Sixteen-dimensional locally compact planes of Lenz-type V on which SU2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{SU}}_2 {\mathbb {H}}$$\end{document} acts as a group of collineations

We explicitly construct all the planes mentioned in the title.


Introduction
We construct all 16-dimensional locally compact topological planes of Lenztype V whose collineation groups contain a subgroup (locally) isomorphic to SU 2 H. If the classical plane over the algebra O of the octonions is put aside, the collineation group of such a plane will be shown to be a Lie group of dimension 37. In fact by results of Löwe these are the only 16-dimensional locally compact translation planes whose collineation groups have this dimension (see H. Löwe, Sixteen-dimensional locally compact translation planes with automorphism groups of dimension at least 36, Preprint, 2019). Note that the sixteen-dimensional locally compact translation planes with collineation groups of dimension at least 38 have been classified in [9], cf. [11, 82.26].
The planes which we determine here fall into three families, which are related among each other by dualization and transposition. The planes of one of these families are self-dual; this invites for further study of their polarities and their unitals, which however will not be pursued here.
The translation line is unique and, hence, fixed by every collineation except if the plane is isomorphic to the classical plane over the algebra O of the octonions (in the latter plane, every line is a translation line). We represent such a translation plane as an affine plane, the translation line playing the rôle of the line L ∞ at infinity. Let G be the group of all affine collineations, that is, of the collineations fixing L ∞ ; it comprises all collineations except for the classical octonion plane, as we have already remarked.
The space of affine points is a 16-dimensional real vector space, the lines through the origin o are certain 8-dimensional R-linear subspaces, and the other affine lines are the images of these under the (vector space) translations. The group G is the semidirect product of the normal subgroup T consisting of these translations by the stabilizer G o of o: In a plane of Lenz type (at least) V, there is a point s at infinity such that for the line S joining o and s the group G [s,S] of collineations with center s and axis S (shears) is sharply transitive on L ∞ \{s}. If the plane is not the classical plane over the octonions, the point s is unique; it will be called the shear center. It is then fixed by every collineation. Hence, the shear group G [s,S] is a normal subgroup of G o , and for a second line W = S through o The action of G S,W on the normal subgroup G [s,S] by conjugation is equivalent to the action on L ∞ \{s}, by sharp transitivity of G [s,S] on L ∞ \{s}.
With respect to W and S as first and second coordinate axis, a 16-dimensional plane of Lenz type V can be coordinatized by an 8-dimensional (non-associative) real division algebra (D, +, •), see [11, 24.7, 25.8 and 64.14]. (In other terminology, a non-associative division algebra is also called a semifield). Its multiplication will be denoted by • in order to distinguish it from the classical multiplication · of the octonions and the quaternions, which will also be used. In affine coordinates over D, the shear group G [s,S] consists precisely of the transformations see [11, 24.7 and 25.4]. Hence, G [s,S] is isomorphic to the additive group (R 8 , +) of D. In particular, it has no nontrivial compact subgroup, and the same is true for the translation group T ∼ = R 16 . Therefore, in view of the semidirect products noted above, a maximal compact subgroup of G S,W is a maximal compact subgroup of G (except in the classical octonion plane).

The planes of Case 1 2.1. The action of Λ on the shear group
We may identify R 16 and H 2 × H 2 as R-vector spaces in such a way that S and W , considered as affine lines, are described as S = {0}×H 2 and W = H 2 ×{0}, and that Λ acts on S and W effectively as the group SU 2 H in its classical action on H 2 . Thus, We now consider an 8-dimensional division algebra (D, +, •) coordinatizing the plane with respect to W and S as first and second coordinate axis and the point 1 0 , 1 0 as unit point. The fact that the shear group G [s,S] is normal in G o and its description in coordinates given in Sect. 1.1 imply that the following set of R-linear transformations of D identified with H 2 is invariant under conjugation by the elements of SU 2 H: for x ∈ D = H 2 , and that this action is equivalent to the action of Λ on the shear group G [s,S] . Moreover, since W and its point at infinity are fixed by Λ and since the shear group G [s,S] acts sharply transitively on L ∞ \{s}, this action is equivalent to the action of Λ on L ∞ , and hence, by assumption of Case 1, to a nontrivial linear action as SO 5 R on R 8 . Hence, there is a 5dimensional R-linear subspace C of D on which Λ acts as SO 5 R in its classical action on R 5 , and a complementary 3-dimensional subspace F of D on which Λ acts trivially. First, we determine the possibilities for the subspace C.

