On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

The automorphism group of certain factorial threefolds and a cancellation problem

A new class of counterexamples to a generalized cancellation problem for affine varieties is presented. Each member of the class is an affine factorial complex threefold admitting a locally trivial action of the additive group, hence the total space for a principal G _a bundle over a quasiaffine base. The automorphism groups for these varieties are also determined.     

G a Invariants and Slices

Criteria for local triviality of algebraic actions of the additive group of complex numbers on complex affine space are extended to more general varieties. Finite generation of rings of invariants of locally trivial actions on factorial affine varieties is dicussed, giving some sufficient conditions for finite generation and examples where finite generation fails. A missing hypothesis in a theorem of Miyanishi is identified, and an example is given to demonstrate the necessity of the hypothesis. The corrected theorem is shown to hold for a class of triangular G _a actions on C⁴, with the consequence that all these actions are conjugate to translations. A new criterion for an action to be conjugate to a global translation is given.     

ON TWIN-TRIANGULAR ADDITIVE GROUP ACTIONS ON C4

This paper concerns some of the conditions satisfied by additive group actions on complex affine space which can be written locally as a translation of a variable. Assume X is the affine variety C ⁿ , G_a = (C, +), and σ : G_a × X → X is the action defined by a group monomorphism G _a → Aut _C X. If σ is locally trivial, then the action satisfies what is termed a “GICO” condition.

It will be shown that a large class of G_a -actions on C ⁴, that is, fixed-point free, “twin-triangular” actions with finitely-generated rings of invariants, are at least GICO. Remaining questions are discussed.     

Triangular actions on

Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine -space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

     

Locally trivial 𝔾a−actions on ℂ5 with singular algebraic quotients

Simple examples are given of proper algebraic actions of the additive group of complex numbers on ?⁵ whose geometric quotients are, respectively, a?ne, strictly quasia?ne, and algebraic spaces which are not schemes. Moreover, a Zariski locally trivial action is given whose ring of invariant regular functions defines a singular factorial a?ne fourfold embedded in ?¹². The geometric quotient for the action embeds as a strictly quasia?ne variety in the smooth locus of the algebraic quotient with complement isomorphic to the normal a?ne surface with the A₂?singularity at the origin.     

Yangians and cohomology rings of Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL _n . We construct the action of the Yangian of \mathfraksl_n{\mathfrak{sl}_n} in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of \mathfraksl_n[s^±1,t]{\mathfrak{sl}_n[s^{\pm1},t]}) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space \mathfrakM_n,d{\mathfrak{M}_{n,d}} of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of \mathfrakgl_n{\mathfrak{gl}_n} naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on \mathfrakM_n,d{\mathfrak{M}_{n,d}} is the image of a noncommutative power sum in Z.     

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

Le groupe fondamental de certains espaces d'orbites regulieres de groupes de Weyl affines

The affine Weyl group W _aof an irreducible root system of rank n acts on the complexification h of a real space of dimension n via the usual (affine) action on the imaginary part and the action through the finite Weyl group on the real part. This group acts freely on the complement h of some complex hyperplanes. We prove a presentation of the fundamental group of the quotient hW _a.

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A Jacques Tits pour son soixantième anniversaire     

Nilpotent and soluble groups with respect to a homomorph

For an excellent ring Awhose real spectrum satisfies some connectedness condition, we give a sensible notion of real analytic component for a Zariski closed subset of Specr A(such a closed subset will also be called a locally Nash set).Indeed we show that the locally Nash sets are the closed subsets of a noetherian topology on an abstract new space G which we introduce.This generalizes the geometric notion of global real analytic component when Ais the ring of global Nash functions on an affine Nash manifold.     

On a family of lie algebras related to homogeneous surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space ?³ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author’s works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups G ? Aff(3, ?) in the complex tangent space T ^? _p M of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups G on the spaces T _p ^? M for affine homogeneous strictly pseudoconvex surfaces of general position in ?³ that are different from quadrics.     

