On the dimensions of automorphism groups of four-dimensional double loops

Bödi, Richard

doi:10.1007/bf02571700

Cited by 7 publications

(10 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The centralizer O leaves the ternary subfield .T~ invariant and the kernel of this action is compact. Thus, the six-dimensional group f~1~22 must act effectively on .To~ which contradicts the results of [1] and [2] or [15]. CASE 3. dim I?…”

Section: Iff Is a Non-compact Semi-simple Lie Group Then Dim F _<mentioning

confidence: 68%

“…But this group contains a central involution w and thus F acts on the Baer ternary fixed field ~. Since F is a quasi-simple group of dimension greater than 4, the group F must fix the Baer ternary subfield .T'~ pointwise, see [1] and [2, (3.3)]. This, however, is a contradiction to [3, 3.1].…”

Section: Iff Is a Non-compact Semi-simple Lie Group Then Dim F _<mentioning

confidence: 78%

“…In the first step we shall define four topologies 7"1 ,..., 7"4 on the set B, where each of these topologies has certain properties, which can easily be verified. In particular, it will be obvious that (B, 7-4) is homeomorphic to M. In steps (2) and (3) we shall verify that the topologies 7"1, 7-2, and 7"4 coincide with the quotient topology 7-on 3,4. This yields three different descriptions of the topology 7-on 34.…”

Section: (3) Lemma Let K Be a Maximal Compact Subgroup Of P Where Fmentioning

confidence: 96%

See 2 more Smart Citations

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

Bdi

1994

Geom Dedicata

View full text Add to dashboard Cite

Abstract. Let 7" be an eight-dimensional, connected, locally compact ternary field and let F denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed field .Tr of F is connected, then P is either isomorphic to one of the compact Lie groups G2 or SUaC, or the (covering) dimension of I ~ is at most 7.Mathematics Subject Classification (1991): 51A35, 12J99, 22E99. This paper continues the study of automorphism groups F of eight-dimensional locally compact connected ternary fields T in the sense of [14] and [3]. Throughout the paper we shall assume that F is a closed subgroup of the (locally compact) automorphism group of T. We shall work with the following definition of a topological temary field, compare [16, 7.2].(1) DEFINITION. A ternary field 7" = (T, 0, 1, 7-) consists of a set T containing the two distinct elements 0 and I and a temary operation 7-: T 3 ~ T: (8, z, t) 7-(8, z, t) which satisfies the following axioms: (T1) 7-(0, x, t) = 7-(x, 0, t) = t andT-(8, 1, 0) = 7-(1, 8, 0) = 8 for all 8, x, t c T.(T2) For any elements 8, x, y E T there exists a unique element t = t (s, x, y) in T such that 7-(8, x, t) = y.(T3) For any elements st, 82, tl, t2 C T with Sl ~ 82 there exist unique elements x = x(81, tl, ~2, t2) and8 = s(81, tl, 82, t2)inTsuchthatT-(si, x, ti) = y and 7-(s, 81, t) = tl holds for some y, t E Tandi = 1, 2.A temary field T is called topological if T is provided with a topology being neither discrete nor indiscrete such that the map 7-and its inverse mappings t, x and s are continuous. Using the terminology and notation of [3] the main results of the articles mentioned above are collected in the following (2) THEOREM. The following conclusions hoM:(a) dim F G 14.

show abstract

Section: Iff Is a Non-compact Semi-simple Lie Group Then Dim F _<mentioning

confidence: 68%

Section: Iff Is a Non-compact Semi-simple Lie Group Then Dim F _<mentioning

confidence: 78%

Section: (3) Lemma Let K Be a Maximal Compact Subgroup Of P Where Fmentioning

confidence: 96%

See 1 more Smart Citation

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

Bdi

1994

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…(A) Dimension formula [5]: For all a E P we have dim A = dim a A + dim A~ (B) The dimension of the stabilizer of a quadrangle A is at most 4 (see [1,Th. …”

