial for the integrability of definite affine spheres: The system (4.13) is invariant under Q A 3 → λ 3 Q A 3 for any λ ∈ S 1 . Thus there exists a one-parameter family of solutions of (4.13) parametrized by λ ∈ S 1 ; The corresponding family {ω λ As a consequence, there exists a one-parameter family of definite affine spheres {f λ For the elliptic case (that is, H = 1), applying the gauge G λ = diag(iλ, iλ −1 , 1) toα λ A 3 , that is, (4.16) For the hyperbolic case (that is, H = −1), applying the gauge G λ − = diag(λ, λ −1 , 1) toα λ A 3 , that is, In both cases α λ takes value in the order 6 twisted loop algebra Λsl 3 C σ , but it is contained in different real forms, namely in the real forms induced by τ (X) = Ad(I 2,1 P 0 )X for the hyperbolic case, and by τ ′ (X) = Ad(P 0 )X for the elliptic case. These two real forms are equivalent and both commute with σ, but, obviously, the associated geometries are very different. (1) Indeed definite affine spheres have two different geometries or elliptic PDE because there are two open cells in the corresponding Iwasawa decomposition, as explained in [16]: To simplify notation, denote this group of twisted loops ΛSL 3 C σ by G. Then G τ and G + denote respectively the subgroups of τ -real loops and the loops with holomorphic extension to the unit disc in C. Iwasawa decomposition means the double coset decomposition G τ \G/G + . The following observation makes it possible to have a unified treatment of elliptic and hyperbolic definite affine spheres. Let s 0 := diag(λ, −λ −1 , −1)P 0 . There are exactly two open τ 2 -Iwasawa cells G τ 2 G + and G τ 2 s 0 G + , which are essentially the same as two open τ ′ 2 -Iwasawa cells (but interchanged): (2) We may conjugate the complex frame to a real SL 3 R-frame: In fact F R = (e 1 , e 2 , ξ) with {e 1 , e 2 } being simply an orthonormal tangent frame with respect to the affine metric. Recall that we obtain the immersion f A 3 = − 1 H ξ A 3 from the last column. It is clear now that we may also simply take the real part of the last column of F λ A 3 ± to get an equivalent affine sphere modulo affine motions.
We now define the two subgroups SL 3 R It is easy to verify that both groups are isomorphic to SL 3 R It is remarkable that a simple condition characterizes the extended frames of proper definite affine spheres: Theorem 4.6 ( [16]). Let f A 3 : D → R 3 be a definite affine sphere in R 3 and let α λ A 3 ± be the family of Maurer-Cartan forms defined in (4.15) or (4.18). Then d + α λ A 3 ± gives a family of flat connections on D × SL 3 R ± .

Conversely, given a family of connections
where α λ A 3 ± is as in (4.15) or (4.18), then d + α λ A 3 ± belongs to an associated family of affine spheres into R 3 if and only if the connection is flat for all λ ∈ S 1 .
Proof. We have discussed the first part of the theorem above. Concerning the converse direction we only show, for simplicity, the hyperbolic (that is H = −1 in the flat connection (4.18)). The positive case is completely parallel.
The reality conditions for σ and τ 2 guarantee that F −1 Fz is affine in λ. So we have (4.20) with A ∈ g −1 , B ∈ g 0 , C = τ (A), and D = τ (B). The fixed points of both σ and τ are of the form diag(e iβ , e −iβ , 1). Gauging by them respects the reality conditions. Let e iβ = ± A 13 |A 13 | . 20 Use it to scale A 13 to a real positive function which then is written in the form e ψ/2 . The rest follows from the equations of flatness. We observe that the equation (4.13) for hyperbolic definite affine spheres is the elliptic version of the above when H = −1 and Q = 1. Both admit the trivial solution ω ≡ 0, and the corresponding surfaces are x 1 x 2 x 3 = 1 and (x 2 1 + x 2 2 )x 3 = 1 respectively. However, the equation (4.13) for elliptic definite affine spheres admits no constant real solution, and some elliptic function examples will be given in [16].

Indefinite proper Affine spheres
In this section, we discuss a loop group formulation of indefinite proper affine spheres. The detailed discussion can be found in [12].

