arXiv subjects

I. Dotti

Publications and source records attributed to I. Dotti.

5 recordsLinked to original sources

arXiv2008

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

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 Cavalcanti-Gualtieri and Fernández-Gotay-Gray, 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.

math.DG↗

arXiv2004

Hyper-Kähler quotients of solvable Lie groups

In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie group structures on $\H^p \times \H^q$ ($\H$: the quaternions). We obtain new complete hyper-Kähler metrics on Euclidean spaces and give their local expressions.

math.DG↗

arXiv2004

Product structures on four dimensional solvable Lie algebras

It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the three dimensional subalgebras.

math.RA↗

arXiv2003

Complex structures on affine motion groups

We study existence of complex structures on semidirect products $\g \oplus_ρ \v$ where $\g$ is a real Lie algebra and $ρ$ is a representation of $\g$ on $\v$. Our first examples, the Euclidean algebra $\e(3)$ and the Poincaré algebra $ \e(2,1)$, carry complex structures obtained by deformation of a regular complex structure on $\sl (2, \c)$. We also exhibit a complex structure on the Galilean algebra $\G(3,1)$. We construct next a complex structure on $\g \oplus_ρ \v$ starting with one on $\g$ under certain compatibility assumptions on $ρ$. As an application of our results we obtain that there exists $k\in \{0,1\}$ such that $(S^1)^k \times E(n)$ admits a left invariant complex structure, where $S^1$ is the circle and E(n) denotes the Euclidean group. We also prove that the Poincaré group $P^{4k+3}$ has a natural left invariant complex structure. In case $\dim \g= \dim \v$, then there is an adapted complex structure on $\g\oplus_ρ \v$ precisely when $ρ$ determines a flat, torsion-free connection on $\g$. If $ρ$ is self-dual, $\g \oplus_ρ\v$ carries a natural symplectic structure as well. If, moreover, $ρ$ comes from a metric connection then $\g\oplus_ρ \v$ possesses a pseudo-Kähler structure. We prove that the tangent bundle $TG$ of a Lie group $G$ carrying a flat torsion free connection $\nabla$ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.

math.DG↗

arXiv2002

Abelian complex structures on solvable Lie algebras

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie groups are complex affine manifolds. It turns out that all 4-dimensional Lie algebras carrying abelian complex structures are central extensions of such affine Lie algebras.

math.RA↗