Locally compact (2, 2)‐transformation groups

We determine all locally compact imprimitive transformation groups acting sharply 2‐transitively on a non‐totally disconnected quotient space of blocks inducing on any block a sharply 2‐transitive group and satisfying the following condition: if Δ₁, Δ₂ are two distinct blocks and P_i, Q_i ∈ Δ_i (i = 1, 2), then there is just one element in the inertia subgroup which maps P_i onto Q_i. These groups are natural generalizations of the group of affine mappings of the line over the algebra of dual numbers over the field of real or complex numbers or over the skew‐field of quaternions. For imprimitive locally compact groups, our results correspond to the classical results of Kalscheuer for primitive locally compact groups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Espaces homogènes génériquement triviaux

We study the homogeneous spaces of a smooth group G over affine spaces or local k-algebra and gerbs locally banded by a semi-simple group H dèfine over k. In particular we show that every gerb locally banded by H and rationally trivial is trivial, and that an homogeneous space of G with semi-simple isotropy which is rationally trivial is trivial in the Zariski topology. This extends a result of Colliot-Thélène and Ojanguren concerning G-torsors.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

Locally symmetric complex affine hypersurfaces

In this paper we study non-degenerate locally symmetric complex affine hypersurfaces Mⁿ of the complex affine space, i.e. hypersurfaces satisfying R=0, where is the affine connection induced on Mⁿ by the complex affine structure on the complex affine space, and R is the curvature tensor of . We classify the non-degenerate locally symmetric hypersurfaces Mⁿ, n > 2, and the minimal non-degenerate locally symmetric hypersurfaces Mⁿ, n > 1.Aspirant N.F.W.O. (Belgium)     

On compression of Bruhat-Tits buildings

Consider an affine Bruhat-Tits building Lat _n of type A_n−1 and the complex distance in Lat _n, i.e., the complete system of invariants of a pair of vertices of the building. An element of the Nazarov semigroup is a lattice in the duplicated p-adic space ℚ _p ⁿ ⊕ ℚ _p ⁿ . We investigate the behavior of the complex distance with respect to the natural action of the Nazarov semigroup on the building. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 163–170.     

On the Manifold of Almost Complex Structures

Let (M, g ₀) be a smooth closed Riemannian manifold of even dimension 2_n admitting an almost complex structure. It is shown that the space of all almost complex structures on M determining the same orientation as the one determined by a fixed almost complex structure J ₀ is a smooth locally trivial fiber bundle over the space of almost complex structures orthogonal with respect to g ₀ and determining the same orientation as J ₀.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 66–71.Original Russian Text Copyright © 2005 by N. A. Daurtseva.     

FAMILIES OF GROUP ACTIONS,GENERIC ISOTRIVIALITY,AND LINEARIZATION

Our base field is the field ? of complex numbers. We study families of reductive group actions on \( {\mathbb A} \) 2 parametrized by curves and show that every faithful action of a non-finite reductive group on \( {\mathbb A} \) 3 is linearizable, i.e., G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine morphisms ??: S ? Y and q : Τ ? Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant étale base change φ: U ? Y . A special case is the following result. Call a morphism φ: X ? Y a fibration with fiber F if φ is at and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an étale dominant morphism μ: U ? Y such that the pull-back is a trivial fiber bundle: U?×?Y X???U?×?F. As an application we give short proofs of the following two (known) results: (a) Every affine A1-_bration over a normal variety is locally trivial in the Zariskitopology (see [KW85]). (b) Every affine A2-_bration over a smooth curve is locally trivial in the ZariskiTopology (see [KZ01]).     

Free

It has been conjectured that every free algebraic action of the additive group of complex numbers on complex affine three space is conjugate to a global translation. The main result lends support to this conjecture by showing that the morphism to the variety defined by the ring of invariants of the associated action on the coordinate ring is smooth. As a consequence, the graph morphism is an open immersion, and simple proofs of certain cases of the conjecture are obtained.

     

Simple left-symmetric algebras with¶solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied. Received: 10 June 1997 / Revised version: 29 September 1997