Blaschke immersions and its Maurer-
Cartan. Let f iA 3 : D → R 3 be a Blaschke immersion, that is, there exists a unique affine normal field ξ iA 3 (up to sign) such that the volume element of the affine metric ds 2 = g (which is determined by the second derivative of f iA 3 and commonly called the Blaschke metric) and the induced volume element on D ⊂ R 2 coincide, that is, In the following we assume that the Blaschke metric ds 2 = g is indefinite. Then there exist null coordinates (u, v) ∈ D [38] or [4,Prop 14. 1. 18], that is, (5.2) ds 2 = 2e ω iA 3 dudv holds for some real valued function ω iA 3 : D → R. Then the affine normal ξ iA 3 can be represented as where ∆ denotes Laplacian of the indefinite Blaschke metric. Combining (5.1) with (5.2), we have Note that the null coordinates can be rephrased as follows: , see (5.2). Moreover, we can introduce two functions . From the definition of Q iA 3 and R iA 3 in (5.5), it is clear that is a cubic differential for the null Blaschke immersion f iA 3 . The shape operator S = [s ij ], which is defined by the Weingarten formula, has relative to the basis {∂ u , ∂ v }, where u and v are null coordinates, the special form: Here H ∈ R is the affine mean curvature of f iA 3 . Then the coordinate frame of f iA 3 is defined by and from (5.1), it is easy to see that F iA 3 takes values in SL 3 R. Moreover, a straightforward computation shows that the following lemma holds.
Lemma 5.1. The Maurer-Cartan form Corollary 5.2. The compatibility conditions for the system of equations stated just above are Theorem 5.3 (Fundamental theorem for indefinite Blaschke immersions). Let f iA 3 : D → R 3 be a Blaschke immersion with affine normal ξ iA 3 , indefinite Blaschke metric in null coordinates u and v, ds 2 = 2e ω iA 3 dudv, affine mean curvature H and cubic differential C iA 3 = Q iA 3 du 3 + R iA 3 dv 3 . Then the coordinate frame satisfies the Maurer-Cartan equation (5.8). Here the coefficient matrices U iA 3 and V iA 3 have the form (5.9) and their coefficients satisfy the equations stated in Corollary 5.2.
Conversely, given functions ω iA 3 , H on D together with a cubic differential Q iA 3 du 3 + R iA 3 dv 3 such that the conditions of Corollary 5.2 are satisfied, then there exists a solution F iA 3 ∈ SL 3 R to the equation (5.8) such that f iA 3 = F iA 3 e 3 is an indefinite Blaschke immersion with null coordinates.

5.2.
Indefinite affine spheres. From here on we will consider affine spheres. As already pointed out in the last section this means that the shape operator s is a multiple of the identity matrix. We will also assume that the Blaschke metric is indefinite. There are still two very different cases: Case H = 0: these affine spheres are called improper. They are very special and well known. We will not consider this case. Case H = 0: such affine spheres are called proper. From now on, we will consider exclusively the proper case, and by a scaling transformation we can assume that H = −1. Affine spheres with this property are called indefinite proper affine spheres. Then the Weingarten formula can be represented as , that is the affine normal ξ iA 3 is the proper affine sphere f iA 3 itself up to a constant vector, that is, ξ iA 3 = f iA 3 + p, where p is some constant vector. By an affine transformation we can assume without loss of generality p = 0, and thus we have If we restrict to affine spheres, then the coefficient matrices of the Maurer-Cartan equation Moreover, the integrability conditions now are 14) The first equation in (3.12) is again a Tzitzéica equation. From the definition of Q iA 3 and R iA 3 in (5.5), it is clear that

As a consequence, there exists a one-parameter family of indefinite affine spheres {f
The family {f λ iA 3 } λ∈R + will be called the associated family of f iA 3 . Let F λ iA 3 be the coordinate frame off λ iA 3 . Then the Maurer-Cartan formα λ iA 3 =Û λ iA 3 du +V λ iA 3 dv of F λ iA 3 for the associated family {f λ iA 3 } λ∈R + is given by U iA 3 and V iA 3 as in (5.13) where we have replaced Q iA 3 and R iA 3 by λ −3 Q iA 3 and λ 3 R iA 3 , respectively.