Lemma.
Let C be a 5-dimensional R-linear subspace of the vector space End R H 2 of R-linear endomorphisms of H 2 (viewed as a real vector space) which is invariant under conjugation by SU 2 H and on which this group acts nontrivially. Then there is a quaternion p ∈ H\{0} of norm 1, pp = 1, such that Proof. The only nontrivial representation of SU 2 H in dimension 5 is the classical action as SO 5 R, the kernel of this representation being generated by the central involution ι.
The involution 1 1 ∈ SU 2 H normalizes Ψ and hence leaves C 1 invariant, but not elementwise fixed, because it does not belong to Ψ. Thus, C 1 is contained in the eigenspace of this involution with eigenvalue −1, so that, in the given description of C, we must have q = −p. aa + cc = 1 = bb + dd, ab + cd = 0, so that A −1 = a c b d , which in turn is equivalent to aa + bb = 1 = cc + dd and ac + bd = 0.
Conjugation of an endomorphism from C 1 by A gives the endomorphism From the relations given above, we obtain that bb − dd = −(aa − cc). Thus, the elements of C have the form stated in the lemma. It is clear that p may be chosen to be of norm 1. That for all s ∈ R, h ∈ H the endomorphisms given in the lemma belong to C follows from the fact that they form a 5-dimensional vector space.

Shears which are invariant under Λ
Now we determine the possibilities for a 3-dimensional subspace F of End R H 2 on which SU 2 H acts trivially by conjugation, which means that the endomorphisms in F commute with the elements of SU 2 H. According to Schur's lemma, the elements of F are given by scalar multiplication from the right by quaternions; hence there is a 3-dimensional R-linear subspace F of H such that In the sequel, we think of F as the orthogonal space of a unit quaternion v, vv = 1 with respect to the scalar product a, b = 1/2(ab + ba) on the R-vector space H:

Lines through the origin
In coordinates over the division algebra D = (H 2 , +, •), the affine lines through o are the subsets The lines L 0,q,0 for q ∈ F are obtained from the endomorphisms in F, which are precisely the elements of D commuting with SU 2 H. Thus, together with the line S = {0} × H 2 , they are the fixed lines of the group Λ ∼ = SU 2 H of collineations. Necessarily, we must have that otherwise, the point 1 0 , p 0 would belong to the two different lines L 1,0,0 and L 0,p,0 through the origin.
The question whether for every pair (v, p) of unit quaternions satisfying these conditions one obtains in this way the system of lines through the origin of an affine plane will be postponed until later, see Sect. 2.9.

Special isomorphisms
The plane for which we have developed this description will be called P. We now consider a second plane P with unit quaternions v and p instead of v and p as defining parameters. The lines through the origin of this plane described as in Sect. 2.4 will be called L s,q,h . We ask under which conditions on v, p, v , p the map ϕ : x 1 of the affine point space of P onto the affine point space of P for unit quaternions c and d is an isomorphism between the two planes.
The answer to this question serves two purposes: First, it will allow to restrict the parameters v and p up to isomorphism, and second, it will later help to determine all collineations of the planes of this type.
The map ϕ is deliberately chosen so as to commute with the action of the group Λ ∼ = SU 2 H on both planes. If ϕ is an isomorphism, it therefore must map fixed lines of Λ to fixed lines. Thus, the line L 0,q,0 for q ∈ F must be mapped to the line L 0,q ,0 for some q ∈ F := v ⊥ . Now, ϕ maps L 0,q,0 to This is a line L 0,q ,0 if and only if c −1 qd = q ∈ F = v ⊥ . If ϕ is an isomorphism, this is true for all q ∈ F = v ⊥ , which means that In Sect. 2.4, the lines L s,0,0 were constructed from the endomorphisms in C 1 , the endomorphisms in C which commute with the elements of Ψ, see 2.2. These lines therefore are fixed lines of the subgroup of Λ ∼ = SU 2 H corresponding to Ψ. But this subgroup has more fixed lines through o than that. They are given by endomorphisms in all of D (not only C) which commute with the elements of Ψ. Now D = C + F, where F consists of the endomorphisms which commute with the whole group SU 2 H, so that the endomorphisms in D commuting with Ψ are the elements of C 1 + F. The corresponding lines through o are the lines L s,q,0 for s ∈ R, q ∈ F . Now, if ϕ is an isomorphism, it must map L 1,0,0 to a line in P which is also a fixed line of the subgroup of Λ ∼ = SU 2 H corresponding to Ψ, that is to a line L s ,q ,0 . The image of L 1,0,0 under ϕ is Sixteen-dimensional locally compact planes Page 7 of 19 46 This is a line L s ,q ,0 if and only if c −1 pd = s p + q and −c −1 pd = −s p + q ; but then q = 0 and c −1 pd = s p . Since c and d are unit quaternions, this means that Conversely, it is now easy to verify that if these conditions are fulfilled, ϕ maps all lines of P to lines of P and thus is an isomorphism of P onto P .

