Free affine actions of unipotent groups on Cn

Püttmann, Annett

doi:10.1007/s00031-005-1133-2

Cited by 3 publications

(4 citation statements)

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“…There is a counterexample to Lipsman's conjecture presented by Yoshino [7] with g (4) = {0}. Essentially the same example serves to show that there is a free nonproper affine C 2 -action on C 5 , which is just the smallest member of a series, n ≥ 5, of free non-proper affine C 2 -actions on C n [5]. We construct a 3-step nilpotent Lie algebra g and 2-dimensional subalgebras h and v, such that the induced V -action on G/H is Winkelmann's [6] the free affine proper C 2 -action on C 6 without global slice.…”

Section: Introduction and Generic Situationmentioning

confidence: 92%

“…Note the similarity to the construction of a pair of counterexamples to Lipsman's conjecture in [7]. The smallest example of a free, affine, non-proper action of a unipotent group on some C n is given by the C 2 -action on C 5 generated by the two derivations [8], [5]. The complex subspaces h = X 1 , X 2 C and v = X 1 + Z 1 , X 2 + Z 2 C are subalgebras of the 4-step nilpotent Lie algebra g constructed in Lemma 7.…”

Section: -Step Nilpotent Lie Algebrasmentioning

confidence: 99%

“…which is an affine action of the commutative group C m on s ∼ = C dim G−m of degree one, i.e., the expression exp( m j=1 t j (X j + Z j )).Y is linear in the variables t j . By [5] this action has a global slice that is algebraically isomorphic to C dim s−m . 3.2.…”

Section: Special Casesmentioning

confidence: 99%

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Biquotient Actions on Unipotent Lie Groups

Püttmann

2008

Int. J. Math.

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We consider pairs (V, H) of subgroups of a connected unipotent complex Lie group G for which the induced V ×H-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra g of G is 3-step nilpotent. If g is 2-step nilpotent, then there is a global slice of the action that is isomorphic to C n . Furthermore, a global slice isomorphic to C n exists if dim V = 1 = dim H or dim V = 1 and g is 3-step nilpotent. We give an explicit example of a 3-step nilpotent Lie group and a pair of 2-dimensional subgroups such that the induced action is proper but the corresponding geometric quotient is not affine.Research supported by SFB/TR12 "Symmetrien und Universalität in mesoskopischen Systemen" of the Deutsche Forschungsgemeinschaft.

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Section: Introduction and Generic Situationmentioning

confidence: 92%

Section: -Step Nilpotent Lie Algebrasmentioning

confidence: 99%

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Biquotient Actions on Unipotent Lie Groups

Püttmann

2008

Int. J. Math.

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“…We can apply this criterion in the affine situation, as follows: Remark 7.4. Püttmann [6,Section 4.2] gives an example of a free action of the abelian group ( 2 , +) on 5 by unipotent affine transformations, such that the quotient is not a Hausdorff space. Hence the action is not proper.…”

Section: Properness Of Actions With Transitive Centralizermentioning

confidence: 99%

Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group

Baues

Globke

2011

Proc. Amer. Math. Soc.

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We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and noncomplete examples, where the linear holonomy is non-abelian, starting in dimensions ≥ 8, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.

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On the moduli of logarithmic connections on elliptic curves

2022

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We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector µ ∈ [0, 1] 5 . The moduli space corresponding to the central weight µ c = ( 1 2 , . . . , 1 2 ) is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the C * -action we obtain a proof of Simpson's foliation conjecture in this case. For each n ≥ 5, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus n points is false.

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Free affine actions of unipotent groups on Cn

Cited by 3 publications

References 10 publications

Biquotient Actions on Unipotent Lie Groups

Biquotient Actions on Unipotent Lie Groups

Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group

On the moduli of logarithmic connections on elliptic curves

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