Then consider the gauge transformation G λ It is easy to see thatF λ iA 3 G λ e 3 =F λ iA 3 e 3 holds. Define f λ iA 3 =F λ iA 3 G λ e 3 . Then we do not distinguish between {f λ iA 3 } λ∈R + and {f λ iA 3 } λ∈R + , and either one will be called the associated family, and F λ iA 3 will also be called the coordinate frame of f λ iA 3 . From the discussion in the previous section, the family of Maurer-Cartan forms α λ iA 3 of the indefinite proper affine sphere f iA 3 : M → R 3 can be computed explicitly as It is clear that α λ iA 3 | λ=1 is the Maurer-Cartan form of the coordinate frame F iA 3 of f iA 3 . Then by the discussion in the previous subsection, we have the following theorem. For each of these surface classes we have introduced natural frames (not always "coordinate frames" in the classical sense) and have characterized them by their "shape". The Maurer-Cartan equations of these frames were (due to the special shape of the coefficient matrices) integrable if and only if a simple set of (highly non-trivial) equations was satisfied.
Inside of each of the classes of surfaces listed above we singled out a special type of surfaces. Respectively these were We showed that for all these special cases either a conformal parameter or a real ("asymptotic line") parameter is natural to choose for a "convenient" treatment. The cases with a preferable conformal parameter are indicated by a • and the other cases by a . Each of 24 the classes of surfaces can be characterized by a Tzitzéica equation: take values in iR and Q iA 3 , R iA 3 take values in R, respectively.
For the conformal cases one can introduce a loop parameter λ ∈ S 1 which produces an associated family of surfaces of the same type. For the asymptotic line cases one can introduce a loop parameter λ ∈ R > 0 which produces an associated family of surfaces of the same type.
For general (non-geometric) purposes one can usually use λ ∈ C × .
The loop parameter was introduced in a special way: Let F denote the frame associated with a surface of one of the special classes listed above. Then we write F −1 dF = α, and write α = F −1 dF = Uda + Vdb, where for the conformal case, (a, b) is given by complex coordinates (a, b) = (z,z) with z = x + iy, and for the asymptotic line case, (a, b) is given by null coordinates, (a, b) = (u, v) with real u, v. Actually, one decomposes naturally in all cases U = U −1 + U 0 and V = V 1 + V 0 and introduces the "loop parameter" λ such that with α 0 = U 0 da + V 0 db. In fact α λ is exactly a family of Maurer-Cartan forms α λ * as in the previous five sections, where * is one of CP 2 , CH 2 , CH 2 1 , A 3 or iA 3 . The 1-form α λ will be called the extended Maurer-Cartan form and a unique solution to the equation with some base point p 0 ∈ D will be called an extended frame. Thus the coordinate frames F λ * of the associated family of f λ * are in all five cases the extended frames up to an initial condition, where * is one of CP 2 , CH 2 , CH 2 1 , A 3 or iA 3 . In all five cases we have stated a theorem saying Since in all our cases the special surface of actual interest can be derived (quite) directly from the extended frame, one of our goals is to construct all these frames.
Corollary 6.2. The construction of all special surfaces listed above is equivalent to the construction of all the 1-forms α λ .

Flat connections and primitive frames.
To find all α λ (at least in an abstract sense) these 1-forms need to be described more specifically. To this end we consider the complex Lie algebra (6.3) g = sl 3 C 25 and the order 6 automorphismσ of g given by (X ∈ sl 3 C): Then on g the automorphismσ has 6 different eigenspaces (6.6) g j ⊂ g, such that [g i , g j ] ⊂ g i+j (mod 6) holds for the eigenvalues ǫ = e 2iπj 6 with j = 0, 1, 2, . . . , 5. Note that we then have for example g −1 = g 5 etc. and we also have 0 ⊂ g 0 . The crucial result for our discussion is: For all special surface classes the matrices U j and V j are contained in the eigenspace ofσ for the eigenvalue e 2iπj/6 , that is, U j , V j ∈ g j . More precisely we have where a and b denote the coordinates of the surface class under consideration. Moreover, for each special surface class there exists an anti-holomorphic involutory automorphismτ of g such that where gτ denotes the real subalgebra of g consisting of all elements in g which are fixed byτ .
