SU(3)-Structures on Hypersurfaces of Manifolds With G 2-Structure

We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G₂-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G₂-structure with the shape tensor of the hypersurface. Additionally, we show how to compute the intrinsic SU(3)-torsion and the intrinsic G₂-torsion by means of the exterior algebra.     

Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G^∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G^∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G, then h is the super Lie algebra of a sub-super Lie group of G. Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G. We also show that if H is a closed sub-super Lie group of a super Lie group G, then G→G/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G^∞ category.     

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g₀ we associate a Clifford algebra over the field of rational functions on O. We find the rank, k(O) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U(g)-module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k(O) is in many cases, equal to the odd dimension of the orbit G⋅O, where G is a Lie supergroup with Lie superalgebra g.     

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH ₁ andH ₂ be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h₁ + Adg ? h₂, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H ₁,rankG/H ₂). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H ₁ xG/H ₂) ^G for the diagonal action ofG. Ifk = ? andG _a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = g_a + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.     

Conformally parallel G 2 structures on a class of solvmanifolds

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G₂-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G₂ structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G₂. In the process we also find a new metric with exceptional holonomy. Received: 20 September     

SHEETS IN SYMMETRIC LIE ALGEBRAS AND SLICE INDUCTION

In this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more detail the sheets of the non-trivial symmetric Lie algebra of type G₂. We characterize their singular loci and provide a nice desingularization lying in so ₇.     

Asymptotic cone of semisimple orbits for symmetric pairs

Let G be a reductive algebraic group over C and denote its Lie algebra by g. Let Oh be a closed G-orbit through a semisimple element h∈g. By a result of Borho and Kraft (1979) [4], it is known that the asymptotic cone of the orbit Oh is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer ZG(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair.More precisely, let θ be an involution of G, and K=Gθ a fixed point subgroup of θ. Then we have a Cartan decomposition g=k+s of the Lie algebra g=Lie(G) which is the eigenspace decomposition of θ on g. Let {x,h,y} be a normal sl₂ triple, where x,y∈s are nilpotent, and h∈k semisimple. In addition, we assume , where denotes the complex conjugation which commutes with θ. Then is a semisimple element in s, and we can consider a semisimple orbit Ad(K)a in s, which is closed. Our main result asserts that the asymptotic cone of Ad(K)a in s coincides with , if x is even nilpotent.     

Two results on centralisers of nilpotent elements

Let g be a semisimple Lie algebra. We provide a short proof of McNinch’s result on centralisers of the sum of commuting nilpotent elements (J. Pure Appl. Algebra 206 (2006), 123-140) and characterise nilpotent elements e∈g having the property that the orbit G⋅e is the largest nilpotent orbit meeting the centraliser of e.     

Deformation rigidity of the rational homogeneous space associated to a long simple root

As a continuation of our previous works we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces G/P of Picard number 1, confirming its validity whenever G/P is associated to a long simple root. For these rational homogeneous spaces the minimal G-invariant holomorphic distribution D is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of G/P is completely determined by the nilpotent symbol algebra of the weak derived differential system of D. The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point o∈G/P we prove that the nilpotent symbol algebra at o is nothing other than the universal Lie algebra generated by D_o subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra G.     

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g^∗), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g^∗. We write down families of examples for the case that g is a classical complex simple Lie algebra.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

Symplectic resolutions for coverings of nilpotent orbits

Let $\text{O}$ be a nilpotent orbit in a semisimple complex Lie algebra $\text{g}$. Denote by G the simply connected Lie group with Lie algebra $\text{g}$. For a G-homogeneous covering $\text{M→}\text{O}$, let X be the normalization of $\text{O}$ in the function field of M. In this Note, we study the existence of symplectic resolutions for such coverings X. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Ricci soliton homogeneous nilmanifolds

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric for some and some derivationD of the Lie algebra ofN, where Ric denotes the Ricci operator of . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set. Received August 24, 1999 / Revised October 2, 2000 / Published online February 5, 2001     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

A Characterization of Stem Algebras in Terms of Central Derivations

Let L be a Lie algebra, and Der _z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der _z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der _z (L) is abelian if and only if L is a stem Lie algebra.     

On the Surjectivity of the Exponential Function of Solvable Lie Groups

For a solvable Lie group G the surjectivity of the exponential function exp_G is equivalent to the connectedness of the near-Cartan subgroups and to the connectedness of the centralizers in a Cartan subgroup of all nilpotent elements in its Lie algebra g. Furthermore, these conditions are satisfied if and only if for all elements g ? G there is an x ? g with g = exp_G x in which exp_G is regular.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820-839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.