Quaternionic representation

Abstract: The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certainN=2 supergravity theories, where dimensional reduction induces a mapping betweenspecial real, Kahler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the Kahler and quaternionic manifolds in homogeneous holomorphic functions of second degree. We classify all cubic polynomials that have an invariance group that acts transitively on the real manifold. The corresponding Kahler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by Alekseevskii (and the corresponding special Kahler spaces given by Cecotti), but there is a new class of rank-3 spaces of quaternionic dimension larger than 3. We also point out that some of the rank-4 Alekseevskii spaces were not fully specified and correspond to a finite variety of inequivalent spaces. A simpler version of the equation that underlies the classification of this paper also emerges in the context ofW 3 algebras.

Abstract: We present a method of reduction of any quaternionic Kahler manifold with isometries to another quaternionic Kahler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kahler case. We compare our results with the known construction for Kahler and hyperKahler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear σ-models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceHP(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kahler and not symmetric.

Abstract: We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n -dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.

Abstract: Among the discrete series representations of a real reductive group G, the simplest family to study are the holomorphic discrete series. These representations exist when the Symmetrie space G/ K has a G-invariant complex structure, and are admissible when restricted to a 1-dimensional torus S in the center of K. They can be constructed äs spaces of analytic sections of certain holomorphic vector bundles on G /K. In the category of (g, K)modules, holomorphic discrete series give rise to unitarizable highest weight modules. Other interesting unitarizable modules with a highest weight can be constructed by "analytic continuation" of the holomorphic discrete series (cf. [Wl]).