The situation in the asymptotic line case is quite different from what we just remarked.
Theorem 6.5. Assume we have an immersion f of split real type with extended frame F λ and Maurer-Cartan form α λ . Letτ be an involutory anti-holomorphic automorphism of g which fixes α λ . Writing it follows thatτ fixes U −1 + U 0 and V 0 + V 1 . Let us assume thatτ actually fixes all U j and all V j . And let us assume also that the Lie algebra generated by generates the Lie algebra gτ . Thenτ andσ satisfy the following relation: (6.9)στσ =τ on g.
More details will be explained in the following section of this paper. An extended frame F λ for which the Maurer-Cartan form α λ satisfies (6.7) and (6.8) will be called primitive relative toσ andτ .
Corollary 6.6. In all our special surface classes the extended frame is primitive relative tô σ and the real form (anti-holomorphic) automorphismτ chosen for the special surface class.
6.3. The loop group method for primitive extended frames. It is most convenient to explain the procedure for the conformal case and for the asymptotic line case separately.
Letσ be as above and letτ be the anti-holomorphic involutory automorphism associated with the chosen surface class. Let By Gτ and gτ we denote the corresponding fixed point group and algebra respectively. Actually, for Gτ one could also use any Lie group between Gτ and its connected component.
From what was said above, the extended frame F λ of an immersion of our special class is contained in Gτ . The corresponding Maurer-Cartan form is contained in gτ .
By the form of (F λ ) −1 dF λ we infer that all the loop matrices associated with geometric quantities are actually defined for all λ ∈ C × . In particular, all extended frames are defined on S 1 . However, geometric interpretations are usually only possible for λ ∈ S 1 in the case of conformal case or λ ∈ R + in the case of asymptotic line case.
Next one does no longer read the extended frame as a family of frames, parametrized by λ ∈ S 1 , but as a function of z into some loop group.
Here are the basic definitions: (1) The loop group of a Lie group G is Considering G as a matrix group we use the Wiener norm on S 1 and thus has a Banach Lie group structure on ΛG. Since all our geometric frames are defined for λ ∈ C × , we can apply the usual loop group techniques (see, for example [37,Theorem 4.2]). (2) The plus subgroup: g as a holomorphic extension to the open unit disk and g −1 has the same property. , and the normalized plus subgroup: The minus subgroup: g has a holomorphic extension to the open upper unit disk in CP 1 and g −1 has the same property. , 27 and the normalized minus subgroup: We now define automorphisms σ and τ of ΛG as natural extensions ofσ andτ of G: where B(λ) = λ ±1 and −1 is taken in the case of conformal type and +1 is taken in the case of asymptotic line type.
(4) The real subgroup We will actually always use "twisted subgroups" of the groups above. First we have The other twisted groups are defined analogously, like Λ + * G σ = Λ + * G ∩ ΛG σ . Finally, we actually use the twisted real loop group: The twisted real loop group may be defined as (6.11) ΛG τ σ = ΛG σ ∩ ΛG τ , if σ and τ commute, these are the cases of (• * ) in Section 6.1, and if σ and τ do not commute, these are the cases of ( * ) in Section 6.1, then ΛG τ σ cannot be defined as in (6.11).
6.3.1. The loop group method for the conformal case. Let us fix a special surface class of conformal type. To understand the construction procedure mentioned above one considers next again an immersion of conformal type f with primitive extended frame F relative to σ and τ as above.
Here we use the dz-coefficients in F −1 dF = α = λ −1 U −1 dz + U 0 dz + V 0 dz + λV 1 dz and consider z and λ as parameters of the differential equation. Note that U 0 (z,z) + λV 1 (z,z) takes values in the Lie algebra of Λ + G σ , thus L + (z,z, λ) takes values in Λ + G σ . On the one hand, the primitive extended frame F is also a solution of the above differential equation, thus these two solutions should coincide up to an initial condition, that is, there exists C(z, λ) which is holomorphic in z ∈ D and λ ∈ C × such that (6.12) F (z,z, λ) = C(z, λ)L + (z,z, λ) holds.
