In mathematics, Moufang polygons are a generalization by

Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and ...

of the

Moufang plane In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the property ...

s studied by

Ruth Moufang Ruth Moufang (10 January 1905 – 26 November 1977) was a German mathematician. Biography Born to German chemist Eduard Moufang and Else Fecht Moufang. Eduard Moufang was the son of Friedrich Carl Moufang (1848-1885) from Mainz, and Elisa ...

, and are irreducible

building A building, or edifice, is an enclosed structure with a roof and walls standing more or less permanently in one place, such as a house or factory (although there's also portable buildings). Buildings come in a variety of sizes, shapes, and fun ...

s of rank two that admit the action of

root group In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...

s. In a book on the topic, Tits and Richard Weiss classify them all. An earlier theorem, proved independently by Tits and Weiss, showed that a Moufang polygon must be a generalized 3-gon, 4-gon, 6-gon, or 8-gon, so the purpose of the aforementioned book was to analyze these four cases.

Definitions

*A generalized ''n''-gon is a

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...

of diameter ''n'' and girth 2''n''. *A graph is called thick if all vertices have valence at least 3. *A root of a generalized ''n''-gon is a path of length ''n''. *An apartment of a generalized ''n''-gon is a cycle of length 2''n''. *The root subgroup of a root is the subgroup of automorphisms of a graph that fix all vertices adjacent to one of the inner vertices of the root. *A Moufang ''n''-gon is a thick generalized ''n''-gon (with ''n''>2) such that the root subgroup of any root acts transitively on the apartments containing the root.

Moufang 3-gons

A Moufang 3-gon can be identified with the

incidence graph In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we for ...

of a Moufang

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

. In this identification, the points and lines of the plane correspond to the vertices of the building. Real forms of

Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

give rise to examples which are the three main types of Moufang 3-gons. There are four real

division algebras In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

: the real numbers, the

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

, the

quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

, and the

octonions In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...

, of dimensions 1,2,4 and 8, respectively. The projective plane over such a division algebra then gives rise to a Moufang 3-gon. These projective planes correspond to the building attached to SL₃(R), SL₃(C), a real form of A₅ and to a real form of E₆, respectively. In the first diagram the circled nodes represent 1-spaces and 2-spaces in a three-dimensional vector space. In the second diagram the circled nodes represent 1-space and 2-spaces in a 3-dimensional vector space over the

, which in turn represent certain 2-spaces and 4-spaces in a 6-dimensional complex vector space, as expressed by the circled nodes in the A₅ diagram. The fourth case — a form of E₆ — is exceptional, and its analogue for Moufang 4-gons is a major feature of Weiss's book. Going from the real numbers to an arbitrary field, Moufang 3-gons can be divided into three cases as above. The split case in the first diagram exists over any field. The second case extends to all associative, non-commutative division algebras; over the reals these are limited to the algebra of quaternions, which has degree 2 (and dimension 4), but some fields admit central division algebras of other degrees. The third case involves "alternative" division algebras (which satisfy a weakened form of the associative law), and a theorem of

Richard Bruck Richard Hubert Bruck (December 26, 1914 – December 18, 1991) was an American mathematician best known for his work in the field of algebra, especially in its relation to projective geometry and combinatorics. Bruck studied at the University of ...

and Erwin Kleinfeld shows that these are Cayley-Dickson algebras. This concludes the discussion of Moufang 3-gons.

Moufang 4-gons

Moufang 4-gons are also called Moufang quadrangles. The classification of Moufang 4-gons was the hardest of all, and when Tits and Weiss started to write it up, a hitherto unnoticed type came into being, arising from groups of type F4. They can be divided into three classes: * (i) Those arising from classical groups. * (ii) Those arising from "mixed groups" (in which there are two imperfect fields of characteristic 2, K and L, with K2 ⊂ L ⊂ K). * (iii) Those arising from quadrangular algebras. There is some overlap here, in the sense that some classical groups arising from pseudo-quadratic spaces can be obtained from quadrangular algebras (which Weiss calls special), but there are other, non-special ones. The most important of these arise from algebraic groups of types E6, E7, and E8. They are k-forms of algebraic groups belonging to the following diagrams: E6 E7 E8. The E6 one exists over the real numbers, though the E7 and E8 ones do not. Weiss calls the quadrangular algebras in all these cases Weiss regular, but not special. There is a further type that he calls defective arising from groups of type F4. These are the most exotic of all—they involve purely inseparable field extensions in characteristic 2—and Weiss only discovered them during the joint work with Tits on the classification of Moufang 4-gons by investigating a strange lacuna that should not have existed but did. The classification of Moufang 4-gons by Tits and Weiss is related to their intriguing monograph in two ways. One is that the use of quadrangular algebras short-cuts some of the methods known before. The other is that the concept is an analogue to the octonion algebras, and quadratic Jordan division algebras of degree 3, that give rise to Moufang 3-gons and 6-gons. In fact all the exceptional Moufang planes, quadrangles, and hexagons that do not arise from "mixed groups" (of characteristic 2 for quadrangles or characteristic 3 for hexagons) come from octonions, quadrangular algebras, or Jordan algebras.

Moufang 6-gons

Moufang 6-gons are also called Moufang hexagons. A classification of Moufang 6-gons was stated by Tits, though the details remained unproven until the joint work with Weiss on Moufang Polygons.

Moufang 8-gons

Moufang 8-gons are also called Moufang octagons. They were classified by Tits, where he showed that they all arise from

Ree groups In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a dif ...

of type ²F₄.

Quadrangular algebras

A potential use for quadrangular algebras is to analyze two open questions. One is the Kneser-Tits conjecture that concerns the full group of linear transformations of a building (e.g. GL_''n'') factored out by the subgroup generated by root groups (e.g. SL_''n''). The conjecture is proved for all Moufang buildings except the 6-gons and 4-gons of type E8, in which case the group of linear transformations is conjectured to be equal to the subgroup generated by root groups. For the E8 hexagons this can be rephrased as a question on quadratic Jordan algebras, and for the E8 quadrangles it can now be rephrased in terms of quadrangular algebras. Another open question about the E8 quadrangle concerns fields that are complete with respect to a discrete valuation: is there, in such cases, an affine building that yields the quadrangle as its structure at infinity?

Definitions

Moufang 3-gons

Moufang 4-gons

Moufang 6-gons

Moufang 8-gons

Quadrangular algebras

See also

Notes and references

Further reading