Moreno Invitti - Lie rings in finite dimensional theories
112
ICJ
Lie rings are algebraic structures that have recently attracted attention in model theory, following the work of Deloro and Ntsiri. A Lie ring g is an abelian group equipped with a bilinear map [,] that is antisymmetric and satisfies the Jacobi identity. A natural question in this context is an analogue of the Algebraicity Conjecture for Lie rings: if g is a simple Lie ring of finite Morley rank, then is g definably isomorphic to a Lie algebra over an algebraically closed field? Although the conjecture remains open, Deloro and Ntsiri have classified simple connected Lie rings of Morley rank up to 4.
Finite-dimensional theories, introduced by Frank Wagner, are a generalization of theories of finite Morley rank and so provide a broader context in which to study this question. Therefore, one may ask whether the results known for Lie rings of finite Morley rank extend to finite-dimensional Lie rings. While the general case remains unresolved, we show that such extensions hold for connected finite-dimensional Lie rings and for NIP finite-dimensional Lie rings. Specifically, a connected Lie ring of dimension 1 is abelian; of dimension 2, it is solvable; and of dimension 3, it is either isomorphic to sl2(K) for a definable field K of dimension 1, or it is “anisotropic”. In the NIP setting, a Lie ring of dimension 1 is virtually abelian; of dimension 2, virtually solvable; and of dimension 3 or 4 either virtually connected or virtually solvable. In the latter case, we obtain a full classification using the results established for connected finite-dimensional Lie rings. Moreover, we show the existence of definable envelopes for nilpotent and soluble Lie subrings of a hereditarily Mfc-Lie ring. This class of Lie rings extends both finite dimensional Lie rings and simple (in the model theoretic sense) Lie rings. In particular, this holds for stable theories, extending a result of Zamour.