Moufang Sets and Structurable Division Algebras
Moufang Sets and Structurable Division Algebras

Author: Lien Boelaert

Publisher: American Mathematical Soc.

Published: 2019-06-10

Total Pages: 102

ISBN-13: 1470435543

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A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. The authors extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, they show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. The authors also obtain explicit formulas for the root groups, the τ-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.

The Role of Nonassociative Algebra in Projective Geometry
The Role of Nonassociative Algebra in Projective Geometry

Author: John R. Faulkner

Publisher: American Mathematical Soc.

Published: 2014-10-09

Total Pages: 247

ISBN-13: 1470418495

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There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes, with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught. On the geometric side, the book introduces coordinates in projective planes and relates coordinate properties to transitivity properties of certain automorphisms and to configuration conditions. It also classifies higher-dimensional geometries and determines their automorphisms. The exceptional octonion plane is studied in detail in a geometric context that allows nondivision coordinates. An axiomatic version of that context is also provided. Finally, some connections of nonassociative algebra to other geometries, including buildings, are outlined. On the algebraic side, basic properties of alternative algebras are derived, including the classification of alternative division rings. As tools for the study of the geometries, an axiomatic development of dimension, the basics of quadratic forms, a treatment of homogeneous maps and their polarizations, and a study of norm forms on hermitian matrices over composition algebras are included.

Tits Polygons
Tits Polygons

Author: Bernhard Mühlherr

Publisher: American Mathematical Society

Published: 2022-02-02

Total Pages: 114

ISBN-13: 1470451018

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Moufang Loops and Groups with Triality are Essentially the Same Thing
Moufang Loops and Groups with Triality are Essentially the Same Thing

Author: J. I. Hall

Publisher: American Mathematical Soc.

Published: 2019-09-05

Total Pages: 206

ISBN-13: 1470436221

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In 1925 Élie Cartan introduced the principal of triality specifically for the Lie groups of type D4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title in 1978 was made by Stephen Doro, who was in turn motivated by the work of George Glauberman from 1968. Here the author makes the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.”

A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth

Author: Jaroslav Nešetřil

Publisher: American Mathematical Soc.

Published: 2020-04-03

Total Pages: 120

ISBN-13: 1470440652

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In this paper the authors introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. The authors show how the various approaches to graph limits fit to this framework and that the authors naturally appear as “tractable cases” of a general theory. As an outcome of this, the authors provide extensions of known results. The authors believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, the authors consider limits of structures with bounded diameter connected components and prove that in this case the convergence can be “almost” studied component-wise. They also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, they consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of modeling the authors introduce here. Their example is also the first “intermediate class” with explicitly defined limit structures where the inverse problem has been solved.

Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type
Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type

Author: Carles Broto

Publisher: American Mathematical Soc.

Published: 2020-02-13

Total Pages: 176

ISBN-13: 1470437724

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For a finite group G of Lie type and a prime p, the authors compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex but can always be reduced to a case where the natural map from Out(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a group by a group of automorphisms, and in part by wanting to describe self homotopy equivalences of BG∧p in terms of Out(G).

Compact Quotients of Cahen-Wallach Spaces
Compact Quotients of Cahen-Wallach Spaces

Author: Ines Kath

Publisher: American Mathematical Soc.

Published: 2020-02-13

Total Pages: 96

ISBN-13: 1470441039

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Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. The authors derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these parameters.

A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side
A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side

Author: Chen Wan

Publisher: American Mathematical Soc.

Published: 2019-12-02

Total Pages: 102

ISBN-13: 1470436868

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Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Time-Like Graphical Models
Time-Like Graphical Models

Author: Tvrtko Tadić

Publisher: American Mathematical Soc.

Published: 2019-12-02

Total Pages: 184

ISBN-13: 147043685X

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The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure— so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.