Hermitian Structures on Cotangent Bundles of Four Dimensional Solvable Lie Groups
نویسندگان
چکیده
We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle T ∗G of a 2n-dimensional Lie group G, which are left invariant with respect to the Lie group structure on T ∗G induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on G. Using this correspondence and results of [8] and [10], it turns out that when G is nilpotent and four or six dimensional, the cotangent bundle T ∗G always has a hermitian structure. However, we prove that if G is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then T ∗G has no hermitian structure or, equivalently, G has no left invariant generalized complex structure.
منابع مشابه
Einstein structures on four-dimensional nutral Lie groups
When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to...
متن کاملOn permutably complemented subalgebras of finite dimensional Lie algebras
Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...
متن کاملDimensional Reduction of Invariant Fields and Differential Operators. I. Reduction of Invariant Fields
Problems related to symmetries and dimensional reduction are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group (“reducing group”) and its orbits (“external dimensions”) are compact, and this is essential in models where the volume of the orbits is related to physical quantities. But this case is only a part of the natural pr...
متن کاملLie Bialgebras of Complex Type and Associated Poisson Lie Groups
In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G∗ are complex Lie groups. We also prove that a Hermitian structure on g with ad-invariant metric induces a structure of the sa...
متن کاملPara-Kahler tangent bundles of constant para-holomorphic sectional curvature
We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...
متن کامل