Parameter restriction
The maps H → H : x → c −1 xd for unit quaternions c, d appearing in the conditions which have just been obtained describe all elements of SO 4 R in its classical action on H ∼ = R 4 . They allow to transform any pair v, p of unit quaternions to any other such pair having the same scalar product. By the result of Sect. 2.5, we therefore may assume, up to isomorphism of planes, that The last restriction comes from the condition that 1 = p / ∈ v ⊥ , see Sect. 2.4. Furthermore, one obtains an isomorphic plane if one replaces v by cos α − sin α · i, since this quaternion has the same scalar product with 1 as v. Also, it is clear from the description of lines in Sect. 2.4 that −v instead of v defines the same plane. So, finally, we may restrict v further by demanding that The plane obtained in this way shall be denoted by P α .
In the sequel, we shall always use this choice of parameters. (The idea for this particular choice was given to us by H. Löwe in his preprint cited in the introduction, although one would have been tempted at first sight to use a choice with 1 ∈ v ⊥ in order to ensure that the diagonal {(x, x); x ∈ H 2 } is a line of the plane, which is not the case with p = 1. This slight disadvantage will be compensated, however, by far greater advantages.) We shall see that different choices for α yield nonisomorphic planes; indeed we shall prove in Sect. 2.11 that no other isomorphisms than those already produced here can be found. With these parameters, the lines through the origin besides S are the following subsets of the point space H 2 × H 2 : {(x, y) → (xc, yc); c ∈ R · 1 + Ri, cc = 1}, a 1-dimensional torus whose intersection with Λ is generated by (x, y) → (−x, −y).

A coordinatizing division algebra
In order to obtain a coordinatizing division algebra (D, +, •), we change the identification of the affine point set with H 2 × H 2 by the map x1 x2 , y1 . In new coordinates, the lines through the origin except S take the form The new coordinates present an advantage for coordinatization, namely that the diagonal {(x, x); x ∈ H 2 } is one of the lines through the origin. There is a disadvantage however: The introduction of these coordinates deforms the action of the group Λ and of the torus group described in Sect. 2.7. Therefore we shall return to the old coordinates after this digression.
In coordinates over D, a line through the origin different from S is given by If this is the line described above, then d is obtained in terms of s, q, h by feeding the unit 1 0 instead of x = x1 x2 into the second coordinate of this line: so that d 1 = s + q, d 2 = −h. By forming the scalar product of d 1 = s + q with v = cos α + sin α · i and keeping in mind that q ∈ v ⊥ and that cos α = 0 we obtain conversely that Thus, the second coordinate of the point on the line described above with first coordinate This can be simplified: where xd is the classical octonion product and ρ = 2 tan α.

