THE PROJECTIVE GEOMETRY OF A GROUP

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THE PROJECTIVE GEOMETRY OF A GROUP WOLFGANG BERTRAM Abstract. We show that the pair (P(?),Gras(?)) given by the power set P = P(?) and by the “Grassmannian” Gras(?) of all subgroups of an arbitrary group ? behaves very much like a projective space P(W ) and its dual projective space P(W ?) of a vector space W . More precisely, we generalize several results from the case of the abelian group ? = (W,+) (cf. [BeKi10a]) to the case of a general group ?. Most notably, pairs of subgroups (a, b) of ? parametrize torsor and semitorsor structures on P. The role of associative algebras and -pairs from [BeKi10a] is now taken by analogs of near-rings. 1. Introduction and statement of main results 1.1. Projective geometry of an abelian group. Before explaining our general results, let us briefly recall the classical case of projective geometry of a vector space W : let X = P(W ) be the projective space of W and X ? = P(W ?) be its dual projective space (space of hyperplanes). The “duality” between X and X ? is encoded on two levels (1) on the level of incidence structures: an element x = [v] ? PW is incident with an element a = [?] ? PW ? if “x lies on a”, i.

• distributive law


• group


• pointwise


• torsor laws


• projective geometry


• g3 ?


• uab


• neglect sign


• left distributivity

Sujets

• Grosse
• Klein four-group
• Distributivity
• Pointwise
• Projective geometry

Informations

• Publié par: pefav
• Nombre de lectures: 13
• Langue: English

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THE PROJECTIVE GEOMETRY OF A GROUP

WOLFGANG BERTRAM

Abstract.We show that the pair (P(Ω),Gras(Ω)) given by the power setP= P(Ω) and by the “Grassmannian”Gras(Ω) of all subgroups of an arbitrary group Ω behaves very much like a projective spaceP(W) and its dual projective space P(W ∗ ) of a vector spaceW precisely, we generalize several results from the. More case of the abelian group Ω = (W,+) (cf. [BeKi10a]) to the case of a general group Ω. Most notably, pairs of subgroups (a, b) of Ω parametrizetorsorandsemitorsor structures onPasbrgealvetiiaocssafoeloˆrehT.nowa]isKi10[meBfsoraprina-d taken by analogs ofsirgnear-n.

1.Introduction and statement of main results

1.1.Projective geometry of an abelian group.Before explaining our general results, let us briefly recall the classical case of projective geometry of a vector spaceW: letX=P(W) be the projective space ofWandX 0 =P(W ∗ ) be its dual projective space (space of hyperplanes). The “duality” betweenXandX 0 is encoded on two levels (1) on the level ofincidence structures: an elementx= [v]∈PWisincident with an elementa= [α]∈PW ∗ if “xlies ona”, i.e., ifα(v) = 0 ; otherwise we say that they areremoteortransversal, and we then writex>a; (2) on the level of(linear) algebra: the seta > of elementsx∈ Xthat are transversal toais, in a completely natural way, anaffine space. In [BeKi10a], the second point has been generalized: for any pair (a, b)∈ X 0 × X 0 , the intersectionU ab :=a > ∩b > of two “affine cells” carries a natural torsor structure. Recall that “torsors are for groups what affine spaces are for vector spaces”: 1 Definition 1.1.Asemitorsoris a setGtogether with a mapG 3 →G,(x, y, z)7→ (xyz)such that the following identity, called thepara-associative law, holds: (T1) (xy(zuv)) = (x(uzy)v) = ((xyz)uv). Atorsoris a semitorsor in which, moreover, the followingidempotent lawholds: (T2) (xxy) =y= (yxx).

Fixing the middle elementyin a torsorG, we get a group lawxz:= (xyz) with neutral elementy semitorsors Similarly,, and every group is obtained in this way. give rise to semigroups, but the converse is more complicated. The torsorsU a :=U aa 2010Mathematics Subject Classification.08A02, 20N10, 16W10, 16Y30 , 20A05, 51N30. Key words and phrases.torsor (heap, groud, principal homogeneous space), semitorsor, rela- tions, projective space, Grassmannian, near-ring, generalized lattice. 1 The concept used here goes back to J. Certaine [Cer43]; there are several equivalent versions, known under various other names such asgroud,heap, orprincipal homogeneous space. 1

2 WOLFGANG BERTRAM are the underlying torsors of the affine spacea > , hence are abelian, whereas for a6=b, the torsorsU ab are in general non-commutative. Thus, in a sense, the torsors U ab aredeformationsof the abelian torsorU a  generally, in [BeKi10a] all this. More is done for a pair (X,X 0 ) ofdual Grassmannians, not only for projective spaces.

1.2.Projective geometry of a general group.In the present work, thecom- mutativegroup (W,+) will be replaced by an arbitrary group Ω (however, in order to keep formulas easily readable, we will still use an additive notation for the group lawofΩ).Itturnsout,then,thattherˆoleofXis taken by the power setP(Ω) ofallsubsets of Ω, and the one ofX 0 by the “Grassmannian” of all subgroups of Ω. We call Ω the “background group”, or justthe background. Its subsets will be denoted by small latin lettersa, b, x, y, . . .and, if possible, elements of such sets by corresponding greek letters:α∈a,ξ∈x said above, “projective As, and so on. geometry on Ω” in our sense has two ingredients which we are going to explain now (1) a (fairly weak)incidence(or rather:encdeciinn-no)structure, and (2) a much more relevantalgebraicstructure consisting of a collection of torsors and semi-torsors.

Definition 1.2.Theprojective geometryof a group(Ω,+)is its power setP:= P(Ω) say that a pair. We(x, y)∈ P 2 isleft transversalif everyω∈Ωadmits a unique decompositionω=ξ+ηwithξ∈xandη∈y. We then writex>y. We say that the pair(x, y)isright transversalify>x, and we let x > :={y∈ P |x>y}, > x:={y∈ P |y>x}.

The “(non-) incidence structure” thus defined is not very interesting in its own right; however, in combination with the algebraic torsor structures it becomes quite powerful. There are two, in a certain sense “pure”, special cases to consider; the general case is a sort of mixture of these two: leta, bbe two subgroups of Ω, (A) thetransversal casea>b: thena > ∩ > bis a torsor of “bijection type”, (B) thesingular casea=b; it corresponds to “pointwise torsors” > aandb > . Protoypes for (A) are torsors of the typeG= Bij(X, Y) (set of bijectionsf:X→Y between two setsXandY), with torsor structure (f gh) :=f◦g −1 ◦h, and prototypes for (B) are torsors of the typeG= Map(X, A) (set of maps fromXtoA), where Ais a torsor andXa set, together with their natural “pointwise product”. Case (A) arises, if, whena>b, we identify Ω with the cartesian producta×b; then elementsz∈ > bcan be identified with “left graphs”{(α, Zα)|α∈a}of mapsZ:a→b map. TheZis bijective iff this graph belongs toa > . Therefore G:= > b∩a >  a torsor, it iscarries a natural torsor structure of “bijection type”: as isomorphic to Bij(a, b). Itmay be empty; if it is non-empty, then it is isomorphic  to (the underlying torsor of) the group Bij(a, a). Note that the structure of this group does not involve the one of Ω, indeed, the group structure of Ω enters here only implicitly, via the identification of Ω witha×b. On the other hand, in the “singular case” (B), the set > ais naturally identified with the set of sections of the canonical projection Ω→Ω/a, and this set is a torsor of pointwise type, modelled on the “pointwise group” of all mapsf: Ω/a→a. It