(2) All η j are contained in g j , where g j is defined in (6.6).
This explains the procedure to obtain a holomorphic potential from a primitive harmonic map. The fortunate point is that this procedure can be reversed.
Theorem 6.8 (The loop group procedure for surfaces of conformal type). Let G,σ andτ as above. Let f be an immersion of conformal type, F (z,z, λ) = F λ (z,z) a primitive extended frame relative toσ andτ . Define C by F (z,z, λ) = C(z, λ) · L + (z,z, λ) and put η = C −1 dC, called a holomorphic potential for f . Then η has the form stated in (6.13), the coefficient functions η j of η are holomorphic on D and we have η j ∈ g j .
Conversely, consider any holomorphic 1-form η satisfying the three conditions just listed for η. Then solve the ODE dC = Cη on D with C ∈ ΛG σ . Next write C = F · V + with F ∈ ΛG τ σ and V + ∈ Λ + G σ . Then F λ (z,z) = F (z,z, λ) is the primitive extended frame of some immersion f of the class of surfaces under consideration. Remark 6.9.
(1) In the the procedure from f to η the decomposition F (z,z, λ) = C(z, λ) · L + (z,z, λ) is always possible. In the converse procedure the decomposition (usually called "Iwasawa decomposition", (see [34,25]) is not always possible. But the set of points, where such a decomposition is not possible is discrete in D.
(2) In the conformal case all geometric quantities like frame, potential etc. are actually real analytic on D and holomorphic in λ ∈ C × . (3) In the conformal case we can start from a real Lie algebra q, say the one generated by the Maurer-Cartan form α(z,z), z ∈ D of the coordinate frame of some immersion of conformal type. This always includes an automorphism κ of this Lie algebra. Then, by carrying out the loop group procedure, we naturally and unavoidably need to use the complexified Lie algebra q C . When extending the automorphism κ complex linear to q C and defining ρ as the anti-holomorphic automorphism of q C which defines q inside q C , then we naturally obtain that κ and ρ commute. Hence immersions of conformal type always have to do with a complex linear automorphism and an antiholomorphic involutory automorphism which commute. (Also see the Remark after Theorem 6.3.) 6.3.2. The loop group method for the asymptotic line case. The loop group method for this case looks at the outset very different. And indeed, there are remarkable differences. Since the scalar second order equation is not elliptic, solutions of low degree of differentiability can occur. In this paper we always use only functions which are as often differentiable as is convenient. Since the loop parameter is for geometric quantities real now, we do not need to use the complex Lie group G nor ΛG etc., but always G replaced by G τ , the real Lie group which is defined by τ and which is characteristic for the frame. 29 The main difference in procedure occurs at equation (6.12). Since the coordinates u and v are on an equal basis (opposite to z andz) we need to carry out the splitting twice (6.14) Note that L + (u, v, λ) can be found by solving the differential equation Here we use the coefficients in F −1 dF = α = λ −1 U −1 du + U 0 du + V 0 dv + λV 1 dv and consider u and λ as parameters. Since V 0 (u, v) + λV 1 (u, v) is given and smooth in u and in v, also L + (u, v, λ) is smooth in u and in v. Moreover, V 0 + λV 1 takes values in the Lie algebra of Λ + G σ , thus L + takes values in Λ + G σ .As a consequence, there exists C 1 (u, λ) only depends on u and is smooth in u and holomorphic in λ ∈ C × such that first equation in (6.14) holds.
The argument for the second equation is, mutatis mutandis, the same. It is also important to observe that the two equations imply: From this discussion we obtain a pair of potentials, Analogous to the conformal case we also need to know what form the potentials η 1 and η 2 take.
We would like to emphasize: (1) All coefficient functions η m,j (j = 1, 2) are smooth on some interval D j ⊂ R.
(2) All the coefficient functions η m,j are contained in gτ j .
Note that here gτ j are defined as gτ j := gτ ∩ g j , where g j is the eigenspace defined in (6.6).
As in the conformal case, one can also reverse the procedure. So let us start from two potentials η 1 (u, λ) and η 2 (v, λ) satisfying the three conditions listed above.
Remark 6.10. Since, in general, the Birkhoff decomposition can not be carried out for any loop matrices, there will be points, maybe curves, where the L ± (u, v, λ) are singular.