Verification of planarity
We still have to verify that for every parameter α ∈ [0, π/2) we obtain in this way a division algebra, so that the lines described in Sect. 2.6 are indeed the lines through the origin of a plane P α of Lenz type V. According to the criterion [11,64.13], which is formulated for a more general situation, it suffices to prove that for x ∈ D, x = 0 the map D → D : d → d • x is bijective. This map is R-linear; so by finite dimension it suffices to show that it is injective, or, in other words, that it has trivial kernel. This means that for x = 0, d • x = 0 implies d = 0. But this is equivalent to saying that d • x = 0 implies d = 0 or x = 0. This is what we are going to prove. Indeed, d • x = 0 means If d 1 , i = 0 or x 2 = 0, then d • x = xd, and our assertion follows from the fact that the octonion algebra has no zero divisors.
So assume d 1 , i = 0 and x 2 = 0. From the first of the two equations above, we then obtain that d 2 = x 2 −1 d 1 x 1 . Inserting this into the second equation and multiplying by x 2 gives x 2 x 2 d 1 + x 2 x 2 ρ d 1 , i + d 1 x 1 x 1 = 0. But then d 1 would be real, in contradiction to d 1 , i = 0.

The classical case
In order to decide isomorphism questions, we first have to determine for which admissible parameters α the plane P α described in Sect. 2.6 is isomorphic to the classical octonion plane, or equivalently, the division algebra in Sect. 2.8 is isomorphic to the classical octonion algebra. The latter is biassociative; so we test biassociativity of the multiplication •.
is isomorphic to the classical octonion algebra, the result has to be the same, so that ρ = 0, which by definition of ρ = 2 tan α means α = 0 and v = 1.
Conversely, it is clear that for ρ = 0 the division algebra (H 2 , +, •) is isomorphic to the octonion algebra (by octonion conjugation). We formulate the result: Proposition. The division algebra constructed in Sect. 2.8 for 0 ≤ α < π/2 is isomorphic to the classical octonion algebra if and only if α = 0. This is equivalent to the plane P α described in Sect. 2.6 being isomorphic to the classical octonion plane.

Isomorphisms and collineations
We now return to the question of isomorphisms between two such planes in the case they are nonclassical.
First, since the translation line L ∞ and the shear center s are unique, see Sect. 1.1, an isomorphism between two such planes P = P α and P = P α maps the translation line to the translation line and the shear center to the shear center. It can be modified by a translation in such a way that the origin o is mapped to the origin. Such an isomorphism is a semilinear transformation of the affine plane of P onto the affine plane of P when the two affine planes are viewed as vector spaces over the respective kernels, see [1], [10,Theorem 1.18]. Now the kernel of such a plane can be obtained from the kernel of the coordinatizing division algebra D, which in the case of an 8-dimensional real division algebra consists of the real multiples of 1 only, see [2, Theorem 1]. Thus, an isomorphism ϕ mapping o to o is R-linear.
As second coordinate axis in P, we have used the line S through o having the shear center s as point at infinity; the isomorphism ϕ therefore maps S to the corresponding line of P . By the transitivity properties of the shear group, ϕ can be further modified by a shear so as to map the first coordinate axis W in P to the first coordinate axis in P , as well. We now represent both affine planes in the same real vector space H 2 × H 2 ; then ϕ may be thought of as an R-linear transformation leaving W = H 2 × {0} and S = {0} × H 2 invariant.
In both planes, the group Λ ∼ = SU 2 H is a subgroup of the stabilizer G S,W of W and S in the collineation group. We assert that in the nonclassical case it is a characteristic subgroup. Since D is a real division algebra, G S,W contains the subgroup Z = {(x, y) → (rx, ty); 0 < r, t ∈ R}.
By [4, 4.2], see also [11, 81.8], G S,W has a largest compact subgroup M, and G S,W is the product of M and Z. The group Λ ∼ = SU 2 H is contained in M. Since it acts irreducibly on W and on S, the same is true for the connected As a consequence, conjugation by the isomorphism ϕ maps the group Λ ∼ = SU 2 H onto itself and induces an automorphism on Λ. Since SU 2 H has only inner automorphisms, ϕ can be modified by composition with an element of Λ such that conjugation by ϕ induces the identity on Λ, in other words, ϕ commutes with the elements of Λ. By Schur's lemma, ϕ is of the form (x, y) → (xc, yd) for x, y ∈ H 2 and some c, d ∈ H\{0}. By composition with one of the collineations in Z, we may assume that c and d are unit quaternions. But then, ϕ is just one of the special isomorphisms considered in Sect. 2.5. This means that we will not find any other isomorphisms and, in particular, collineations than those which we know already from Sects. 2.5 and 2.7. We summarize our findings: 2.12 Theorem. Up to isomorphism, the non-classical sixteen-dimensional locally compact planes of Lenz type V whose collineation groups contain a subgroup Λ ∼ = SU 2 H acting according to Sect. 1.2 Case 1 are the planes P α whose lines through the origin are described in Sect. 2.6 for v = cos α + sin α · i, where 0 < α < π/2. For different choices of α, one obtains non-isomorphic planes.
The collineation group of such a plane is the product of the group of translations, the group of shears with axis S, the group Z given in Sect. 2.11, the group Λ ∼ = SU 2 H consisting of the collineations and the 1-dimensional torus group consisting of the collineations The group of collineations is therefore connected and has dimension 37.

