Tài liệu Quaternionic Structures in Mathematics and Physics docx

Proceedings of the Second Meeting Quaternionic Structures in Mathematics and Physics r Editors Stefano Marchiafava Paolo Piccinni Massimiliano Pontecorvo i 2 =j 2 =k 2 =-l, ij=-ji=kjk=-kj=i, ki=-ik=j Proceedings of die Second Meeting Quaternionic Structures in Mathematics and Physics [...]... coadjoint action Finally we can define orbits and stabilizers of sections of z ——> XRemark 1.13 In the case of an action on a twistor space of a hyperkahler manifold M, we shall also speak of Q acting on M Similarity, if s is a twistor line corresponding to a point m in M, we can speak of the stabilizer of Q at m € M etc We shall be particularly interested in the following types of actions Definition... Geometry J Sawon 349 Quaternionic Group Representations and Their Classifications G Scolarici and L Solombrino 365 Vanishing Theorems for Quaternionic Kahler Manifolds U Semmelmann and G Weingart 377 CONTENTS xv Weakening Holonomy A Swann 405 Maxwell's Vision: Electromagnetism with Hamilton's Quaternions D Sweetser and G Sandri 417 Special Kahler Geometry A Van Proeyen 421 Singularities in HyperKahler Geometry... W) In the present work we study some remarkable properties of a special hyperHermitian metric which corresponds to a four-dimensional solvable Lie group We also sketch a procedure for constructing hypercomplex structures on certain nilpotent and solvable Lie groups, following the lines of [3] Acknowledgement The author would like to thank the organizers of the meeting Quaternionic Structures in Mathematics. .. HyperKahler Geometry M Verbitsky 439 Second Meeting on Quaternionic Structures in Mathematics and Physics Roma, 6-10 September 1999 HYPERCOMPLEX STRUCTURES ON SPECIAL CLASSES OF NILPOTENT A N D SOLVABLE LIE GROUPS MARIA LAURA BARBERIS 1 INTRODUCTION A hypercomplex structure on a manifold M is a family {JQ}a=i,2,3 of complex structures on M satisfying the following relations: (1.1) Jl = ~I, a = 1,2,3, J3... g] < 2 carrying hypercomplex structures (cf [1]) it remained to give a characterization in the case when g is 2-step nilpotent and dim[g, g] = 2: this is obtained by taking m — 2 in Theorem 3.1 below It is a result of [7] that the only 8-dimensional non-abelian nilpotent Lie algebras carrying abelian hypercomplex structures are trivial central extensions of //-type Lie algebras We show in [3] that... and new) hyperkahler manifolds arise as quotients in this setting For example, all hyperkahler structures on semisimple coadjoint orbits of a complex semisimple Lie group G arise as such quotients of T*G The generalized Legendre transform construction of Lindstrom and Rocek is also explained in this framework INTRODUCTION The motivation for this work stems from two problems The first is the following... —> X is smooth if Z is smooth and n is a submersion The category of spaces over X is a category with products (fiber product) and Id a final object (x > x)- ^n a n y category with such properties we can define a group as an object Q together with morphisms defining group multiplication, inverse, and the identity Thus we define: Definition 1.1 A group over X is a group in the category of spaces over... [group inverse) and 1 : X —> Q [identity section) which commute with 7r and satisfy the group axioms In particular, for each x € X (7r_1(:r), • ,i\ _^ , l(aO) ls a group Remark 1.2 Even if one is interested (as we are) primarily in smooth groups over X, one cannot avoid the singular ones, since a subgroup of a smooth group can be singular In particular the stabilizers of smooth group actions can be singular... Lawrynowicz and 0 Suzuki 241 Differential Geometry of Circles in a Complex Projective Space S Maeda and T Adachi 253 On Special 4-Planar Mappings of Almost Hermitian Quaternionic Spaces J Mikes, J Belohldvkovd and 0 Pokornd 265 Special Spinors and Contact Geometry A Moroianu 273 Generalized ADHM-Construction on Wolf Spaces Y Nagatomo 285 £p(l) n -Invariant Quaternionic Kahler Metric T Nitta and T Taniguchi... automorphisms of p*Q and for local sections of p*Q coincide, and this concludes the proof • 1.5 Orbits and homogeneous spaces The definition of an orbit of a twistor group as an orbit of a section is quite inadequate We remark that in the case of an action - : G x M —• M of a Lie group G on a manifold M, an orbit can be defined > in two ways: 1) as the image of a point m under the mapping -m : G — M, or . first meeting " ;Quaternionic Struc- tures in Mathematics and Physics " interest in quaternionic geometry and its appli- cations continued to increase Proceedings of die Second Meeting Quaternionic Structures in Mathematics and Physics