Section: Every Locally Compact Connected Group a With Dim A < Oo Has mentioning

confidence: 99%

Large automorphism groups of 8-dimensional projective planes are Lie groups

Priwitzer

1994

Geom Dedicata

View full text Add to dashboard Cite

Abstract. We prove the following theorem: Let 7 ~ be an 8-dimensional compact topological projective plane. If the connected component A of its automorphism group has dimension at least 12, then A is a Lie group.Mathematics Subject Classification (1991): 51H 10.For any compact connected topological projective plane 7', whose point space P has dimension at most 4, we know that P is a manifold (in fact P ~ P2~ or P ~ P2C respectively) and that the automorphism group of 7" is a Lie group (see [13]). If the point space has dimension greater than 4, it is not known whether or not it is a manifold. The automorphism group of the projective plane is always a locally compact transformation group of P, but it is still an open question if it is always a Lie group. For the most interesting cases, however, i.e. for automorphism groups of sufficiently large dimension, we can show that they are Lie groups. Thus, the purpose of this paper is to prove the following theorem: THEOREM. The connected component A of the automorphism group of an 8-dimensional compact connected projective plane is a Lie group if dim A > 12.Remark. An analogous result for 16-dimensional planes was published by LOwen and Salzmann [8]. They proved that an automorphism group is a Lie group if its dimension is at least 36. Using methods of the present paper, this bound can be improved considerably: 28 is already sufficient. PreliminariesThe notation in this paper is as follows: 79 will always denote a compact connected 8-dimensional projective plane with point space P and line space/2. The connected component of its automorphism group is called A, and A _< A is the stabilizer of a nondegenerate quadrangle. For X C P we denote by Ax the stabilizer of X, and by A[x ] its pointwise stabilizer. The set of fixed points of an element E A is denoted by Fs. For the dimension of a coset space we write dim A/F = dimA-dimP = A.F.

show abstract

“…We have to use [18] 55.21b) and [1] Theorem 2.7 to see that the almost simple group A has dimension at most 8 or 20 in case dim ~ = 4 or 8, respectively. By [17] w Corollary or 1.3 the stabilizer B~ is at most 8-or 3-dimensional.…”

Section: Groups With a Centre That Consists Of Homologies Or Elationsmentioning

confidence: 99%

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

Priwitzer

1997

Arch. Math.

View full text Add to dashboard Cite

The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E6(-26). A 16-dimensional, compact projective plane ~ admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group d acting on ~ the same result can be obtained if dimd _-> 37, see [i6]. Our aim is to lower this bound. We show: if d is semisimple and dim d _-> 29, then ~ is either classical or a Moufang-Hughes plane or A is isomorphic to Spin 9 (~., r), r E {0, 1}.The underlying paper contains the first part of the proof showing that A is in fact almost simple.O. Introduction. Let ~ be a 16-dimensional compact projective plane with point space P and line space ~. An example of such a plane is the Moufang plane ~2 9 over the octonions, the so-called classical plane of dimension 16. In order to classify such planes we consider a subgroup d of the automorphism group Aut(~) of ~'. In the compact-open topology, Aut (~) is a locally compact transformation group, [18] 44.3. The group A shall be closed and connected. The larger the topological dimension of A, the better are the chances to obtain a complete classification of planes ~ with d < Aut (~).We also take into account the structure of A. Semisimple and non-semisimple groups behave in a completely different way. For projective planes the main difference between these two types is based on the fact that semisimple groups (with one exception) always contain involutions. Here, we will restrict the discussion to semisimple groups. Now we can formulate the problem: classify all 16-dimensional compact projective planes ~' admitting a semisimple group A < Aut (~') with dimA => d for some suitably chosen d. In [16] this problem was solved for d = 37. In this case ~ is always classical. In the present paper and in [9] we will lower this bound to d = 29. Our aim is to prove the following Theorem. Let ~ be a 16-dimensional compact projective plane and .4 a connected and closed subgroup of the automorphism group of ~. If .4 is semisimple and at least 29-dimensional, then one of the following statements holds:Mathematics Subject Classification (1991): 51H10.

show abstract

On the dimensions of automorphism groups of four-dimensional double loops

Cited by 7 publications

References 15 publications

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

Large automorphism groups of 8-dimensional projective planes are Lie groups

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

Contact Info

Product

Resources

About