On affine groups admitting invariant two-point sets

A two-point set is a subset of the plane which meets every line in exactly two points. We discuss previous work on the topological symmetries of a two-point set, and show that there exist subgroups of S¹ which do not leave any two-point set invariant. Further, we show that two-point sets may be chosen to be topological groups, in which case they are also homogeneous.     

Uniqueness of left invariant symplectic structures on the affine Lie group

We show the uniqueness of left invariant symplectic structures on the affine Lie group under the adjoint action of , by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with , and also by giving an unexpected identity on it which relates two left invariant symplectic structures. As an application of this result, we classify maximum rank left invariant Poisson structures on the simple Lie groups and . This result is a generalization of Stolin's classification of constant solutions of the classical Yang-Baxter equation for and .

     

On a class of Lie groups with a left invariant flat pseudo-metric

The study of the Lie groups with a left invariant flat pseudo-metric is equivalent to the study of the left-symmetric algebras with a nondegenerate left invariant bilinear form. In this paper, we consider such a structure satisfying an additional condition that there is a decomposition into a direct sum of the underlying vector spaces of two isotropic subalgebras. Moreover, there is a new underlying algebraic structure, namely, a special L-dendriform algebra and then there is a bialgebra structure which is equivalent to the above structure. The study of coboundary cases leads to a construction from an analogue of the classical Yang–Baxter equation.     

Left invariant contact structures on Lie groups

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

Classical Yang-Baxter equation and left invariant affine geometry on lie groups

Let G be a Lie group, T*G=Lie(G)*G its cotangent bundle considered as a Lie group, where G acts on Lie(G)* via the coadjoint action. Each solution r of the Classical Yang Baxter Equation on G, corresponds to a connected Lie subgroup H of T*G such that Lie(H) is a Lagrangian graph in Lie(G)Lie(G)* and H carries a left invariant affine structure. If r is invertible, the Poisson Lie tensor given by r on G is polynomial of degree at most 2 and every double Lie group of (G,) is endowed with an affine and a complex structures and J, both left invariant and given by r, such that J=0.Mathematics Subject Classification (2000): 53D17, 53A15, 17B62Acknowledgements. The first author was partially supported by Enterprise Ireland. He wishes to thank the mathematical department of NUI Maynooth for their kind welcome, during his stay.Revised version: 5 April 2004     

Left invariant poisson structures on classical non-compact simple Lie groups

A Poisson structure on a Lie group is called left invariant if the contravariant 2-tensor field π corresponding to the Poisson structure is left invariant. Explicit examples of such structures were known only for few cases, and in this paper, we give new examples of high rank left invariant Poisson structures for all non-compact classical real simple Lie groups. This result is equivalent to give constant solutions of the classical Yang-Baxter equation [r ¹²,r ¹³]+[r ¹²,r ²³]+[r ¹³,r ²³]=0 taking values in the space ^²g for these Lie groups.     

Holomorphic flat projective structures on projective threefolds

Here we classify projective 3-folds with a holomorphic flat projective structure and Kodaira dimension 1 or 2.The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Symplectic or contact structures on Lie groups

We study left invariant contact forms and left invariant symplectic forms on Lie groups. In the case of filiform Lie groups we give a necessary and sufficient condition for the existence of a left invariant contact form and we prove the uniqueness of this contact form up to a nonzero scalar multiple. As an application we classify all symplectic structures on nilpotent Lie algebras of dimension ?6.     

An affine invariant characterization of flat gravity curves

The study of affine differential geometry is partially motivated by the fact that the direction of the affine normal coincides with the tangent direction of the gravity curve at the base point. A way to prove this is to compute the Taylor expansion of the hypersurface up to third order derivatives. By computing fourth and fifth order coefficients we characterize the second order behavior of the gravity curve at its base point in terms of affine invariants.     

Homogeneous spaces with invariant projectively flat affine connections

We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.

     

Six-dimensional product Lie algebras admitting integrable complex structures

We classify the 6-dimensional Lie algebras of the form $\mathfrak{g}\times \mathfrak{g}$ that admit an integrable complex structure. We also endow a Lie algebra of the kind $\mathfrak{o}(n)\times \mathfrak{o}(n)$ ($n\ge 2$) with such a complex structure. The motivation comes from geometric structures à la Sasaki on $\mathfrak{g}$-manifolds.     

Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

For each simply connected three‐dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three‐dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three‐dimensional case, and Kowalski and Nikv?cevi?'s results [6, Theorems 3.1 and 4.1] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On projective invariant subgroups of Abelian groups

The properties of projective invariant subgroups are studied. The structure of these subgroups in nonreduced groups is described. The conditions under which projective invariant subgroups are fully invariant are considered.     

Algebraically flat or projective algebras

We define and study algebraically flat algebras in order to have a better understanding of algebraically projective algebras of finite type (the projective algebras of literature). A close examination of the differential properties of these algebras leads to our main structure theorem. As a corollary, we get that an algebraically projective algebra of finite type over a field is either a polynomial ring or the affine algebra of a complete intersection.