| Résume | Lie rings are algebraic structures that have recently attracted attention in model the-
ory, 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 alge-
bra 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 “bad.” 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 soluble. In the
latter case, we obtain a full classification using the results established for connected
finite-dimensional Lie rings.
Moreover, we will show the existence of definable envelopes for nilpotent and soluble
Lie subrings of a hereditarily Mc-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. |
