Binding groups, permutations groups and modules of finite Morley rank

The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank, although the same questions can be asked about other classes of objects, for example, groups definable in $omega$-stable and $o$-minimal theories. In many cases, answers are not known even in the classical category of algebraic groups over algebraically closed fields.

The geometry of involutions in ranked groups with a TI-subgroup

We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $operatorname{PGL}_2(mathbb{K})$.

Simple groups of Morley rank 5 are bad

By exploiting the geometry of involutions in $N_circ^circ$-groups of finite Morley rank, we show that any simple group of Morley rank $5$ is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. The main result is then used to catalog the nonsoluble connected groups of Morley rank $5$.

Locally quadratic modules and minuscule representations

We give a new proof of a theorem by Timmesfeld showing that for simple algebraic groups, abstract modules where all roots act quadratically are direct sums of minuscule representations.

Symmetric powers of Nat SL(2,K)

We identify the representations $mathbb{K}[X^k, X^{k-1}Y, dots, Y^k]$ among abstract $mathbb{Z}[mathrm{SL}_2(mathbb{K})]$-modules. One result is on $mathbb{Q}[mathrm{SL}_2(mathbb{Z})]$-modules of short nilpotence length and generalises a classical quadratic theorem by Smith and Timmesfeld. Another one is on extending the linear structure on the module from the prime field to $mathbb{K}$. All proofs are by computation in the group ring using the Steinberg relations.

Involutive automorphisms of $N_circ^circ$ groups of finite Morley rank

We classify a large class of small groups of finite Morley rank: $N_circ^circ$-groups which are the infinite analogues of Thompsons $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_circ^circ$-subgroup.

Rank 3 Bingo

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3.

Lie rank in groups of finite Morley rank with solvable local subgroups

We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Prufer p-rank at least 2, leading either to some p-strong embedding, or to the Prufer p-rank being exactly 2.

Veranderungen uber einen Satz von Timmesfeld - II. Symmetric powers of Nat sl(2,K)

We identify the spaces of homogeneous polynomials in two variables K[Y^k, XY^{k-1}, ..., X^k] among representations of the Lie ring sl(2,K). This amounts to constructing a compatible K-linear structure on some abstract sl(2,K)-modules, where sl(2,K) is viewed as a Lie ring.

Small groups of finite Morley rank with involutions

We consider groups of finite Morley rank with solvable local subgroups of even and mixed types. We also consider miscellaneous aspects of small groups of finite Morley rank of odd type.