The planes of Case 2
These are obtained from the planes of Case 1 by passing to the dual planes. Indeed, the line at infinity L ∞ of a plane P α of Case 1 plays the rôle of the pencil of lines through the shear center in the dual plane P * α , so that the group Λ ∼ = SU 2 H acts on this pencil as SO 5 R, and this action corresponds to the action on the first coordinate axis. It is well known and easy to see that the dual plane of a plane of Lenz type V is again of Lenz type V.

Dualization
The affine points of P * α are the affine lines of P α which do not pass through the shear center s, that is which are not parallel to S. According to Sect. 2.6, these are the lines of the affine point set of the dual plane P * α with H 2 × H 2 , which we shall use in the sequel.

The division algebra for the dual plane
In order to express s − q in terms of s + q, recall that q ∈ v ⊥ and that v = cos α + sin α · i. Thus, with the real part Re q of q, one has 0 = q, v = Re q · cos α + q, i sin α = Re q · cos α + s + q, i sin α, so that Re q = − tan α s + q, i .
Vol. 111 (2020) Sixteen-dimensional locally compact planes Page 13 of 19 46 The coordinatizing division algebra for the dual plane thus has multiplication where dx is the classical octonion product.
In preparation for Sect. 4, where we will use transposition to produce the planes of Case 3, we rewrite the multiplication * by the real 8 With respect to the R-basis 1, i, j, k of H, the matrices when applied to a quaternion q, give Kq = q and T q = q, i . Furthermore, let L a and R a for a ∈ H be the real 4 × 4-matrices describing the endomorphisms q → aq and q → qa (q ∈ H) of the real vector space H given by the classical multiplication of quaternions. With these matrices In order to express this in quaternal coordinates, that is, in terms of s + q and h only, we note that s = Re(s + q) − Re q = Re(s + q) + tan α s + q, i and q = s + q − s = s + q − Re(s + q) − tan α s + q, i , so that s + q = (Re(s + q) + tan α s + q, i )(aa − bb) + s + q − Re(s + q) − tan α s + q, i + 2Re(bha) = −2(Re(s + q) + tan α s + q, i )bb + s + q + 2Re(bha) (one uses that aa − bb − 1 = aa − bb − (aa + bb) = −2bb). Also, one obtains that h = 2(Re(s + q) + tan α s + q, i )ca + dha + ch b.

Collineations
Again, for later use in Sect. 4, we rewrite the result by a real 8 × 8-matrix using the 4 × 4-blocks defined above and the matrices L a , R t a = R a ; this is easily established by a little computation with quaternions. With these block matrices, one obtains where ρ = 2 tan α. The corresponding line through the origin of the transposed plane P * t α is For d 1 = 1, d 2 = 0 this is the diagonal of H 2 × H 2 . The line through the origin and the point 1 0 , u1 u2 is obtained for d 1 = u 1 , d 2 = −u 2 . Therefore the multiplication of the coordinatizing division algebra is where ux is the classical octonion product.
We note that the transposed plane is self-dual. Indeed, its dual plane is known to be coordinatized by the converse division algebra, with multiplication u x = x u = xu − ρRe(x 2 u 2 ) i 0 .
There are involutory antiautomorphisms of the octonion algebra which fix i, leave {0} × H invariant, and commute with conjugation so that they preserve real parts, and it is readily seen that such an antiautomorphism establishes an isomorphism beween (H 2 , +, ) and (H 2 , +, ). For instance,