But away from singularities (6.18) implies that there exists a matrix function W (u, v, λ) satisfying Theorem 6.11 (The loop group procedure for surfaces of asymptotic line type). Let G,σ and τ as above. Let f be an immersion of asymptotic line type, F (u, v, λ) = F λ (u, v) a primitive extended frame relative toσ andτ . Define C 1 and C 2 by F (u, v, λ) = C 1 (u, λ) · L + (u, v, λ) and F (u, v, λ) = C 2 (v, λ) · L − (u, v, λ) and put η i = C −1 i dC i (i = 1, 2), called a pair of potential for f . Then η i has the form stated in (6.16) and (6.17), the coefficient functions η i,j of η i depends only on one variable and we have η i,j ∈ gτ j .
Conversely, consider any pair of 1-forms (η 1 , η 2 ) satisfying the three conditions just listed for η i (i = 1, 2). Then solve the ODEs Then there exist a gauge F 0 ∈ Gτ 0 3 such that F λ (u, v) = F (u, v, λ)F 0 takes values in ΛG τ σ is the primitive extended frame of some immersion f of the class of surfaces under consideration.

Complexification and real forms
This section is a brief digression which is intended to help to put this survey into a larger context. It is clear that the extended frames F introduced in the previous sections take values in the loop groups of For more details about these frames we refer to Section 6.1 and the corresponding subsections of the first five sections. We show that their Maurer-Cartan forms correspond to different real forms of Λsl 3 C σ or, more generally, of the affine Kac-Moody Lie algebra of type A 2 . Moreover, by using the classification of real forms of type A (2) 2 in [22], we obtain a rough classification of all surface classes associated with specific real forms of Λsl 3 C σ . 7.1. Real forms of Λsl 3 C σ and the surface classes considered in this paper. In the following discussion the Maurer-Cartan form α λ denotes α λ CP 2 , α λ CH 2 , α λ CH 2 1 , α λ A 3 + , α λ A 3 − , and α λ iA 3 in (1.14), (2.12), (3.8), (4.15), (4.18) and (5.16), respectively. Accordingly, the extended frame , and F iA 3 in (1.13), (2.10), (3.13), (4.14), (4.17) and (5.15), respectively. A straightforward computation shows that the Maurer-Cartan form α λ of the extended frame F λ satisfies the following two equations (where we write α(λ) for α λ if it is convenient): σ(α)(λ) = α(λ), τ (α)(λ) = α(λ), where σ is the order 6 linear outer automorphism of sl 3 C given by with ǫ = e πi/3 the natural primitive sixth root of unity and and τ is a complex anti-linear involution of sl 3 C varying with the surface class considered.
Note, for simplicity we will sometimes write σ(X) = − Ad(P )X T .
More precisely, the family of Maurer-Cartan form α λ takes values in the following loop algebra: , τ (g)(λ) = g(λ) and g ∈ W }, where W denotes the set of all 3 × 3−matrices with coefficients in the Wiener algebra on the unit circle which extend to all of C × .
Note, by abuse of language we use the same notation for the Lie group automorphisms σ and τ and their differentials. The order 6 automorphism σ is in all cases the same.
From the first five sections of this paper we obtain by inspection Theorem 7.1. The five surface classes discussed in the first five sections of this survey are related to complex anti-linear involutions τ as follows: τ (g)(λ) is given by −g(1/λ) 7.3. Real form involutions. It is known that all real form involutions τ of Λsl 3 C σ are induced from some complex anti-linear involution of sl 3 C, see [22]. Since we restrict for now our concentration on sl 3 C it is fairly easy to reduce the possibilities.
Remark 7.2. It is known [22] that some real forms of "untwisted" loop algebras such as A (1) n are not coming from any real form involutions on underlining finite dimensional Lie algebras.
