Large semisimple groups on 16-dimensional compact projective planes are almost simple

Priwitzer, Barbara

doi:10.1007/s000130050075

Cited by 6 publications

(5 citation statements)

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“…[15, 94.26]. For semi-simple groups the claim is an immediate consequence of Priwitzer's classi®cation [5,6]. If is compact, the assertion follows from Salzmann [13].…”

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confidence: 84%

Near-homogeneous 16-dimensional planes

2001

Advg

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A projective plane P PY L with point set P and line set L is a (compact) topological plane if P and L are (compact) topological spaces and the geometric operations of joining two distinct points and intersecting two distinct lines are continuous.The classical examples are the Desarguesian planes over the real or complex numbers or the quaternions and the Moufang plane over the octonions, each taken with its natural topology on P and on L. These planes are also connected. Within the vast class of compact connected topological projective planes they are distinguished by their high degree of homogeneity: their automorphism groups are transitive on quadrangles. A detailed discussion of the classical planes can be found in Chapter I of the book Compact Projective Planes [15].Each compact projective plane P with a point space P of positive (covering) dimension dim P`y has lines which are homotopy equivalent to an l-sphere S l , where l j 8 and dim P 2l, see [15, 54.11]. In all known examples, the lines are actually homeomorphic to S l . The automorphism group Aut P consists of all continuous collineations of PY L. Taken with the compact-open topology (the topology of uniform convergence), is a locally compact transformation group of P of ®nite dimension ([15, 83.2]). Let h denote a connected closed subgroup of . If dim h b 5l, then P is a classical plane and is a simple Lie group, see Salzmann [8], Theorem II,and [15, 87.7] for the cases l 4. Suppose now that 3l dim h 5l. Under suitable additional assumptions on the structure of h and its action on P, the classi®cation problem requires to determine all possible pairs PY h. The cases l j 4 are understood fairly well. In particular, the classi®cation has been completed in the following cases:

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“…[15, 94.26]. For semi-simple groups the claim is an immediate consequence of Priwitzer's classi®cation [5,6]. If is compact, the assertion follows from Salzmann [13].…”

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confidence: 84%

Near-homogeneous 16-dimensional planes

2001

Advg

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“…Without the restriction |F ∆ | = 1, Priwitzer [35], [36] has treated semi-simple groups of dimension ≥ 29. She proved in particular:…”

Section: Exactly One Fixed Elementmentioning

confidence: 99%

“…The group Γ = Aut O is the 14-dimensional compact simple Lie group G 2 , it is transitive on the unit sphere S 6 in R ⊥ , and Γ i is transitive on S 5 ⊆ C ⊥ ([54] 11. [31][32][33][34][35]. Obviously, λ : (x, y) → (x, −y) ∈ Γ, and easy verification shows that…”

Section: Introductionmentioning

confidence: 99%

Compact 16-dimensional projective planes

Salzmann¹

1999

Results. Math.

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This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in the book (which was published 20 years ago) is a complete description of all planes admitting an automorphism group of dimension at least 40. Newer results on 8-dimensional planes have been collected in [52]. Here, we present a classification of 16-dimensional planes with a group of dimension ≥35, provided the group does not fix exactly one flag, and we prove several further theorems, among them criteria for a connected group of automorphisms to be a Lie group. My sincere thanks are due to Hermann Hähl for many fruitful discussions. closure of an affine plane coordinatized by a mutation O (t) = (O, +, •) of the octonions, where c•z = t·cz + (1−t)·zc with t > 1 2 ; either t = 1, O (t) = O, and P ∼ = O , or dim Σ = 40 and Σ fixes the line at infinity and some point on this line ([54] 87.7).The ultimate goal is to describe all pairs (P, ∆), where ∆ is a connected closed subgroup of Aut P and dim ∆ ≥ b for a suitable bound b in the range 27 ≤ b < 40. This bound varies with the structure of ∆ and the configuration F ∆ of the fixed elements (points and lines) of ∆. In some cases, only ∆ can be determined, but there seems to be no way to find an explicit description of all the corresponding planes.

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“…The following is already known and will be used: Vol. 90 (2008) 16-dimensional compact projective planes 285 Theorem S. If ∆ is semi-simple, dim ∆ > 28 , and if ∆ fixes some element (point or line), then ∆ ∼ = Spin 9 (R, r) with r ≤ 1 , or P is the classical Moufang plane, see [4], [5].…”

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16-dimensional compact projective planes with a collineation group of dimension ≥ 35

Salzmann

2008

Arch. Math.

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A compact projective plane is isomorphic to the classical Moufang plane if it admits an automorphism group of dimension at least 38 which fixes a non-incident point-line pair.Let P = (P, L) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0 . A systematic treatment of such planes can be found in the book Compact Projective Planes [11]. Each line L ∈ L is homotopy equivalent to a sphere S with | 8 , and d = 2 , see [11] (54.11). In all known examples, L is in fact homeomorphic to S . Taken with the compact-open topology, the automorphism group Σ = Aut P (of all continuous collineations) is a locally compact transformation group of P with a countable basis, the dimension dim Σ is finite [11] (44.3 and 83.2). In many cases, Σ is known to be even a Lie group, cf. [11] (87.1) and [6]. Here, we deal exclusively with 16-dimensional planes; then the full automorphism group Σ is a Lie group whenever dim Σ ≥ 29 , see [9].The last theorem in the book [11] states that a plane with dim Σ ≥ 40 can be coordinatized by a "mutation" of the octonions with a multiplication c • z = (1−τ )cz +τ zc, where 1 2 = τ ∈ R . In the meantime, considerable progress has been made towards a classification of compact planes admitting a connected group ∆ of automorphisms with dim ∆ ≥ 35 . All such planes have been determined in which ∆ does not fix exactly one point and one line, see [2], [3] and the references given there. Here we will deal with the case that ∆ fixes exactly two non-incident elements. The following is already known and will be used:

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Large semisimple groups on 16-dimensional compact projective planes are almost simple

Cited by 6 publications

References 14 publications

Near-homogeneous 16-dimensional planes

Near-homogeneous 16-dimensional planes

Compact 16-dimensional projective planes

16-dimensional compact projective planes with a collineation group of dimension ≥ 35

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