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1
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Finite-dimensional differential algebraic groups and the Picard-Vessiot theory

100%
Pillay A.
Banach Center Publications
|
2002
|
tom 58
|
nr 1
189-199
EN
We make some observations relating the theory of finite-dimensional differential algebraic groups (the ∂₀-groups of [2]) to the Galois theory of linear differential equations. Given a differential field (K,∂), we exhibit a surjective functor from (absolutely) split (in the sense of Buium) ∂₀-groups G over K to Picard-Vessiot extensions L of K, such that G is K-split iff L = K. In fact we give a generalization to "K-good" ∂₀-groups. We also point out that the "Katz group" (a certain linear algebraic group over K) associated to the linear differential equation ∂Y = AY over K, when equipped with its natural connection ∂ - [A,-], is K-split just if it is commutative.
2
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Model-theoretic consequences of a theorem of Campana and Fujiki

100%
Pillay A.
Fundamenta Mathematicae
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2002
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tom 174
|
nr 2
187-192
EN
We give a model-theoretic interpretation of a result by Campana and Fujiki on the algebraicity of certain spaces of cycles on compact complex spaces. The model-theoretic interpretation is in the language of canonical bases, and says that if b,c are tuples in an elementary extension 𝓐* of the structure 𝓐 of compact complex manifolds, and b is the canonical base of tp(c/b), then tp(b/c) is internal to the sort (ℙ¹)*. The Zilber dichotomy in 𝓐* follows immediately (a type of U-rank 1 is locally modular or nonorthogonal to the field ℂ*), as well as the "algebraicity" of any subvariety X of a group G definable in 𝓐* such that Stab(X) is trivial.
3
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Generic sets in definably compact groups

64%
Peterzil Y. a., Pillay A.
Fundamenta Mathematicae
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2007
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tom 193
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nr 2
153-170
EN
A subset X of a group G is called left genericif finitely many left translates of X cover G. Our main result is that if G is a definably compact group in an o-minimal structure and a definable X ⊆ G is not right generic then its complement is left generic. Among our additional results are (i) a new condition equivalent to definable compactness, (ii) the existence of a finitely additive invariant measure on definable sets in a definably compact group G in the case where G = *H for some compact Lie group H (generalizing results from [1]), and (iii) in a definably compact group every definable subsemi-group is a subgroup. Our main result uses recent work of Alf Dolich on forking in o-minimal stuctures.
4
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On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures

64%
Conversano A., Pillay A.
Fundamenta Mathematicae
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2013
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tom 222
|
nr 1
49-62
EN
We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With a rather strong definition of ind-definable semisimple subgroup, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R· S where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.
5
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Some model theory of SL(2,ℝ)

51%
Gismatullin J., Penazzi D., Pillay A.
Fundamenta Mathematicae
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2015
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tom 229
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nr 2
117-128
EN
We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space $S_{G}(M)$. We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on $S_{G}(M)$). We also show that the "Ellis group" (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.
6
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Countable modules

23%
Pillay A.
Fundamenta Mathematicae
|
1984
|
tom 121
|
nr 2
125-132
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