7.3.1. Real form involutions commuting with σ. We now classify real form involutions commuting with σ.
(a) If τ = Ad(B) • β, then B is a generalized permutation matrix coinciding with P 0 after setting all non-zero coefficients equal to 1. More precisely, after removing appropriate cubic roots and after possibly a conjugation by Ad(D) with some diagonal matrix D such that Ad(D) commutes with σ we obtain B = P 0 or B = I 21 P 0 . (b) If τ is of the form τ = ψ • β with ψ an outer automorphism of sl 3 C, then we write τ = Ad(Q) • τ 0 . Then Q is without loss of generality a diagonal matrix of the form Q = diag(q, q −1 , 1), q ∈ R × .
Proof. In the following we denote the restrictions of the σ and τ on Λsl 3 C σ to the finite dimensional Lie algebra sl 3 C by the same symbols.
(a) Since τ commutes with σ, it also commutes with σ 2 = Ad(Ω), where Ω = diag(ǫ 4 , ǫ 2 , 1). A direct evaluation yields Finally we need to evaluate the commutation relation with σ directly. Writing this out yields the equivalent equation Replacing all non-zero coefficients in this equation by 1 still yields a correct equation. Since now the "reduced equation" readsP (B T ) −1 =BP , it followsB =P . Hence B has non-zero entries exactly, where P has them. Evaluating (7.5) explicitly yields four equations and one infers B 3 33 = −1. Hence B 33 = ǫ, ǫ 3 or ǫ 5 . For these cases one pulls out of B the matrix (−ǫ m )I and obtains without loss of generality B 33 = −1. Putting x = B 12 , then the (7.5) also implies B 21 = x −1 .

34
Evaluating the involution property of τ implies that x is real. Now we put A straightforward computation yieldsσ = σ andB = I 2,1 P 0 orB = −P 0 . Clearly the minus sign is irrelevant and we obtain the claim.
(b) By evaluating the first line in Theorem 7.1 we know that τ 0 commutes with σ. Hence the C-linear automorphism Ad(Q) commutes with σ, whence it also commutes with σ 2 and therefore Q is a diagonal matrix. A direct evaluation of the commutation property now yields QP = µP Q −1 . Taking the determinant yields µ 3 = 1 and the equation yields µ = Q 2 33 and µ = Q 11 Q 22 . Hence Q 3 33 = 1 and we can pull out without loss of generality Q 33 I from Q. Finally we evaluate the consequence of τ being an involution and obtain the claim.

7.3.2.
Real form involutions satisfying στ σ = τ . In this case we proceed very similarly to the previous case.
(1) If τ = Ad(B) • β, then B is a diagonal matrix coinciding with I after removing appropriate cubic roots and after possibly a conjugation by Ad(D) with some diagonal matrix D = diag(δ, δ −1 , 1) such that Ad(D) commutes with σ. (2) If τ is of the form τ = ψ • β with ψ an outer automorphism of sl 3 C, then writing τ = Ad(Q) • τ 0 we obtain that Q is, up to manipulations as in the proof of the last proposition, the matrix P 0 .
Evaluating now the relation (7.6) one obtains with little effort the equation B 3 33 = 1. Hence, after pulling out B 33 I from B we can assume without loss of generality that B 33 = 1 holds. 35 Evaluating all this we see that B is, without loss of generality, a diagonal matrix of the form B = (b, b −1 , 1) with b ∈ S 1 .
But now it is straightforward to verify that D = √ b This proves the claim.
(b) By evaluating the first line in Theorem 1.1 we know that τ 0 commutes with σ. Hence we obtain σ • Ad(Q)σ = Ad(Q). But then we also obtain σ 2 • Ad(Q)σ 2 = Ad(Q). Similar to the proof of the last proposition we conclude from his that Q is a generalized permutation matrix, more precisely belonging to a transposition. Moreover, the equation σ • Ad(Q)σ = Ad(Q). leads to P = νQP T Q T . For the underlying permutation matrices this impliesP =QP TQT .
Since P and Q are transpositions we concludeP =Q. Evaluating now σ • Ad(Q)σ = Ad(Q) one obtains that all entries of Q are sixth roots of unity and have the same square. Finally evaluating that τ is an involution we obtain after a simple computation Q 33 = −1 and the other two entries are equal and ±1. If they are equal to 1, then we have shown Q = P 0 . If they are −1, then we conjugate τ and σ by Ad diag(−1, −1, 1) and observe that this does not change σ and brings τ into the form Ad(P 0 )τ 0 .