Note on the holonomy groups of pseudo-Riemannian manifolds

For an arbitrary subalgebra h ? so(r, s) a polynomial pseudo-Riemannian metric of signature (r + 2, s + 2) is constructed, the holonomy algebra of this metric contains h as a subalgebra. This result shows the essential distinction between the holonomy algebras of pseudo-Riemannian manifolds of index greater than or equal to 2 and the holonomy algebras of Riemannian and Lorentzian manifolds.     

Pseudo-Riemannian manifolds with recurrent spinor fields

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold (M,g) is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of (M,g). We characterize the following simply connected pseudo-Riemannian manifolds that admit these subbundles in terms of their holonomy algebras: Riemannian manifolds, Lorentzian manifolds, pseudo-Riemannian manifolds with irreducible holonomy algebras, and pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.     

Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras \mathfrakg ì \mathfrakosp(p, q|2m){\mathfrak{g}\subset\mathfrak{osp}(p, q|2m)} of Riemannian supermanifolds under the assumption that \mathfrakg{\mathfrak{g}} is a direct sum of simple Lie superalgebras of classical type and possibly of a 1-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

Convex bodies and multiplicities of ideals

We associate convex regions in ? ⁿ to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on the one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.     

A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.     

Associative conformal algebras with finite faithful representation

The classification of irreducible subalgebras of the associative conformal algebra Cend_N is presented in this paper. The structure theory of associative conformal algebras with finite faithful representation is developed.     

New HKT manifolds arising from quaternionic representations

We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K?hler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in ${SL(n,\mathbb{H})}$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K?hler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K?hler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.     

On the non existence of some rational 3-folds in ℙ5

In this paper we show that there are no smooth rational 3-folds in ?⁵ (C) which are rational conic bundles, over minimal surfaces, whose generic fibre is embedded as a rational curve of degreeh≥3, (ifh=2 there is a complete classification for these 3-folds as well as for the case of ?¹-bundles). Except for conic bundles, we also give the complete list of rational 3-folds in ?⁵ which are minimal according to Mori’s theory. These are little steps towards the classification of all smooth 3-folds in ?⁵ not of general type.     

On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

Invariant complex structures on four-dimensional solvable real lie groups

In this paper, invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups are classified where the dimension of the commutator is less than three. The resulting complex surfaces corresponding to these structures are also determined. The classification problem is reduced to determining certain complex “structure” subalgebras of the complexifications of the four-dimensional, solvable real Lie algebras. Most of the eleven types of non-abelian solvable real Lie algebras do have complex structure subalgebras; three do not. Only three types of algebras have solvable complex structure subalgebras, and only one possesses both abelian and solvable complex structure subalgebras. Each of the possible homogeneous surfaces is represented in the list of resulting manifolds.     

Projective holonomy II: cones and complete classifications

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a ‘projective cone’, a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra.     

Einstein metrics via intrinsic or parallel torsion

The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The classifications for these give isolated examples that are isotropy irreducible spaces, and known families that are the nearly Kähler manifolds in dimension 6 and Grays weak holonomy G₂ structures in dimension 7.Mathematics Subject Classification (2000): 53C10, 17B10, 53C25, 53C29in final form: 11 June 2003     

Typical ℤn-actions can be inserted only in injective ℝn-actions

We study actions of the groups ?ⁿ and ?ⁿ by Lebesgue space automorphisms. We prove that a typical ?ⁿ-action can be inserted only in injective actions of ?ⁿ, n ∈ ?. We give a simple proof of the fact that a typical ?²-action cannot be inserted in an ?-action.     

Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g^∗ related to an algebraic Nijenhuis operator on a finite-dimensional Lie algebra g. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related to Borel subalgebras and a new proof of the complete integrability of the free rigid body system on gln.     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?     

A Dixmier-Douady theorem for Fell algebras

We generalise the Dixmier-Douady classification of continuous-trace C^?-algebras to Fell algebras. To do so, we show that C^?-diagonals in Fell algebras are precisely abelian subalgebras with the extension property, and use this to prove that every Fell algebra is Morita equivalent to one containing a diagonal subalgebra. We then use the machinery of twisted groupoid C^?-algebras and equivariant sheaf cohomology to define an analogue of the Dixmier-Douady invariant for Fell algebras, and to prove our classification theorem.