Papers by Dmitri Alekseevsky
An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together... more An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in \Gamma({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras.
Journal of Algebra, 2007
Let g be a real semisimple Lie algebra with Killing form B and k a B-nondegenerate subalgebra of ... more Let g be a real semisimple Lie algebra with Killing form B and k a B-nondegenerate subalgebra of g of maximal rank. We give a description of all ad k -invariant decompositions g = k + m + + m − such that B| m ± = 0, B(k, m + + m − ) = 0 and k + m ± are subalgebras. It is reduced to a description of parabolic subalgebras of g with given reductive part k. This is obtained in terms of crossed Satake diagrams. As an application, we get a classification of invariant bi-Lagrangian (or equivalently para-Kähler) structures on homogeneous manifolds G/K of a semisimple group G.
We study a geometric structure on a (4n + 3)-dimensional smooth manifold M which is an integrable... more We study a geometric structure on a (4n + 3)-dimensional smooth manifold M which is an integrable, nondegenerate codimension 3-subbundle D on M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of the quaternionic CR structure. In order to study this geometric structure on M , we single out an sp(1)-valued 1-form ω locally on a neighborhood U of M such that Nullω = D|U . We shall construct the invariants on the pair (M, ω) whose vanishing implies that M is uniformized with respect to a finite dimensional flat quaternionic CR geometry. The invariants obtained on (4n+3)-M have the same formula as the curvature tensor of quaternionic (indefinite) Kähler manifolds. From this viewpoint, we shall exhibit a quaternionic analogue of Chern-Moser's CR structure.
Annali Di Matematica Pura Ed Applicata, 2008
We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}$ on a (4n + 3)-manif... more We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}$ on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an ${\mathfrak{s}}{\mathfrak{p}}(1)$ -valued 1-form ω locally on a neighborhood U such that ${\rm Null}\omega = \mathcal D|U$ and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic Kähler 4n-manifolds. From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure.
Central European Journal of Mathematics, 2004
We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional man... more We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.
We study a geometric structure on a (4n + 3)-dimensional smooth manifold M which is an integrable... more We study a geometric structure on a (4n + 3)-dimensional smooth manifold M which is an integrable, nondegenerate codimension 3-subbundle D on M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of the quaternionic CR structure. In order to study this geometric structure on M , we single out an sp(1)-valued 1-form ω locally on a neighborhood U of M such that Nullω = D|U . We shall construct the invariants on the pair (M, ω) whose vanishing implies that M is uniformized with respect to a finite dimensional flat quaternionic CR geometry. The invariants obtained on (4n+3)-M have the same formula as the curvature tensor of quaternionic (indefinite) Kähler manifolds. From this viewpoint, we shall exhibit a quaternionic analogue of Chern-Moser's CR structure.
Annali Di Matematica Pura Ed Applicata, 2008
We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}$ on a (4n + 3)-manif... more We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}$ on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an ${\mathfrak{s}}{\mathfrak{p}}(1)$ -valued 1-form ω locally on a neighborhood U such that ${\rm Null}\omega = \mathcal D|U$ and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic Kähler 4n-manifolds. From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure.
Central European Journal of Mathematics, 2004
We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional man... more We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non homogeneous (para)-quaternionic CR manifolds is described.
Communications in Mathematical Physics, 2005
A class of ℤ2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra... more A class of ℤ2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form where the algebra of generalized translations W=W 0+W 1 is the maximal solvable ideal of W 0 is generated by W 1 and commutes with W. Choosing W 1 to be a spinorial module (a sum of an arbitrary number of spinors and semispinors), we prove that W 0 consists of polyvectors, i.e.all the irreducible submodules of W 0 are submodules of We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of invariant valued bilinear forms on the spinor module S.
Transactions of The American Mathematical Society, 2007
ABSTRACT
Osaka Journal of Mathematics, 2008
We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure... more We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space Z of such a G-structure is endowed with a field of involutions J ∈ ¼(End T Z ) and a J -invariant distribution H Z . We study the conditions for the integrability of J and for the (para-)holomorphicity of H Z . Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces (Z¯, J , H Z ),¯= ±1, such that J 2 =¯Id. We prove that in both cases J is integrable (recovering results of Blair, Davidov and Muskarov) and that H Z defines a holomorphic (¯= −1) or para-holomorphic (¯= +1) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on Z¯.
Communications in Mathematical Physics, 2003
BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M... more BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M × N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n ≥ 0. Such gradient flows are generated by the "energy function" f = P 2 , where P is a (bundle-valued) moment map associated to n + 1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p ∈ M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f , for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima.
Journal of Geometry and Physics, 2005
We prove that a symmetric space is Osserman if its complexification is a complex hyper-Kähler sym... more We prove that a symmetric space is Osserman if its complexification is a complex hyper-Kähler symmetric space. This includes all pseudo-hyper-Kähler as well as para-hyper-Kähler symmetric spaces. We extend the classification of pseudo-hyper-Kähler symmetric spaces obtained by the first and the third author to the class of para-hyper-Kähler symmetric spaces. These manifolds are possible targets for the scalars of rigid N = 2 supersymmetric field theories with hypermultiplets on fourdimensional space-times with Euclidean signature.
Journal of Geometry and Physics, 2002
We introduce the notion of a special complex manifold: a complex manifold (M, J) with a flat tors... more We introduce the notion of a special complex manifold: a complex manifold (M, J) with a flat torsionfree connection ∇ such that ∇J is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a ∇-parallel symplectic form ω. This generalises Freed's definition of (affine) special Kähler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and Kähler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form α : C n → T * C n . Such a realisation induces a canonical ∇-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special Kähler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms α. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-Kähler structure on the cotangent bundle of a special Kähler manifold.
We classify indefinite simply connected hyper-Kähler symmetric spaces. Any such space without fla... more We classify indefinite simply connected hyper-Kähler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m, 4m).
Communications in Mathematical Physics, 2005
A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebr... more A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W. Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W_0 are submodules of \Lambda V. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant \Lambda^k V-valued bilinear forms on the spinor module S.
We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without fl... more We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H) over the quaternions on the space (S^4C^n)^{\tau} of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = \tau S, where \tau is the real structure induced by the quaternionic structure of C^{2m} = H^m. We define and classify also complex hyper-Kaehler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.
Let M be a manifold with Grassmann structure, i.e., with an isomorphism of the cotangent bundle T... more Let M be a manifold with Grassmann structure, i.e., with an isomorphism of the cotangent bundle T*M ХE H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection ٌ W in a vector bundle W →M as a connection whose curvature
Israel Journal of Mathematics, 1998
We clarify the question whether for a smooth curve of polynomials one can choose the roots smooth... more We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.
We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological... more We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. 2000 Mathematics Subject Classification. Primary 17B05, 17B56. Key words and phrases. Extensions of Lie algebras, cohomology of Lie algebras. P.W.M. was supported by 'Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MAT'. Typeset by A M S-T E X
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Papers by Dmitri Alekseevsky